Intrinsic spin–orbit interaction in carbon nanotubes and curved nanoribbons

Solid State Communications 152 (2012) 1477–1482

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Intrinsic spin–orbit interaction in carbon nanotubes and curved nanoribbons ˜ oz, M. Pilar Lo´pez-Sancho Leonor Chico n, Herna´n Santos, M. Carmen Mun Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Cientı´ficas, Cantoblanco, 28049 Madrid, Spain

a r t i c l e i n f o

abstract

Article history: Accepted 12 April 2012 Accepted by L. Brey Available online 21 April 2012

We present a theoretical study of spin–orbit interaction effects on single wall carbon nanotubes and curved graphene nanoribbons by means of a realistic multiorbital tight-binding model, which takes into account the full symmetry of the honeycomb lattice. Several effects relevant to spin–orbit interaction, namely, the importance of chirality, curvature, and a family-dependent anisotropic conduction and valence band splitting are identified. We show that chiral nanotubes and nanoribbons exhibit spin-split states. Curvature-induced orbital hybridization is crucial to understand the experimentally observed anisotropic spin–orbit splittings in carbon nanotubes. In fact, spin–orbit interaction is important in curved graphene nanoribbons, since the induced spin-splitting on the edge states gives rise to spinfiltered states. & 2012 Elsevier Ltd. All rights reserved.

Keywords: A. Graphene nanoribbons A. Carbon nanotubes D. Spin-orbit

1. Introduction Recent advances in nanofabrication techniques have resulted in the production of high quality carbon nanotubes (CNTs) and graphene nanoribbons (GNRs). They have also allowed to manipulate graphene sheets of different geometries [1] and to unzip CNTs forming combined GNR–CNT structures [2–4]. These novel carbon materials are natural candidates for developing new nanoelectronic devices and present promising features for spintronics [5]. CNTs and GNRs are interesting objects for both, experimental and theoretical research, due to their distinct properties and can be considered as basic ingredients of possible graphene-based nanostructures. The importance of spin states in CNTs was evidenced earlier on by single-electron transport experiments [6–9], although due to the low atomic number of carbon, it was widely accepted that the spin–orbit interaction (SOI) was weak in carbon-based materials and thus disregarded. The independence of the spin and orbital symmetries seemed to be corroborated by observations done in defect-free CNTs, from which electron-hole symmetry was also deduced [10]. Later experimental improvements have permitted the observation of the SOI effects on the single electron states of ultra-clean CNT quantum dots [11]. The reported measurements demonstrated the coupling of the spin and the orbital motion of the electrons and evidenced the splitting of the fourfold degenerated single electron energy levels close to EF at zero magnetic field. Besides proving the relevance of spin–orbit effects in CNTs, this

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experiment shows an anisotropic splitting of electrons and holes. In addition, recent experiments have corroborated the importance of SOI effects. For example, spin-polarized currents have been measured on chiral single-wall CNTs in pulsed magnetic fields, the spin splitting being connected to the SOI [12]. Also, a systematic dependence of the SOI on the electron occupation of CNT quantum dots has been observed and attributed to the curvature-induced spin–orbit split Dirac spectrum [13]. These findings have enhanced the interest of SOI on CNTs and promoted theoretical works considering external electrical and magnetic fields [14–16]. In fact, the bipartite character of the graphene lattice, alongside the different symmetries of carbon systems make the SOI effects particularly interesting in low-dimensional carbon-based materials [17,18]. From the theoretical viewpoint, the effects of the spin–orbit coupling were first investigated to the lowest order in perturbation theory in a continuous k  p scheme [19]. Band splitting and the opening of an energy gap for the bands crossing at the Fermi energy were found in several works considering curvature effects and using either the perturbative approach [19,20] or low-energy field theory [21]. Based on the empirical tight-binding (ETB) approach, we have shown that the inclusion of the full lattice symmetry is essential to study spin–orbit interactions in CNTs, demonstrating that SOI effects present an intrinsic symmetry dependence [22]. We showed that the SOI lifts the spin-degeneracy of the energy bands for chiral nanotubes. Furthermore, we also proved that the observed electron-hole asymmetry was directly related to the induced s–p hybridization [23]. The study of the electronic properties of graphene nanoribbons has been mainly based on the k  p scheme and on the effective p-orbital tight-binding approximation, models known to capture the low-energy physics of graphene. First-principles and LCAO

