NANO LETTERS
Phase Control of Graphene Nanoribbon by Carrier Doping: Appearance of Noncollinear Magnetism
2009 Vol. 9, No. 1 269-272
Keisuke Sawada,† Fumiyuki Ishii,†,‡ Mineo Saito,*,† Susumu Okada,§,| and Takazumi Kawai⊥ DiVision of Mathematical and Physical Science, Graduate School of Natural Science and Technology, Kanazawa UniVersity, Kakuma, Kanazawa 920-1192, Japan, Research Institute for Computational Sciences (RICS), National Institute of AdVanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan, Institute of Physics and Center for Computational Sciences, UniVersity of Tsukuba, Tennodai, Tsukuba 305-8571, Japan, Core Research for EVolutional Science and Technology (CREST), Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan, and Nano Electronics Research Laboratories, NEC Corporation, 34 Miyukigaoka, Tsukuba 305-8501, Japan Received September 20, 2008; Revised Manuscript Received November 14, 2008
ABSTRACT The zigzag graphene nanoribbon (ZGNR) has an antiferromagnetic property, that is, the relative spin angle θ between the two edges is 180°. By using noncollinear first-principles calculations, we find that the magnetic phase of the ZGNR can be controlled by injecting either electrons or holes: as the carrier density increases, θ continuously decreases from 180 to 0°, which indicates that the net magnetization is possible. Either FET doping or chemical doping is found to be possible.
Carbon materials, which have various atomic structures (fullerenes, nanotubes, graphenes, and graphene nanoribbons), exhibit useful properties of semiconductors,1 metals,1 half-metals,2-4 superconductors,5,6 and magnets.7-9 Among these properties, magnetism has not yet been fully elucidated, even though it is expected to be useful for spintronics applications. Zigzag graphene nanoribbons (ZGNRs) are potential candidates for achieving the magnetism of carbon nanomaterials. When graphene flakes or nanoribbons have zigzag edges, wave functions are localized around the edges and their energies are close to the Fermi level. Although the ZGNR is a p-electron system, the states whose dispersion is very small along the edge direction appear in a part of the Brillouin Zone.8,10 These localized states near the Fermi level have in fact been observed by a susceptibility measurement11 and a scanning tunneling microscope.12 Density of states at * To whom correspondence should be addressed. E-mail: m-saito@ cphys.s.kanazawa-u.ac.jp. † Kanazawa University. ‡ National Institute of Advanced Industrial Science and Technology (AIST). § University of Tsukuba. | Core Research for Evolutional Science and Technology (CREST), Japan Science and Technology Agency. ⊥ NEC Corporation. 10.1021/nl8028569 CCC: $40.75 Published on Web 12/19/2008
2009 American Chemical Society
the Fermi level [D(EF)] and electron-electron repulsion are two major factors that induce magnetic ordering in itinerant electron systems. In the case of ZGNR, a large D(EF) originating from the flat band states causes ferromagnetic (FM) spin ordering along each edge line.13 Two edges have opposite spin directions and accordingly the ground-state is the antiferromagnetic (AFM) state and not the FM state.8 However, FM or ferrimagnetic properties are required for spintronics applications. The AFM property originates from the cancelations of magnetic moments at the two sublattices in the honeycomb structure of the ZGNR. Since atoms at different edges belong to different sublattices, an antiparallel spin configuration at the two edges always appears. The cancelation of spin is supported by a theorem based on the Hubbard model under some conditions,14 that is, according to this theorem, the total spin moment is given by (NA - NB)/2, where NA and NB are the numbers of sublattices A and B, respectively. This theorem strongly supports the antiparallel interedge spin (APIES) state, but the conclusion is invalid when carriers are injected into the ZGNR. In this study, we perform first-principles noncollinear density functional theory (DFT) calculations and elucidate that a variety of magnetic phases of the ZGNR can be
Figure 1. Lattice and magnetic structures of ZGNR. The yellow and light blue spheres denote C and H atoms, respectively. The blue rectangle denotes the unit cell. The ZGNR shown in this figure has the ribbon width N ) 10. The black arrows denote the magnetic moments SL and SR at the two edges. The relative angle between SL and SR is represented by θ.
