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Spin-orbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps.

Daniel Huertas-Hernando,1 F. Guinea,2 and Arne Brataas1, 3

1

arXiv:cond-mat/0606580v3 [c

ond-mat.mes-hall] 25 Oct 2006

Department of Physics, Norwegian University of Science and Technology, N-7491, Trondheim, Norway 2 Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco E28049 Madrid, Spain 3 Centre for Advanced Study, Drammensveien 78, 0271 Oslo, Norway A continuum model for the e?ective spin orbit interaction in graphene is derived from a tightbinding model which includes the π and σ bands. We analyze the combined e?ects of the intraatomic spin orbit coupling, curvature, and applied electric ?eld, using perturbation theory. We recover the e?ective spin-orbit Hamiltonian derived recently from group theoretical arguments by Kane and Mele. We ?nd, for ?at graphene, that the intrinsic spin-orbit coupling ?int ∝ ?2 and the Rashba coupling due to an perpendicular electric ?eld E , ?E ∝ ?, where ? is the intraatomic spin-orbit coupling constant for carbon. Moreover we show that local curvature of the graphene sheet induces an extra spin-orbit coupling term ?curv ∝ ?. For the values of E and curvature pro?le reported in actual samples of graphene, we ?nd that ?int < ?E ?curv . The e?ect of spin orbit coupling on derived materials of graphene like fullerenes, nanotubes, and nanotube caps, is also studied. For fullerenes, only ?int is important. Both for nanotubes and nanotube caps ?curv is in the order of a few Kelvins. We reproduce the known appearance of a gap and spin-splitting in the energy spectrum of nanotubes due to the spin-orbit coupling. For nanotube caps, spin-orbit coupling causes spin-splitting of the localized states at the cap, which could allow spin-dependent ?eld-e?ect emission. INTRODUCTION.

A single layer of carbon atoms in a honeycomb lattice, graphene, is an interesting two-dimensional system due to its remarkable low energy electronic properties[1, 2, 3], e.g. a zero density of states at the Fermi level without an energy gap, and a linear, rather than parabolic, energy dispersion around the Fermi level. The electronic properties of the many realizations of the honeycomb lattice of carbon such, e.g. bulk graphite (3D), carbon nanotube wires (1D), carbon nanotube quantum dots (0D), and curved surfaces such as fullerenes, have been studied extensively during the last decade. However, its two dimensional (2D) version, graphene, a stable atomic layer of carbon atoms, remained for long ellusive among the known crystalline structures of carbon. Only recently, the experimental realization of stable, highly crystalline, single layer samples of graphene [4, 5, 6, 7], have been possible. Such experimental developments have generated a renewed interest in the ?eld of two dimensional mesoscopic systems. The peculiar electronic properties of graphene are quite di?erent from that of 2D semiconducting heterostructures samples. It has been found that the integer Hall e?ect in graphene is di?erent than the “usual” Quantum Hall e?ect in semiconducting structures[8, 9, 10, 11]. Moreover, it has been theoretically suggested that a variety of properties, e.g. weak (anti)localisation[12, 13, 14, 15, 16, 17], shot-noise[18] and anomalous tunneling-Klein’s paradox[19], are qualitatively di?erent from the behavior found in other 2D systems during the last decades. All these predictions can now be directly investigated by experiments. The activity in graphene, both theoretically and experimentally, is at present very intense. However so far, the work has mainly focused on i) the fact that the unit cell is described by two inequivalent triangular sublattices A and B intercalated, and ii) there are two independent k -points, K and K ′ , corresponding to the two inequivalent corners of the Brillouin zone of graphene. The Fermi level is located at these K and K’ points and crosses the π bands of graphene (see Fig.[1] for details). These two features provide an exotic fourfold degeneracy of the low energy (spin-degenerate) states of graphene. These states can be described by two sets of two-dimensional chiral spinors which follow the massless Dirac-Weyl equation and describe the electronic states of the system near the K and K’ points where the Fermi level is located. Neutral graphene has one electron per carbon atom in the π band, so the band below the Fermi level is full (electron-like states) and the band above it is empty (hole-like states). Electrons and holes in graphene behave like relativistic Dirac fermions. The Fermi level can be moved by a gate voltage underneath the graphene sample[4]. State-of-the-art samples are very clean, with mobilities ? ? 15000cm2V ? 1s? 1[6], so charge transport can be ballistic for long distances across the sample. From the mobilities of the actual samples, it is believed that impurity scattering is weak. Furthermore, it has been recently suggested that the chiral nature of graphene carriers makes disordered regions transparent for these carriers independently of the disorder, as long as it is smooth on the scale of the lattice constant[13, 14, 20]. Less attention has been given to the spin so far. The main interactions that could a?ect the spin degree of freedom in graphene seem to be the spin-orbit coupling and exchange interaction. It is not known to which extent magnetic impurities are present in actual graphene samples. Their e?ect seem small though, as noticed recently