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calculations have also been performed [24,25]. The intrinsic SOI effects on the transport properties of GNRs have been investigated within the Dirac model [26–28]. Earlier theoretical works based on the ETB approach have focused mostly on the study of achiral ribbons, namely of zigzag and armchair type [29,30]. Here we go further, studying SO coupling in chiral ribbons. We address the effects of the intrinsic SOI on the low energy properties of both, chiral and achiral, CNTs and GNRs of curved geometries. We focus on the interplay of the spin–orbit coupling and the two main parameters of these carbon nanostructures: chirality and curvature. We use an ETB model with a four-orbital sp3 basis set including the intrinsic atomic-like spin–orbit contribution. Within this description the Hamiltonian is exactly solved for all the points of the one-dimensional (1D) Brillouin zone (BZ). Therefore, our approach is specially appropriate to investigate chiral GNRs, since the description of edge states of arbitrary geometry requires a model that provides the energy bands on the whole BZ. Comparison of the CNTs and GNRs properties allows one to understand the important role played by symmetry when the spin–orbit coupling is considered. The paper is outlined as follows. The structure of the different types of CNTs and GNRs are depicted in Section 2. In Section 3 we introduce the model Hamiltonian and the calculation method. Results are presented in Section 4.1 for CNTs while those corresponding to curved GNRs are analyzed in Section 4.2. Finally, in Section 5 some conclusions are drawn.

2. Structure and geometry of CNTs and GNRs

width as a number N multiplying the width vector, which is the smallest graphene lattice vector perpendicular to T [35]. Curved ribbons of constant curvature are formed by bending the ribbon along its width. For a given ribbon, different curvatures can be obtained by changing the angles and the diameter of the cylindrical configurations [33]. In the present work, we focus on ribbons with maximal curvature. The structures studied have the shape of open cylinders, thus lacking of the rotational symmetry of their CNT counterparts. We assume that the bending is realized without stretching and reconstruction or relaxation of the edges are not allowed [29]. In the calculations we take into account the 1D periodicity of both CNT and GNR, and the atomic positions are those of the three-dimensional cylindrical structures. They are given in the XYZ-coordinate system, with the Z axis in the direction of the corresponding translation vectors of the 1D unit cells. Thus, the CNT and GNR axis are along the Z direction. Since the full atomic unit cell is considered, the actual discrete nature of the symmetries and the curvature of the CNTs and curved GNRs are automatically included.

3. Theoretical model and method of calculation The electronic properties have been calculated within the empirical tight-binding (ETB) approach using the Slater–Koster (SK) orbital projections [37]. We consider an orthogonal four-orbital 2s, 2px , 2py , 2pz basis set and the Toma´nek–Louie parametrization for graphite up to nearest-neighbour interactions [38]. Within the ETB approach the one-electron Hamiltonian is given by H ¼ H0 þ HSO ,