Figure 2. Noncollinear magnetic phase diagram of ZGNR. This figure shows the interedge exchange coupling Ex(θ) per unit cell as a function of carrier doping x (e/nm) and θ (degree).
achieved by carrier doping. We find that the doping varies the angle θ between the spin directions at the two edges. As the carrier density increases, θ decreases continuously from 180° (APIES). Therefore, a canted interedge spin (CIES) state (0° < θ < 180°) and parallel interedge spin (PIES) state (θ ) 0°) can be attained. It is found that this doping can be carried out using a field-effect transistor (FET) structure or by chemical doping. Our first-principles electronic-structure calculations are based on the noncollinear DFT with two-component spinor wave functions for the Kohn-Sham Bloch orbitals.15,16 Exchange correlation energy is based on the generalized gradient approximation.17 The norm-conserving pseudopotential method18 is used and the wave functions are expanded by linear combination of multiple pseudo atomic orbitals.19,20 Two valence orbitals (s- and p-orbitals) for C atoms and a single valence orbital (s-orbital) plus a polarization orbital (p-orbital) for H atoms were used as a basis set. It is found that the total energy converges within 1 meV/cell when this basis set is used. The lattice constant along the periodic direction (x-direction) is 0.246 nm, which is the same as the experimental value of graphite. The geometry optimization has been performed for the nonmagnetic (NM) states. We confirm that the optimized structures of the other magnetic states are close to that of the NM state: the optimized bond lengths differ within 0.0001 nm. We sample 60 k points along the x-direction. We confirm that the total energy differences between APIES and PIES states are converged within 1 meV/ 270
Figure 3. Single-particle energies and band gap for nondoped case. The red circles and blue triangles correspond to hole doping and electron doping, respectively. ε(θ) denotes εlul(θ) and εhol(θ) in the cases of electron doping and hole doping, respectively. The inset shows the band gap as a function of θ when x ) 0.
cell when 60 k points are used. The length of the vacuum region along the y- and z- directions is larger than 2.5 nm. By performing fully relativistic calculations, we estimate that the spin-orbit interaction (SOI) effect shifts the total energies within 0.1 meV/cell, thus the SOI is neglected in our calculations. The spin canting angle θ is defined as the relative angle between the magnetic moments SL and SR at the two edges (Figure 1). By using a constrained DFT, where penalty functions are introduced in the total-energy functional,21,22 we obtain the total energy for a fixed value of θ. All the calculations are performed by using the OPENMX code.23 Nano Lett., Vol. 9, No. 1, 2009
Figure 4. Spatial distribution of spin density in CIES state of ZGNR. This figure shows the spatial distribution of the spin density in the CIES (θ ) 90°) state in the case of x ) 0.16 e/nm. Spin density is defined as the difference between the diagonalized density matrix ∆n(r) ) Fvv(r) - FVV(r). The black arrows show the directions of the magnetic moments of C atoms.
Table 1. Total Energy Difference between APIES and PIES States (in meV/cell)a FLS x ) 0.41 e/nm
B-doped
FLS x ) -0.41 e/nm
3.88 3.24 3.73 EAPIES - EPIES a The calculated results in the case of the Fermi-level shift (FLS) method and chemical doping are tabulated.
We first study the ZGNR whose width N is given by 10. In this case, the unit cell includes 20 C and 2 H atoms (Figure 1). We find that the APIES state is the ground-state for the nondoped case. The total energy difference between the APIES and PIES states is found to be 4.6 meV/cell. As θ decreases from 180 to 0 °, the energy is found to decreases monotonically, previously reported by us.24 Next, we study the carrier doped ZGNR. The calculation is performed by the Fermi-level shift (FLS) method, where we introduce the uniform background charge so that the system is neutral. The number of doped carriers per unit length is expressed as x (e/nm), and the positive and negative values correspond to hole and electron doping, respectively. To elucidate the ground magnetic state for x, we calculate the interedge exchange coupling Ex(θ), which is the total energy measured from the ground-state for a given x (Figure 2). As shown in the phase diagram in Figure 2, the CIES state (0° < θ < 180°) becomes the ground state in the regions of -0.24 e/nm < x < 0 and 0 < x < 0.24 e/nm. As |x| increases from 0 to 0.24 e/nm, θ in the ground state decreases continuously from 180 to 0°; thus, when |x| equals to 0.24 e/nm, the PIES state appears. We further find that the PIES state is the ground-state in the regions of -1.22 e/nm e x e -0.24 e/nm and 0.24 e/nm e x e 0.81 e/nm. In the regions of -2.85 e/nm < x < -1.22 e/nm and 0.81 e/nm < x < 2.85 e/nm, the total energies of the APIES and PIES states almost degenerate to within 0.4 meV/cell. We further find that the NM state becomes the ground-state when 2.85 e/nm e |x|. Therefore, we conclude that the phase transition APIES f CIES f PIES occurs when |x| increases from 0 and that the NM state becomes the ground-state when the level of doping is high. Next, we discuss why the APIES state transforms into the CIES state as a result of carrier doping. When x ) 0, we Nano Lett., Vol. 9, No. 1, 2009