2 when investigating weak localization and universal conductance ?uctuations in graphene[14]. Spin-orbit interaction in graphene is supposed to be weak, due to the low atomic number Z = 6 of carbon. Therefore both spin splitting and spin-?ip due to the combination of spin-orbit and scattering due to disorder is supposed to be not very important. As a result, the spin degree of freedom is assumed to have a minor importance and spin degenerate states are assumed. Besides, the spin degeneracy is considered to be “trivial” in comparison to the fourfold degeneracy previously mentioned, described by a pseudo-spin degree of freedom. At present, there is a large activity in the study of the dynamics of this pseudo-spin degree of freedom[8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. We think that the physics of the electronic spin in graphene must be investigated in some detail, however. Although it could be that the electronic spin is not as important/exotic as the pseudo-spin when studying bulk properties, edge states may be quite di?erent. Induced magnetism at the edges of the surface of graphite samples irradiated with protons have been reported[27]. Moreover, perspectives of spintronic applications in graphene could be very promising, so it is important to clarify the role of the electronic spin. This is one of the main purposes of the present paper. Moreover, we feel that the existent knowledge about the spin-orbit interaction in graphene is not yet complete[24] and that certain, both quantitative and qualitative, points must be discussed in more detail. That is why we focus our discussion on the spin-orbit coupling. The e?ect of other interactions as the exchange interaction will be discussed elsewhere. Spin-orbit coupling in graphene has an intrinsic part, completely determined from the symmetry properties of the honeycomb lattice. This is similar to the Dresselhauss spin-orbit interaction in semiconducting heterostructures[28]. Group theoretical arguments allow to obtain the form of the e?ective Hamiltonian for the intrinsic spin-orbit coupling around the K , K ′ points[24, 29, 30]. It was predicted that this interaction opens-up a gap ?int in the energy dispersion. However, the strength of this intrinsic spin-orbit coupling is still a subject of discussion, although it is believed to be rather small, due to the weakness of the atomic intraatomic spin orbit coupling of carbon ?. If an electric ?eld E is applied perpendicular to the sample, a Rashba interaction[31] ?E will also be present in graphene. Analogous to the intrinsic coupling, group theoretical arguments allow deducing the form of the Rashba interaction[24, 25]. The strength of this Rashba spin-orbit coupling is also still under discussion. We follow a di?erent approach. We set up a tight binding model where we consider both the π and σ bands of graphene and the intraatomic spin-orbit coupling ?. We also include curvature e?ects of the graphene surface and the presence of a perpendicular electric ?eld E . Starting from this model, we obtain an e?ective Hamiltonian for the π bands, by second order perturbation theory, which is formally the same as the e?ective Hamiltonian obtained previously from group theoretical methods[29, 30] by Kane and Mele[24]. Moreover, we show that curvature e?ects between nearest neighbor atoms introduce an extra term ?curv into the e?ective spin-orbit interaction of graphene, similar to the Rashba interaction due to the electric ?eld ?E . We obtain explicit expressions for these three couplings in terms of band structure parameters. Analytical expressions and numerical estimates are given in Table[I]. We ?nd that the intrinsic interaction ?int ? 10mK is two orders of magnitude smaller than what was recently estimated[24]. Similar estimates for ?int have been reported recently[26, 32, 33]. Moreover, we ?nd that for typical values of the electric ?eld as e.g. used by Kane and Mele[24] ?E ? 70mK. Similar discussion for ?E has appeared also recently[33]. So spin-orbit coupling for ?at graphene is rather weak. Graphene samples seem to have an undulating surface however[14]. Our estimate for the typical observed ripples indicates that ?curv ? 0.2K. It seems that curvature e?ects on the scale of the distance between neighbouring atoms could increase the strength of the spin-orbit coupling at least one order of magnitude with respect that obtained for a ?at surface. More importantly, this type of “intrinsic” coupling will be present in graphene as long as its surface is corrugated even if E =0 when ?E =0. The paper is organized as follows: The next section presents a tight binding hamiltonian for the band structure and the intraatomic spin-orbit coupling, curvature e?ects and a perpendicular electric ?eld. Then, the three e?ective spin-orbit couplings ?int , ?E , ?curv for a continuum model of the spin-orbit interaction for the π bands in graphene at the K and K ′ points are derived. Estimates of the values are given at the end of the section. The next section applies the e?ective spin-orbit hamiltonian to: i) fullerenes, where it is shown that spin-orbit coupling e?ects play a small role at low energies ii) nanotubes, where known results are recovered and iii) nanotubes capped by semispherical fullerenes, where it is shown that the spin orbit coupling can lead to localized states at the edges of the bulk subbands. The last calculation includes also a continuum model for the electronic structure of nanotube caps, which, to our knowledge, has not been discussed previously. A section with the main conclusions completes the paper.