Both carbon nanotubes and graphene nanoribbons exhibit peculiar properties directly related to their geometry. The underlying structure is the honeycomb lattice, but boundary conditions and distinct symmetries mark out important differences [31]. Carbon nanotubes (CNTs) consist of a wrapped-up graphene sheet with cylindrical symmetry. They are very close to ideal 1D systems and are labeled by the numbers (n,m), which define the unrolled circumference chiral vector Ch ¼ na1 þ ma2 in the 2D graphene lattice, where a1 , a2 are the two lattice vectors of graphene at 601; a ¼ 9ai 9. The angle formed by Ch and a1 is known as the chiral angle [32]. According to its spatial symmetry, CNTs are classified as achiral (belonging to a symmorphic group) and chiral (non-symmorphic). Achiral tubes present an inversion center and can be an armchair (n,n) and a zigzag ðn,0Þ, with chiral angle y of p=6 and 01, respectively. Chiral tubes (n,m), with n a 0 and m a 0 and 01 o y o 301, do not possess an inversion center and exhibit spiral symmetry operations. A graphene nanoribbon consists of a stripe of graphene of infinite length and finite width. In the same way that chirality and diameter determine the properties of CNTs, the width and the edge shape set those of graphene nanoribbons [31,33]. There are two prototypical shapes of periodic GNR edges, namely, zigzag and armchair, which differ in 301 in the cutting direction of the graphene sheet. GNR with these types of edges are achiral. In general, any periodic minimal edge is characterized by a translation vector T ¼ na1 þ ma2 . The low-energy states of a minimal edge are unambiguously fixed by T, or equivalently, by (n,m), which label the edge. An arbitrary minimal edge (n,m) can be described as a combination of a zigzag and an armchair edge [34,35]. Henceforth, chiral ribbons are denoted by two indices, which also indicate the number of armchair (m) and zigzag (nm) units forming the minimal repeat cell along the edge. In achiral ribbons, the width is customarily given by the number N of dimers (armchair GNR) or zigzag chains (zigzag GNR) along the transversal direction [36]. For chiral ribbons, we will denote their

ð1Þ

where H0 is the SK-ETB spin-independent term and HSO is the atomic-like spin–orbit interaction contribution. H0 is written as X X a, b Ea þ t ij cai,sþ cbj,s þ h:c:, ð2Þ H0 ¼ /ijS, b,s

i, a,s

where Ea represents the atomic energy of the orbital a, i and j stand for the atomic sites in the honeycomb lattice and cai,sþ and cai,s are the creation and annihilation operators of an electron at site i, orbital a and spin s, respectively. The summation on /ijS is taken over all the carbon atoms in the CNT (GNR) 1D unit cell. The spin–orbit interaction arises from the coupling of the electron spin with the magnetic field resulting in the rest frame of the electron due to its orbital motion in an electrostatic potential. In a crystalline environment, in the absence of external fields, the major internal contribution arises from the microscopic crystal potential. The magnetic field acting on the electron is thus determined by the atomic potential modified by the presence of neighboring atoms, and therefore the SOI effects are influenced by the crystal symmetry. Since the major contribution of the crystal potential to the spin–orbit interaction is close to the atomic cores, spherical symmetry can be assumed and HSO can be expressed by X ð3Þ HSO ¼ l li  si ¼ l L  S, i

where l is a renormalized atomic SO coupling constant, and L and S stand for the total orbital angular momentum and electron spin operators, respectively. Correspondingly, li and si are the orbital angular momentum and spin operators acting on each atomic side and the summation is over the atoms in the unit cell. Therefore, taking into account the spin parts of the atomic wavefunctions, the Hamiltonian of a nanotube or nanoribbon with N atoms in the unit cell is a matrix of 8N  8N dimensions. If the spin quantization direction of the spinor wavefunction is chosen along Z parallel to the CNT (GNR) axis, using the raising and lowering operators, L þ ¼ LX þiLY , L ¼ LX iLY , H is given in

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the 2  2 block spinor structure by ! lL H0 þ lLZ : H¼ lL þ H0 lLZ

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ð4Þ

The HSO term of the Hamiltonian couples states of equal angular momentum in the same atom [39,40], i.e., px, py and pz orbitals in the present case. Hence, to solve the problem by direct diagonalization of the Hamiltonian, all the p states must be included in the model. Although our discussion is focussed in the bands around the Fermi level, our calculation is valid for the entire energy scale, not only in the region close to the Fermi energy, as in the effective models for p electrons. In fact, SOI effects are larger for states involving s bands. The spin–orbit interaction considered in our model is the so-called intra-atomic SOI, so extrinsic effects due to external potentials are not taken into account. H includes all the contributions—spin conservation and spinflip processes—arising from the discrete symmetries of the underlying crystal potential. Even more, since the SOI matrix elements between different atomic p orbitals are nonzero, SOI also induces the hybridization of the s and p bands. Thus, in our model both, curvature and SOI, contribute to the s–p hybridization. In fact, the effects of the intrinsic Dint and curvature Dcurv contributions of the effective p SOI model derived by perturbation theory [19,41] are naturally obtained in our calculation. Different estimations of l have been done [19,41], always assuming a very small value, considerably reduced in graphite, graphene and CNTs with respect to that for atomic C (  12 meV). The recent experiments of Ref. [11] point out to a large enhancement of SOI in CNTs with respect to that of graphene. Nevertheless, since the exact value of the SO coupling is still unknown, SOI-induced energy splittings shown in the present work are given relative to the strength of the SOI, and only in the figures we have chosen an artificially large value of l, for the sake of clarity.

