Electronic and magnetic properties of zigzag-edged hexagonal graphene ring nanojunctions

CARBON 94 (2015) 996–1002

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CARBON journal homepage: www.elsevier.com/locate/carbon

Electronic and magnetic properties of zigzag-edged hexagonal graphene ring nanojunctions D. Wang a, Z. Zhang a,b,⇑, J. Zhang a,⇑, X. Deng a, Z. Fan a, G. Tang a a

Institute of Nanomaterial & Nanostructure, Changsha University of Science and Technology, Changsha 410114, China Hunan Province Higher Education Key Laboratory of Modeling and Monitoring on the Near-Earth Electromagnetic Environments (Changsha University of Science and Technology), Changsha 410114, China b

a r t i c l e

i n f o

Article history: Received 11 June 2015 Received in revised form 23 July 2015 Accepted 25 July 2015 Available online 29 July 2015

a b s t r a c t The zigzag-edged hexagonal graphene quantum rings (ZHGQRs) are sandwiched between two Au electrodes to construct nanojunctions, and their electronic and magnetic properties are investigated by the first-principles method. It shows that there are always significant transmission gaps (semiconductor behavior) for all ZHGQRs but changing in an obvious oscillating manner with variations of the edge width. The mechanism for such a feature is analyzed deeply, and it is found that the weak localization for the quantum interference correction to conductance and the parity of wave function play the crucial roles. No intrinsic magnetism is observed in the ZHGQRs, this is because there always exist local armchair-edge-like defects at its corners to weaken the spin polarization strongly. And also shown is that the magnetic field can effectively suppress the weak localization behaviors leading to a negative magnetoresistance for the ZHGQR, which is in good agreement with experimental findings in graphene. These findings are of very important for understanding the electromagnetic properties of graphene quantum ring devices. Ó 2015 Published by Elsevier Ltd.

1. Introduction Graphene, a two-dimensional single atomic layer composed of sp2 hybridized carbon atoms, has attracted intense research interest due to unique material properties and promising applications in nanoelectronics [1,2]. The recent graphene fabrication and patterning technology demonstrated that graphene can be cut in many different shapes and sizes, including graphene nanoribbons (GNRs) [3], graphene nanoflakes (GNFs) [4], graphene quantum rings (GQRs) [5], antidot arrays [6], and so on. Particularly, precision-controlled formation of zigzag and armchair edges have been achieved [7]. These advances in experiments further reinforce the feasibility of the technical use of graphene. A lot of studies so far have shown that electronic properties of graphene can strongly be influenced by its geometries, such as shapes, sizes, and edge symmetries, which open possibilities to obtain tailored electronic features for developing special electromagnetic devices only by structuring desirable geometries. GNFs are arbitrarily shaped finite graphene fragments consisting of carbon hexagonal rings and bounded by a single topological ⇑ Corresponding authors at: Institute of Nanomaterial & Nanostructure, Changsha University of Science and Technology, Changsha 410114, China (Z. Zhang). E-mail addresses: [email protected] (Z. Zhang), [email protected] (J. Zhang). http://dx.doi.org/10.1016/j.carbon.2015.07.082 0008-6223/Ó 2015 Published by Elsevier Ltd.

circuit, where all in-plane dangling r bonds at the edge are assumed to be passivated. Among them, triangular, rectangular and hexagonal GNFs are the most principal structures investigated because they feature a simple structure and show unique electronic and magnetic properties [8–12]. For example, spin magnetism of GNFs depends on its shape due to topological frustration of the p-bonds [13]. In particular, Coulomb blockade [10], current rectifications [14], Kondo effects [15], spin valve effects [15], and logic gate effects [15] can be found in GNFs, and particularly, the zigzag-edge triangular GNFs (ZTGs) with the ferromagnetic ground state can be viewed as an interesting new class of nanomagnets [12] by a spin value scaling with its linear size [13]. GQRs are intimately related to GNFs and can be viewed as GNFs containing an antidot, which introduces an additional inner edge to the system. One expects that the interedge coupling of states and enhanced size effects in a GQR will lead to some new physical properties, therefore, it may be one of the most novel building blocks for future nanodevices. Currently, there are several groups contributing their work to studies of GQRs, either experimentally [5,16,17] or theoretically [18–24]. The experimental investigations mainly focus on external magnetically induced Aharonov–Bohm (AB) effects [5,16,17] and Shubnikov–de Haas (SdH) oscillations [16] in GQRs, and theoretical studies basically concentrate on using the usual tight-binding (TB) model to explore the AB oscillation [18–20], persistent currents [21], and energy spectra as a function