3

DERIVATION OF CONTINUUM MODELS FROM INTRAATOMIC INTERACTIONS. Electronic bands.

The orbitals corresponding to the σ bands of graphene are made by linear combinations of the 2s, 2px and 2py atomic orbitals, whereas the orbitals of the π bands are just the pz orbitals. We consider the following Hamiltonian: H = HSO + Hatom + Hπ + Hσ , where the atomic hamiltonian in the absence of spin orbit coupling is: Hatom = ?p c? is′ cis′ + ?s

i=x,y,z ;s′ =↑,↓ s;s′ =↑,↓

(1) c? s,s′ cs,s′ .

(2)

where ?p,s denote the atomic energy for the 2p and 2s atomic orbitals of carbon, the operators ci;s′ and cs;s′ refer to pz , px , py and s atomic orbitals respectively and s′ =↑, ↓ denote the electronic spin. HSO refers to the atomic spin-orbit coupling occuring at the carbon atoms and the terms Hπ , Hσ describe the π and σ bands. In the following, we will set our origin of energies such that ?π = 0. We use a nearest neighbor hopping model between the pz orbitals for Hπ , using one parameter Vppπ . The rest of the intraatomic hoppings are the nearest neighbor interactions Vppσ , Vspσ and Vssσ between the atomic orbitals s, px , py of the σ band. We describe the σ bands using a variation of an analytical model used for three dimensional semiconductors with the diamond structure[34], and which was generalized to the related problem of the calculation of the acoustical modes of graphene[35]. The model for the sigma bands is described in Appendix A. The bands can be calculated analytically as function of the parameters: ?s ? ?p V1 = 3 √ 2Vppσ + 2 2Vspσ + Vssσ V2 = . (3) 3 The band structure for graphene is shown in Fig.(1).

Intraatomic spin-orbit coupling.

The intraatomic spin orbit coupling is given by HSO = ?Ls [36] where L and s are, the total atomic angular momentum operator and total electronic spin operator respectively, and ? is the intraatomic spin-orbit coupling constant. We de?ne: 0 1 s+ ≡ 0 0 s? ≡ sz ≡ L+ 0 0 1 0

1 2

L?

Lz |pz |px |py

0 1 0 ?2 ? √ ? 0 0 2 √ ≡ ?0 0 2? 0 0 0 ? ? 0 0 0 √ 0 0? ≡ ? 2 √ 0 2 0 ? ? 1 0 0 ≡ ?0 0 0 ? 0 0 ?1

≡ |L = 1, Lz = 0 1 ≡ √ (|L = 1, Lz = 1 + |L = 1, Lz = ?1 ) 2 +i ≡ √ (|L = 1, Lz = 1 ? |L = 1, Lz = ?1 ) , 2

(4)

4

10 8 6 4 2 E(eV) 0 -2 -4 -6 -8 -10

Γ

Κ

M

Γ

k

FIG. 1: “(Color online)”: Black (Full) curves: σ bands. Red (Dashed) curves: π bands. The dark and light green (grey) arrows give contributions to the up and down spins at the A sublattice respectively. The opposite contributions can be de?ned for the B sublattice. These interband transitions are equivalent to the processes depicted in Fig.[3].

Using these de?nitions, the intraatomic spin-orbit hamiltonian becomes: HSO = ? L + s? + L ? s+ + L z sz 2 (5)

The Hamiltonian Eq. (5) can be written in second quantization language as:

? ? ? ? ? HSO = ? c? z ↑ cx↓ ? cz ↓ cx↑ + icz ↑ cy ↓ ? icz ↓ cy ↑ + icx↓ cy ↓ ? icx↑ cy ↑ + h.c. .