4. Results and discussion 4.1. Carbon nanotubes The electronic properties of CNTs, which have been intensively investigated both theoretically and experimentally [42,43], are determined by the graphene band structure and the boundary conditions. They show metallic or semiconducting behaviour depending on their diameter and chirality [32]. We concentrate on metallic CNTs. As we consider tubes of different unit cell sizes, we denote the edge of the BZ as X, and we define wavevectors with respect to the size of half the BZ, i.e., GX. Armchair tubes (n,n) are always metals even after including curvature effects. Their bands cross at the Fermi level at two thirds from G and they are spin-degenerated. Zigzag tubes ðn,0Þ are metallic only if n ¼ 3q, q being an integer; the band crossing at EF occurs at G. Due to the valley K and K 0 contributions and to the spin degeneracy, they are fourfold degenerate, which is the maximum degeneracy of CNT bands [32]. Chiral tubes are metals if nm ¼ 3q and semiconductor otherwise. The band crossing at Fermi can occur either at 2 3 GX or at G. Similarly to achiral tubes, the former have twofold- and the second fourfold-degenerate bands. 4.1.1. Spin–orbit interaction and chirality In Fig. 1, the energy bands of the achiral ð10; 10Þ armchair and ð12; 0Þ zigzag nanotubes are shown around the Fermi level. SOI removes all the degeneracies except those due to the spin. In the armchair tubes, since the bands that cross at Fermi are

Fig. 1. Band structures of two achiral carbon nanotubes calculated with SOI ðl ¼ 0:2Þ. Top: (10,10) armchair nanotube. Bottom: (12,0) zigzag nanotube.

twofold spin-degenerate, SOI only lifts the accidental degeneracy at EF opening a small gap, as it has been previously reported [19,22]. Hence, the metallic character of armchair CNTs, which is robust under curvature effect, is destroyed by SOI. In zigzag tubes, SOI eliminates the degeneracy, but only partially, because the bands are still spin-degenerate. Therefore, in achiral CNTs inclusion of SOI does not lift the spin-degeneracy, albeit a splitting of the bands occurs whenever their degeneracy is due to the K and K 0 contributions. In addition, the states are no longer spin eigenstates, although they present a dominant spin contribution. The calculated spin-polarization direction is almost parallel to the tube axis. We focus now on chiral tubes. The band structures of the (8,2) and (9,3) CNTs calculated with SOI are depicted in Fig. 2. In this case SOI lifts all degeneracies and yields an energy splitting between states with opposite spin orientations. Thus, the states are spin eigenstates and, analogously to achiral tubes, the direction of the spin polarization is almost parallel to the CNT axis. This is a general result valid for any chiral tube. This behaviour is due to the interplay of spatial and timereversal symmetries. Because SOI does not break the timereversal symmetry, Kramers’ theorem states that Ek,s ¼ Ek,s , where Ek,s is the energy of the eigenstate with wavevector k and spin s. Besides, achiral tubes also have spatial inversion symmetry, so Ek,s ¼ Ek,s , and then Ek,s ¼ Ek,s . Thus in achiral tubes the spin degeneracy cannot be lifted, as shown in the left panel of Fig. 3. However, in chiral tubes only time-reversal symmetry holds, so Ek,m ¼ Ek,k and the bands are non-degenerated at any k point of the BZ, except at the time-reversal invariant G and X points [22]. In the right panel of Fig. 3, the spin-resolved bands of the (6,3) CNT are represented around the G point of the BZ close to the Fermi energy. The asymmetry between up and down electron bands shows that electron velocities become dependent upon both the spin and the direction of the motion. The slight shift of the Fermi wave vector is induced by curvature effects.