D. Wang et al. / CARBON 94 (2015) 996–1002

of the magnetic flux [22]. However, little theoretical works have been done for electronic and magnetic properties of GQRs with an attachment by metal electrodes. Under such an environment, to understand their electrical properties, doping effects, quantum interference, and magnetism properties is essential for developing GQR-based devices, and thus it is highly desirable for these issues to be investigated in depth. In this present work, we use the first-principles calculations based on the density functional theory (DFT) combined with the non-equilibrium Green’s function (NEGF) technique to investigate the electronic and magnetic properties of zigzag-edged hexagonal graphene quantum nanorings (ZHGQRs) connected with two Au electrodes. It shows that all ZHGQRs under study are semiconductor and their energy gaps change in an obvious oscillating manner with the edge width, which can be mainly attributed to the quantum interference correction to conductance. Especially, we find that the weak localization and the parity of wave functions play an important role in the electronic properties. And no significantly intrinsic magnetisms are observed in ZHGQRs, which is strikingly distinct from zigzag-edged GNRs. 2. Model and method To investigate unique electronic behaviors and device features for the ZHGQR as shown in Fig. 1(a), the device model exhibited in Fig. 1(b) is made by putting a ZHGQR between two Au electrodes. Both zigzag-shaped outer and inner edges of the ZHGQR are terminated by H atoms to eliminate the dangling bonds on edge carbon atoms. The total number of carbon atoms in an   ZHGQR is 6 N 20  N 2r , where N0 and Nr are the number of hexagonal cells for one outer-side of the ZHGQR and the removed hexagonal GNF, respectively. The geometries of the ZHGQR are determined by two structural parameters: diameter D and edge width W, which are characterized by the number of zigzag carbon chains Nz. Therefore, the ZHGQR is denoted as ZHGQR(D,W). Here, ZHGQR(14,4) as an example is shown in Fig. 1. ZHGQR is a sixfold rotational symmetric structure and can thus be viewed to be composed of six arms. Each arm is a trapezoidal GNF, which is a region between two red dotted lines. When W is an even number, there exists the r mirror plane in the region between two blue dotted lines, indicating the geometrical symmetry of lateral direction of an arm in ZHGQR. Our studies focus on the changing regularities of electronic structure and electronic transport with variations of D or/and W, and the finding of underlying physical origins. To investigate these issues, each optimized structure by a separate calculation based on the density function theory (DFT) is positioned between two Au (1 0 0) electrodes with a typical Au–C

997

distance of 2.0 Å, which makes it consistent with the experiment result [25], to construct a nanojunction, namely, taking electrode effects into account. Subsequently, the geometries of the scattering region are optimized further until all residual forces on each atom are smaller than 0.05 eV/Å. After optimization, we find that ZHGQRs remain their planar structures, the C–H bond length is 1.16–1.17 Å, and C–C bond length in the ZTG appears in a range of 1.42–1.45 Å with larger values internally than those externally. In our studies, geometric optimizations of the device region and calculations of electronic structure, transport properties, and spin magnetism are performed by using the density function theory (DFT) combined with the non-equilibrium Green’s function (NEGF) method [26–30]. We employ Troullier–Martins norm-conserving pseudopotentials to represent the atom core and linear combinations of local atomic orbitals to expand the valence states of electrons. The generalized gradient approximation (GGA) or the spin-dependent generalized gradient approximation (SGGA) is used as the exchange–correlation functional. Single-zeta plus polarization (SZP) basis set for Au and H atoms and double-zeta plus polarization (DZP) basis set for other atoms are adopted. The k-point sampling is 3, 3, and 200 in the x, y, z direction, respectively, where the z is the electronic transport one, and the cutoff energy is set to 150 Ry. Open boundary conditions are used to describe the electronic and the transport properties of nanojunctions. In our following calculations, the average Fermi level, an average value of the chemical potential of the left and right electrodes, is set as zero.