N-doped 3.32
find that the ground APIES state has the maximum band gap of 0.42 eV. As θ decreases from 180°, the band gap becomes small and is zero when θ ) 0° (Figure 3 and Supporting Information Figure S1). Consequently, the oneelectron energy of the lowest unoccupied level in the conduction band [εlul(θ)] is reduced as θ decreases from 180° (Figure 3). This reduction in the one-electron energy causes the total energy of the CIES state to be lower than that of the APIES state when electron doping is carried out. In the case of hole doping, the energy of the highest occupied level [εhol(θ)] increases as θ decreases from 180° (Figure 3). This increase in the one-electron energy causes the θ value in the ground-state to decrease monotonically from 180° as x increases from 0. Now, we analyze the novel CIES state in detail. In Figure 4, we show the spin density distributions in space and spin directions at each atomic site in the case of θ ) 90°. In this state, the spin configurations in the neighboring C atoms are almost AFM everywhere, whereas in the case of the APIES state, we find that the spin configuration is exactly AFM. Because of the slight deviation from the AFM configurations between the neighboring sites in the CIES state, we find that the spin direction gradually varies from side L to side R (Figure 4). As N increases, we find that the total energy difference between the APIES and PIES states becomes small (Figure 5). In the case of the nondoped system, the total energy difference between the APIES and PIES states decreases with increasing N. For a certain value of x, the energy of the PIES state becomes lower than that of the APIES state. This critical value of |x| tends to decrease as N increases, for example, |x| ) 0.16 e/nm and |x| ) 0.06 e/nm for the cases of N ) 10 and N ) 24, respectively, as shown in Figure 5. Thus, the PIES ground-state would be attained by a low level of carrier doping when N increases. 271
Acknowledgment. The authors would like to thank T. Ozaki for his development of codes on constraint DFT for noncollinear spin orientations. This work was partly supported by Grants-in-Aid for Scientific Research (Nos. 19740182 and 19560021) from the JSPS and by the Next Generation Super Computing Project, Nanoscience Program, MEXT Japan. The computations in this research have been performed using the AIST Super Cluster facility at Tsukuba and the supercomputers at the ISSP, University of Tokyo, and the RCCS, Okazaki National Institute. Supporting Information Available: This material is available free of charge via the Internet at http://pubs.acs.org. Figure 5. Magnetic stability of various N-ZGNRs. This figure shows the total energy difference (EAPIES - EPIES) between the APIES and PIES states as a function of x. The squares, circles, triangles and diamonds correspond to N ) 6, 10, 16, and 24, respectively.
The abovementioned carrier doping can be performed by the FET doping technique. Here, we show that chemical doping can also be performed. We study the cases of substitutions of B and N atoms. In the ZGNR (N ) 10), we introduce single B and N atoms in each super cell that includes 200 host atom sites, and this doping corresponds to the case |x| ) 0.41 e/nm. We optimize the geometry and find that the substituted atoms are located in the graphene plane. The calculated energy difference between the APIES and PIES states is shown in Table 1. We find that the energy difference in the case of chemical substitution is close to that in the case of the FLS method, indicating that chemical substitution using either B or N atoms is effective. In conclusion, our understanding of the magnetic properties of nanocarbon materials advanced by noncollinear firstprinciples calculations. Carrier injections, which were used to induce the superconductivity of nanocarbon materials,5,6 were found to result in new magnetic phases of a ZGNR. We found that the spin canting angle θ decreased monotonically from 180 to 0° with increasing carrier density. Therefore, both the novel CIES state and the PIES state were achieved, which indicates that the net magnetization of the ZGNR was obtained. We demonstrated that carrier injection could be carried out by either FET doping or chemical doping, suggesting that it is practically possible to control the magnetic phase.
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