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(6,3)

Energy (eV)

1

Fig. 4. (Color online) Normalized spin–orbit splitting of the top valence and bottom conduction bands as a function of the tube diameter for the three CNT zigzag families.

0

-1 X

Γ

X

Fig. 2. Band structures of two chiral carbon nanotubes calculated with SOI ðl ¼ 0:2Þ. Top: (8,2) nanotube. Bottom: (6,3) nanotube.

Fig. 3. Zoom of the band structure around the G point close to EF calculated with SOI ðl ¼ 0:2Þ for the (12,0)—left panel—and (6,3)—right panel—carbon nanotubes. Arrows indicate the spin polarization of the bands.

4.1.2. Spin–orbit interaction and curvature Curvature is determinant in CNTs and must be taken into account to understand the behaviour of carriers. The combined effect of curvature and SOI in the honeycomb lattice [41], as well as in CNTs and quantum dots, has been previously investigated [20,23,41,44,45,46]. In general, CNTs (n,m) can be classified in three families, nm ¼ 3qn, where n ¼ 0, 7 1 and q is an integer. If n ¼ 0 the CNTs are metallic and semiconducting otherwise, without including curvature. Several properties such as band gaps or band splittings present a family dependence [47–49]. This behaviour can be related to the trigonal warping effect of the honeycomb lattice. SOI effects also obey this family rule. To study the effect of curvature, we first consider zigzag tubes ðn,0Þ, with a chiral angle of 0o , which present the largest curvature. Zigzag tubes have the minimum band gap at G and even in the absence of SOI, metallic tubes have a very small energy gap due to the curvature. We have calculated the band structure in the presence of SOI for tubules with varying diameter and family index n. In Fig. 4 the normalized spin–orbit induced splitting of the top valence band and bottom conduction band are represented as a function of the tube diameters for the three zigzag families. Curvature effects are clearly shown: the absolute values of the energy splittings become larger with decreasing tube diameters [23]. Furthermore, in agreement with the experimental results [11], for a given tube the splittings of the highest valence band (VB) and lowest conduction band (CB) are asymmetric [22]. However,

Fig. 5. (Color online) Contour plot of the graphene bandstructure calculated around the Fermi K point with the quantization lines corresponding to the (7,0) (green, dashed) and (8,0) (blue, dotted) CNTs.

while in the n ¼ 0, þ 1 series the splitting is larger for the valence states, in the n ¼ þ 1 it is larger for the conduction band. In order to understand this behaviour, we focus on three CNTs representative of the different families, namely (9,0), (8,0) and (7,0), with n ¼ 0, n ¼ 1 and n ¼ þ 1 respectively. The constant energy contours of the graphene p-band around the K point of coordinates ð4p=3a,0Þ are depicted in Fig. 5, together with the allowed quantization lines closer to K for the (8,0) and (7,0) CNTs. For the sake of clarity, those corresponding to the (9,0) CNT, whose behaviour is analogous to that of the (8,0), are not shown. The quantization line closest to K determines the band gap and lies to the left of K, on the GK line, for the (8,0) NT (n ¼ 1 family), whereas for the (7,0) tube (n ¼ þ1 family) it lies to the right over the KM line. Their relative positions around the K 0 Fermi point are reversed. Because of the trigonal warping effect, which deforms the energy contours, as can be seen in Fig. 5, quantization lines at opposite sides of K intersect graphene bands with different slopes even if they are at a similar distance to the point [23]. Therefore, the quantization lines of CNTs of the n ¼ 1 and n ¼ þ 1 families intersect bands of different slopes and, most importantly, of different symmetries. Due to the honeycomb lattice topology, the bands crossing at EF have different symmetries. The lowest CB