3. Results and discussions 3.1. Electronic properties and quantum interference correction effects The calculated transmission spectra for ZHGQR(14,W) at equilibrium state are displayed in Fig. 2(a) and (b), where taking W = 2–7, respectively, and Fig. 2(c) shows that their energy gaps change with different values of W. Obviously, two major features can be observed: (1) all of ZHGQR have an energy gap–transport gap (an energy interval between transmission peaks for the highest occupied molecular orbital (HOMO) and the lowest unoccupied orbital (LUMO)) regardless of values of W, that is, ZHGQRs always exhibit semiconductoring behaviors, this result is different from TB calculations for isolated ZHGQRs [21], which show the metallicity when W is an odd number, and (2) the transmission gaps change in an oscillatory manner with values of W, namely, a large (small) gap occurs when W is an even (odd) number. To understand underlying physical origins of these features, we first compute the electronic structure of one arm which is a

Fig. 1. (a) A geometrical structure for a zigzag-edged hexagonal graphene quantum nanoring (ZHGQR). (b) The ZHGQR is sandwiched between two Au electrodes to construct a nanojunction.

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HZGQR(14,6) HZGQR(14,4) HZGQR(14,2)

(a) 1.0

0.8 Transmission

Transmission

0.8 0.6 0.4

0.6 0.4 0.2

0.2 0.0

HZGGR(14,7) HZGQR(14,5) HZGQR(14,3)

(b) 1.0

-0.50

-0.25 0.00 Energy(eV)

gap(eV)

(c)

0.25

0.50

0.0 -0.50

-0.25

0.00 0.25 Energy(eV)

0.50

0 .7 0 .6 0 .5 0 .4 0 .3 0 .2 0 .1 0 .0

2

3

4

5

6

7

(1 4 ,W ) Fig. 2. (a and b) Transmission spectra at equilibrium state for ZHGQR(14,W) with W = 2–7, respectively. (c) The change of transmission gaps in an oscillatory manner with the value of W.

trapezoidal GNF with two H-passivated zigzag-shaped edges, as mentioned above. The calculated results for an isolated trapezoidal GNF or when it is attached with Au electrodes show that it is a gapless metal with several zero-energy states localized at the Fermi level, similar to zigzag-edged GNRs. Therefore, it is very interesting to clarify what factors make an energy gap opened after six arms are connected to form a ZHGQR. This might be related to a combination of multiple causes. When six arms are merged to a ring-shaped structure, there always exist interactions between arms, in particular, stronger couplings occurs for one site in one arm with other five equivalent sites. As a result, the degeneracy for energy level located at the Fermi level might be eliminated and a small gap occurs. Additionally, there exist 60° turns for the crystal direction in the ZHGQR, the propagating direction of electron waves have thus to be altered in succession and must be scattered unambiguously when the electron transfers in such a ring structure, namely, the corner scattering results in a gap. However, other more important causes might be the weak localization (WL) effect and the different-path quantum interference effect, which plays a vital role in the larger gap opening and the oscillatory changing gaps versus the ring width W. The WL behavior can be well described by the standard theory [31]. It originates from the quantum-mechanical treatment of the backscattered partial electron waves traveling back to the original point along the time-reversed path, and contains an interference term adding up constructively to causes an increase in resistance. The WL correction to conductivity has been found in 2D graphene experimentally [32–34]. Thereby, for a particular structural graphene, ring-shaped graphene, the WL effect should be expected to be more possible. When an electron enters into the left extremity of a ZHGQR from the source, its partial waves will propagate along various directions. However, they will be scattered by ring edges to form the closed paths in large numbers. Fig. 3(a) shows a schematic of the ring and simplified two propagating paths of electron partial waves, marked 1 and 2. Their time-reversal paths are indicated as 10 and 20 , respectively. The electron partial waves