L. Chico et al. / Solid State Communications 152 (2012) 1477–1482

p-band changes from antibonding along GK to bonding character along KM. Therefore, the VB and CB of tubes belonging to n ¼ 1 and n ¼ þ1 families show an opposite behaviour, because the character of the electron and hole bands is exchanged at the Fermi points. The metallic n ¼ 0 tubes present a similar behaviour to the semiconductor n ¼ 1 tubes, since curvature induces a shift of the Fermi vectors K ðK 0 Þ to the left (right) in metallic zigzag CNTs. The absolute values of the VB and CB SOI energy splittings are directly related to the sp hybridization induced by both curvature and SOI. Our model Hamiltonian includes a four orbital sp3 basis set and therefore the s and p contributions to each band can be calculated. Table 1 shows these contributions at G, summed over all the atoms of the unit cell, for the top VB and bottom CB states of the (7,0), (8,0), (9,0) and (10,0) zigzag tubes. There is a clear correlation between the proportion of s orbitals, that is s–p hybridization, and the spin–orbit splittings. There is a larger contribution of s orbitals to the CB than to the VB states for NTs belonging to the n ¼ þ 1 family, which corresponds with larger CB splittings. The opposite occurs for the n ¼ 0,1 families. Moreover, within a given family, for increasing diameter the hybridization decreases, as can be inferred from Table 1 by comparing the (7,0) and (10,0) NTs which belong to the same n ¼ þ1 family. In the case of chiral tubes, the allowed k quantization lines present different orientations around K due to the chiral angle, 01 o y o 301, so the gathering in three families is not so clear as for zigzag tubes. Also, due to the lack of inversion symmetry, SOI lifts all the degeneracies, including spin. Nevertheless, the SOI energy splittings follow the same tendency, although the deviation from the zigzag behavior is greater for higher chiral angles [23]. The splitting is larger for CNTs of small diameter and it is larger for the VB of n ¼ 1 and n ¼ 0 tubes and for the CB of n ¼ þ1 tubes. Accordingly, the asymmetric energy splitting of electrons and holes is the result of the different s–p hybridization induced in the corresponding bands by both curvature and SOI. 4.2. Graphene nanoribbons The electronic properties of GNRs are derived from the band structure of graphene subject to a stripe geometry. The confinement due to the finite-size and the presence of boundaries yield the peculiar band structure of GNRs. Edge states appear due to the truncation of inter-atomic bonds caused by the borders; therefore, they are strongly dependent on the atomic termination of the GNR. This strong dependence of the electronic properties on the geometry allows to tune different behaviours by changing the ribbon size and edge shape. Zigzag ribbons have zero-energy edge states, whereas armchair ribbons present a strong dependence on their width: they are metallic when its width N ¼ ð3M1Þd, where M is an integer and d is the C–C atom distance. For other values of N the GNRs are semiconductors [24,33]. Chiral ribbons, as they have a zigzag component, also present edge states. We analyze the interplay of SOI with the two main parameters of the curved ribbons, chirality and curvature. We focus first on zigzag metallic GNRs with maximum curvature. In the zigzag termination of the honeycomb lattice all atoms belong to the Table 1 Orbital electronic densities of the states closest to the gaps. Orbital

s p

(7,0)

(8,0)

(9,0)

(10,0)

CB

VB

CB

VB

CB

VB

CB

VB

0.1734 0.8263

0.0873 0.9127

0.057 0.943

0.087 0.913

0.058 0.941

0.103 0.897

0.121 0.879

0.061 0.939

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Fig. 6. Band structure of the curved zigzag nanoribbon N ¼60 calculated with SOI ðl ¼ 0:2Þ.

Fig. 7. Band structure of the curved nanoribbon with edge (3,2) and width N¼ 4 calculated with SOI ðl ¼ 0:2Þ.

same sublattice. Thus, the atoms at opposite edges of a zigzag ribbon belong to different sublattices. In order to avoid the coupling of the edge states, a critical value of the ribbon width, slightly greater than in the flat geometry, is needed [29]. In Fig. 6 the band structure of the achiral curved zigzag N ¼60 GNR is shown. The curved geometry induces small changes with respect to the flat ribbon band structure. The zero-energy p-orbital states at the 23 GX o ko X interval, which are dispersionless in the flat geometry, present a small contribution of s orbitals and a very weak dispersion is observed. Nevertheless, the localized character of the edge states in a k interval around the X point remains. The contribution of the s orbitals is larger around 23 GX, thus the merging of edge states with the bulk bands occurs at a k value smaller than for the corresponding flat GNR. Besides the small dispersion of the edge states, curvature induces a downshift in energy. However, SOI effects are basically the same than in the flat geometry, but larger. SOI partially lifts the fourfold degeneracy of the p edge states except at the G and X points, which are protected by timereversal symmetry. Each edge state splits into two degenerate Kramers doublets with linear dispersion in a very small region around the crossing points, so forward and backward movers have opposite spin polarization. These twofold degenerate states are confined at the borders of the ribbon. Therefore, for a given energy the zigzag GNR has four conducting channels spatially separated, i.e., two spin-filtered edge states are at each boundary of the ribbon. We have calculated the expectation value of the spin /SS for the edge states, and found that it is no longer perpendicular to the graphene sheet as in the flat ribbons. An inplane contribution changes the spin axis to form an angle with the ribbon plan; the in-plane contribution is proportional to the coupling constant [29,30]. We consider now a metallic chiral nanoribbon with termination (3,2) and width N ¼4, the 1D unit cell contains 304 carbon atoms. Fig. 7 represents the corresponding band structure.

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Changes induced by the curvature are similar to those described for the zigzag ribbon. However, in analogy to chiral CNTs, due to the lack of inversion symmetry, SOI lifts all the degeneracies of the system and therefore, edge states are nondegenerate in spin. The spin-polarization direction of edge states also deviates from the normal to the surface of the GNR. The spatial distribution of the edge states in chiral ribbons is mainly at the sublattice where the majority of zigzag edge atoms belong to, and their penetration into the ribbon is greater than for pure zigzag ribbons of similar length.

5. Conclusions In this work we have investigated the SOI effects on carbon nanotubes and nanoribbons. We have proved that the spin–orbit interaction has an intrinsic dependence on the symmetry of the CNTs. SOI removes the spin degeneracy in chiral tubes, whereas in achiral tubes the spin degeneracy is not lifted due to the inversion symmetry of the crystal potential. This symmetry dependence has important consequences for spin relaxation times in CNTs: in chiral tubules the motion of electrons with opposite spin polarization are completely decoupled, e.g. electrons in spin-up states cannot scatter to states with spin-down, thereafter, spin-independent scatterers can not induce spin scattering. On the contrary, in achiral tubes spin-independent impurities can cause spin relaxation. We have shown that the s–p hybridization due to curvature enhances spin–orbit effects in CNTs. The s–p hybridization, which is specially important for small-diameter tubes, depends on the symmetry of the band and thus it is different for the electrons and holes bands crossing at Fermi. We found that the SOI energy splitting is related to the sp hybridization and is larger for bands with greater s contribution. Therefore, the asymmetric energy splitting reported by the experiment [11] results from the interplay of the trigonal warping effect of the graphene band structure and the sp hybridization induced by curvature and SOI. In curved GNRs, the rotational symmetry present in CNTs is broken, affecting the degeneracy of the bands. Although the interplay of curvature and SOI effects is fundamentally similar to that in CNTs, the enhancement of the SOI strength, because of symmetry considerations, is weaker. Curvature does not affect the localized character of the edge states, but due to the increase in the s contribution it induces a small deviation in the spin orientation from the direction normal to the surface of the GNR. Our results give an explanation to the experimental observations, in particular to the asymmetric splitting of the valence and conduction bands due to SOI. Our approach allows for a complete vision of the connections among curvature, hybridization and symmetry effects related to spin–orbit interaction in carbon nanostructures.

Acknowledgements This work has been partially supported by the Spanish Ministry of Science and Innovation (MICINN) DGES under grants FIS2009-08744, MAT2009-14578-C03-03, PIB2010BZ-00512 and FIS2011-23713.

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