along the path 1 (or 2) and its time-reversal path 10 (or 20 ) will give rise to a constructive interference with each other at the original point, leading thus to a negative correction to conductivity, i.e., further opening an energy gap for a ZHGQR. This is because there exists an inverse relationship between carrier mobility and bandgap in graphene [35]. Generally speaking, the phases of different paths, such as paths 10 and 20 , are all random, thus this interference effect, on the average, can be canceled out [36]. However, we find the ZHGQR(14,4) is a particular exception. Fig. 3(b) shows its wave function distribution (the red and blue corresponding to different phase) of the LUMO, which is the molecular projected self-consistent Hamiltonian (MPSH) eigenstates containing the ZHGQR–electrode coupling effects during the self-consistent iteration. The reason to present the LUMO wave function distribution here is that the LUMO is very close to the Fermi level compared with the HOMO, as shown in Fig. 2(a) and (b). Accordingly, the LUMO feature would play a crucial role for the electron tunneling through a ZHGQR-based nanojunction. It is clearly seen that such a wave function are completely symmetric with respect to the mirror plane r (the even symmetry for wave functions) in each section, that is, the phases of different paths are the same. But for ZHGQR(14,3), the case is different, the phases of different paths are not identical because there does not exist the mirror plane r geometrically. In order to more profoundly understand the quantum interference effects as mentioned above, we here perform a general derivation. First, supposing the total number of paths is N, the electron partial wave function for the nth path is An = |An|eih, where h is a phase, and then, the probability that an electron stays at the original point can be written as

2   X N N X   P ¼  An þ An0    n n0 ¼4

N N X N X X jAn j2 þ 4 jAn jjAm j cosðhn  hm Þ; n

n –m

ð1Þ

D. Wang et al. / CARBON 94 (2015) 996–1002

999

(b)

(a)

Fig. 3. (a) A schematic of the ring, and simplified two propagating paths of electron partial waves and their time-reversal paths. (b) The wave function distribution of the LUMO for the ZHGQR(14,4).

where n0 indicates the time-reversal path related to nth path with jAn0 j ¼ jAn j, and the first term corresponds the WL effect, obviously, P this value is twice as large as 2 Nn jAn j2 , without considering quantum interference, and the second term represents the different-path quantum interference. As stated above, for ZHGQRs with W = an even number, there exists hn = hm due to the even symmetry for wave functions, thus the probability of an electron at the original point is

Peven ¼ 4

N N X N X X 2 jAn j þ 4 jAn jjAm j; n

ð2Þ

n –m

but for ZHGQRs with W = an odd number, hn – hm owing to a lack of the even symmetry for wave functions, hence the average value for cos(hn  hm) should be proximately zero [35], namely, interference effects of different paths can be negligible on the average. As a result, its probability Podd is much smaller than Peven. In other words, the interference effects make an additional positive correction to resistance, i.e., opening an additional gap [35] for the ZHGQR with W = an even number. Therefore, this is an essential origin for the conciliatorily changing gaps versus the ring width W. To further confirm quantum interference correction effects, we consider the influence of doping in the ZHGQR, and positions of single B-atom substituted doping are indicated in Fig. 1(a) as site1 and site 2. The calculated transmission spectra for the B-doping ZHGQR(14,4) with an optimized structure at equilibrium state are exhibited in Fig. 4(a), and for the purpose of comparison, the transmission spectrum for the undoped ZHGQR(14,4) is also drawn in the same figure. As can be seen, the energy gap (HLG) is narrowed remarkably in the presence of one heteroatom regardless of its sites. To understand this issue, in Fig. 4(b) and (c), we plot the electron wave function distribution of the LUMO for the ZHGQR(14,4) with B-doping at sites 1 and 2. Particular obviously, in the proximity of doping sites, the symmetry of the electron wave function is destroyed strictly. Correspondingly, it will affect the phase coherent propagation of electrons and makes quantum interference correction effects weakened. This is easily understood, the B-doping involves a small lattice deformation around the dopant atom. Just with this topological lattice defect and a heteroatom, the propagating electron wave will suffer an inelastic scattering to cause a dephasing. In addition, one can see that very small transmission peaks appear at the Fermi level. For finding their origin, we calculate the PDOS, the density of states (DOS) projected on the B atom. Clearly, PDOS peaks are just located at the Fermi level. This means that transmission peaks at the Fermi level originate from the dopant atom, unrelated to the HLG of ZHGQRs.

Finally, we demonstrate the quantum size-effect by fixing the edge width of ZHGQRs as W = 2 and 3 to calculate transmission spectra for ZHGQR(D,2) and ZHGQR(D,3) at equilibrium state, as displayed in Fig. 5(a) and (b), where taking ring diameters as D = 8, 10, and 12, respectively. As a comparison, transmission spectra for ZHGQR(14,2) and ZHGQR(14,3) are also plotted in the same figures. Obviously, one finds that energy gaps increase with decreasing the ring diameter D, which can be attributed to an enhanced quantum confinement and corner scattering for a small-size ZHGQR. Additionally, from Fig. 5(a) and (b), the parity behavior of gaps (HLG) can be clearly detected for a small-size ZHGQR, just like that found in ZHGQR(14,W). 3.2. Magnetic properties We now turn our studies to the spin magnetism of ZHGQRs. It has been shown [11,37] that the ground-state magnetic ordering in graphene-based nanostructures (GBNs) is consistent with the theorem of itinerant magnetism in a bipartite lattice within the one-orbital Hubbard model [38]. Graphene consists of two atomic sublattices: A and B, and each carbon atom in graphene is connected to the nearest neighbors in the different sublattice by three covalent bonds, and remaining pz-orbital electrons contribute to the spin magnetic moment. However, pz-orbital electrons in A and B sublattices have opposite spin directions, thus, net magnetic moments for GBNs at the ground state originating from imbalance in the number of carbon atoms in two sets of sublattices can be calculated by Lieb’s theorem [38]

M¼

1 jN A  NB jg lB ; 2

ð3Þ

where g = 2 for the electron, lB is the Bohr magneton, and NA and NB denote the number of carbon atoms in A and B sublattices, respectively. As we know, for the zigzag-edged GNRs, these exists the global balance of NA = NB. As a result, such GNRs do not have any intrinsic net magnetic moments at the ground state, but showing distinct magnetic (or spin) states at their edges, namely, ferromagnetically ordered edge states at each zigzag edge and an antiferromagnetic arrangement of spins between two zigzag edges. Similarly, ZHGQRs also possess NA = NB, then, what is their spin-polarized anti-ferromagnetic (AFM) states? Here, we take ZHGQR(14,3) as an example and calculate its spatial distribution of spin polarization density. The isosurface plot for the spin density (rq = qa  qb) at the anti-ferromagnetic (AFM) state are shown in Fig. 6(a), where qa and qb denote the electron densities of a-spin and b-spin states,

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D. Wang et al. / CARBON 94 (2015) 996–1002

(b)

1.2

no-doping site1-B site2-B

site1-B site2-B

6

1.5 Transmission

8

1.8 PDOS

(a)

4

B

2

0 -0.25

0.00 0.25 Energy(eV)

0.9

(c)

0.6 0.3

B 0.0 -0.50

-0.25

0.00

0.25

0.50

Energy(eV)

Fig. 4. (a) The transmission spectra for the B-doping ZHGQR(14,4) at equilibrium state sites 1 and 2. (c and d) The electron wave function distribution of the LUMO for the ZHGQR(14,4) with B-doping at sites 1 and 2.

(a)

Transmission

0.8

ZHGQR(14,2) ZHGQR(12,2) ZHGQR(10,2) ZHGQR(8,2)

(D,2)

1.0

(D,3)

0.8

0.6 0.4 0.2 0.0 -1.0

(b)

Transmission

1.0

HZGQR(14,3) HZGQR(12,3) HZGQR(10,3) HZGQR(8,3)

0.6 0.4 0.2

-0.5

0.0 0.5 Energy(eV)

1.0

0.0 -0.50

-0.25

0.00 0.25 Energy(eV)

0.50

Fig. 5. The calculated transmission spectra at equilibrium state for ZHGQR (D,2) (a) and ZHGQR (D,3) (b), where taking ring diameters as D = 8, 10, 12, and 14, respectively.

respectively. From such an isosurface plot, two important properties can be found: (1) for each arm in ZHGQR, ferromagnetically ordered edge states at each zigzag edge and an anti-ferromagnetic arrangement of spins between two zigzag edges occur ambiguously, similar completely to the zigzag-edged GNRs. Nevertheless, the spin direction on carbon atoms for one zigzag edge (inner or outer) changes alternatively from one arm to the next. This is easily understood because carbon atoms on the edge (inner or outer) of two neighboring arms belong to two different sublattices, and (2) generally speaking, the spin density |rq| for the zigzag-edged GNRs can reach 102|e|/Å3 at the edges. However, from Fig. 6(a), only the spin density |rq| 104|e|/Å3 at the edges can be observed. This means that the spin magnetism of the ZHGQRs is two orders of magnitude smaller than the zigzag-edged GNRs. Fig. 6(b) exhibits the density of states (DOS) for a-spin and b-spin states. No spin polarization arises, as expected. Next, we explore the ferromagnetic (FM) features of ZHGQRs. The previous works [39,40] have shown that the FM state for the GNRs is able to be realized just by the application of a magnetic field, which can be simulated by simply arranging the magnetic moments on all the edge carbon atoms parallel to the direction of the external magnetic field [41]. Based on this method, we achieve the FM state for ZHGQR(14,3), and the calculated isosurface plot for the spin density is shown in Fig. 6(c). It is interesting

to note that although the spin of all carbon atoms on the inner or outer edges points the same direction, the magnetism for ZHGQR has not been enhanced yet and only takes the extremely small spin density |rq| 104|e|/Å3. Additionally, the total energy calculations for systems unveil that the ZHGQR has the same energy value for the AFM and the FM states, which are all the ground state of system. In short, the spin magnetism (AFM or FM) of ZHGQRs is negligible small as compared with that of the GNRs, and the ZHGQR can thus be viewed to be at a non-magnetic state or a paramagnetic state at most. How to understand this issue? We consider that the most likely cause for such a reduction of spin magnetism is edge defects, local armchair-edge-like defects at the corners, which weaken strongly the degree of the spin polarization and suppress the magnetism. We have also calculated the magnetic distribution and spin-resolved DOS for larger-size ZHGQRs such as ZHGQR(25,3), and found that it has almost similar results as those for ZHGQR(14,3), as shown in Fig. 6. This seems to imply that the nonmagnetic feature of ZHGQRs is independent of the armchair-edge ‘‘concentration’’. It is worth noting that the spin-resolved DOS in Fig. 6(d) shows a very interesting phenomenon: the non-zero DOS appears at the Fermi level, implying the transition from a semiconductor to a metal for the ZHGQR due to the application of a magnetic field perpendicular to the ring plane. This increased metallicity is not originating from the spin-magnetism effect since such an effect is able

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60

(a)

α β

(b) DOS

30 0

-30 -60 -2

(c)

(d) 6 0

-1

0 1 Energy(eV)

2

α β

DOS

30 0

-3 0 -6 0 -2

-1

0 1 E n e rg y (e V )

2

Fig. 6. Isosurface plots of the spin density (rq = qa  qb) and spin-resolved density of states (DOSs) of the ZHGQR(14,3), and values for red (a-spin) and blue (b-spin) isosurfaces are ±104|e|/Å3, respectively. (a and b) For anti-ferromagnetic (AFM) state. (c and d) For the ferromagnetic (FM) state. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

to be neglected, as stated above, and is just an indication of the suppression of WL because the magnetic field breaks down the time-reversal symmetry of electron propagating paths, leading to a positive correction to conductance and thus a negative magnetoresistance for the ZHGQR, which is observed experimentally in 2D graphene [32–34]. In other words, our calculated result shown in Fig. 6(d) further confirms that the strong WL indeed exists in the ZHGQR. In the results presented above, only one possible combination of electrode attachment to the ZHGQR with the highest possible symmetry is considered. Other connecting manners for the ZHGQR to electrodes, similarly to a benzene suggested by different anchoring sites [42–45] such as quasi-para, meta, ortho positions, are also investigated. We find that their electromagnetic features and changing regularity, such as the odd–even oscillating behavior of gap, size effects, and non-magnetic property, are essentially similar to each other. This might is because the size of ZHGQR is much larger than that of one benzene, leading to a very low sensitivity to anchoring sites. 4. Conclusion ZHGQRs are sandwiched between two Au electrodes to construct nanojunctions and their electronic structures and spin magnetisms are calculated by using the first-principles method based on the DFT combined with the NEGF to. It is found that all ZHGQRs are semiconductor and their transmission gap changes in an obvious oscillating manner with the edge width. To explain this issue, a new mechanism is proposed, namely, the parity of wave functions play an important role for the quantum interference correction to conductance, which is completely different from previous studies on other quantum systems, where the phases of different paths are all random, thus this interference effect, on the average, can be canceled out. And no intrinsic magnetisms are observed in the ZHGQR due to local armchair-edge-like defects

at its corners to weaken strongly the spin polarization. Furthermore, the magnetic field is found to be able to effectively suppress WL leading to a negative magnetoresistance for the ZHGQR, which is in good agreement with experiments in graphene. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant Nos. 61371065, 61201080, and 51302022), the Hunan Provincial Natural Science Foundation of China (Grant Nos. 2015JJ3002, 2015JJ2009, 2015JJ2013), the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 14A013), the Construct Program of the Key Discipline in Hunan Province, and Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province. References [1] D. Elias, R. Nair, T. Mohiuddin, S. Morozov, P. Blake, M. Halsall, et al., Control of graphene’s properties by reversible hydrogenation: evidence for graphane, Science 323 (5914) (2009) 610–613. [2] J. Zeng, K.-Q. Chen, J. He, X.-J. Zhang, W. Hu, Rectifying and successive switch behaviors induced by weak intermolecular interaction, Org. Electron. 12 (10) (2011) 1606–1611. [3] M.Y. Han, B. Ozyilmaz, Y. Zhang, P. Kim, Energy band-gap engineering of graphene nanoribbons, Phys. Rev. Lett. 98 (20) (2007) 206805. [4] X. Feng, S. Kwon, J.Y. Park, M. Salmeron, Superlubric sliding of graphene nanoflakes on graphene, ACS Nano 7 (2) (2013) 1718–1724. [5] S. Russo, J.B. Oostinga, D. Wehenkel, H.B. Heersche, S.S. Sobhani, L.M. Vandersypen, et al., Observation of Aharonov–Bohm conductance oscillations in a graphene ring, Phys. Rev. B 77 (8) (2008) 085413. [6] J. Bai, X. Zhong, S. Jiang, Y. Huang, X. Duan, Graphene nanomesh, Nat. Nanotechnol. 5 (3) (2010) 190–194. [7] X. Jia, M. Hofmann, V. Meunier, B.G. Sumpter, J. Campos-Delgado, J.M. RomoHerrera, et al., Controlled formation of sharp zigzag and armchair edges in graphitic nanoribbons, Science 323 (5922) (2009) 1701–1705. [8] W.L. Wang, S. Meng, E. Kaxiras, Graphene nanoFlakes with large spin, Nano Lett. 8 (1) (2008) 241–245.

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