(6)

where the operators c? z,x,y ;s′ and cz,x,y ;s′ refer to the corresponding pz , px and py atomic orbitals. The intraatomic hamiltonian is a 6 × 6 matrix which can be split into two 3 × 3 submatrices: HSO =

11 HSO 0 22 0 HSO

(7)

11 The block HSO acts on the basis states |pz ↑ , |px ↓ and |py ↓ : ? ? 0 1 i ? 11 ? 1 0 ?i ? . HSO = 2 ?i i 0 22 On the other hand HSO :

(8)

22 HSO

? ? 0 ?1 i ?? ?1 0 ?i ? = 2 ?i i 0

(9)

5

1 0 0 1 0 1

1 0 0 1 0 1

pz

+

+ ?

1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111

? ?

pz +

+

?

px

px

θ

FIG. 2: “(Color online)”: Sketch of the relevant orbitals, px and pz needed for the analysis of spin-orbit e?ects in a curved nanotube. The arrows stand for the di?erent hoppings described in the text.

acts on |pz ↓ , |px ↑ and |py ↑ states. The eigenvalues of these 3 × 3 matrices are +? (J = 3/2) which is singly degenerate and ??/2 (J = 1/2) which is doubly degenerate. The term L+ s? + L? s+ of the intraatomic spin orbit coupling Hamiltonian, Eq.(5), allows for transitions between states of the π band near the K and K ′ points of the Brillouin zone, with states from the σ bands at the same points. These transitions imply a change of the electronic degree of freedom, i.e. a “spin-?ip” process. We describe the σ bands by the analytical tight binding model presented in Appendix A. The six σ states at the K and K ′ points can be split into two Dirac doublets, which disperse linearly, starting at energies ?σ (K, K ′ ) = V1 /2 ± 9V12 /4 + V22 and two ?at bands at ?σ (K, K ′ ) = ?V1 ± V2 . We denote the two σ Dirac spinors as ψσ1 and ψσ2 , and the two other “?at” orbitals as φσ1 and φσ2 . The intraatomic spin orbit hamiltonian for the K and K ′ point becomes: HSOK ≡ ? 2 d2 r α 2 d2 r α 2 α 2 cos 3 2

? Ψ? AK ↑ (r)ψσ1AK ↓ (r) + ΨBK ↑ (r)ψσ1BK ↓ (r) +

+ sin HSOK ′ ≡ ? 2

? Ψ? AK ↑ (r)ψσ2AK ↓ (r) + ΨBK ↑ (r)ψσ2BK ↓ (r)

+

2 ? Ψ? AK ↑ (r) + ΨBK ↑ (r) φ1↓ (r) + h.c. 3

α 2 cos 3 2

? Ψ? AK ′ ↑ (r)ψσ2AK ′ ↓ (r) + ΨBK ′ ↑ (r)ψσ2BK ′ ↓ (r) +

+ sin

? ′ ′ Ψ? AK ′ ↑ (r)ψσ1AK ↓ (r) + ΨBK ′ ↑ (r)ψσ1BK ↓ (r)

+

2 ? Ψ? (10) AK ′ ↑ (r) + ΨBK ′ ↑ (r) φ2↓ (r) + h.c. 3

where Ψ stands for the two component spinor of the π band, and cos(α/2) and sin(α/2) are matrix elements given by: α = arctan (3V1 )/2 (9V12 )/4 + V22 . (11)

Next we would like to consider two posibilities: i) A curved graphene surface. ii) The e?ect of a perpendicular electric ?eld applied to ?at graphene. In the latter case we will have to consider another intraatomic process besides the intraatomic spin-orbit coupling, the atomic Stark e?ect.

6

E?ects of curvature.

In a curved graphene sheet, a hopping between the orbitals in the π and σ bands is induced[37]. First we will use a simple geometry to illustrate the e?ect of curvature between neighbouring atoms. This geometry is schematically shown in Fig.[2], where we ?rst consider two atoms at the same height along the axis of the tube. In this geometry we consider that the pz orbitals are oriented normal to the surface of the nanotube, the px orbitals are oriented along the surface circumference (Fig.[2] ) and the py orbitals are parallel to the nanotube axes. The curvature modi?es the hopping between the two atoms compared to the ?at surface for the pz and px orbitals but will not change, for this simple case, the hopping between py orbitals. The (reduced) pz -px hopping hamiltonian is the sum of three terms: HT =

? 2 2 Vppπ cos2 (θ) + Vppσ sin2 (θ) c? z 1s′ cz 0s′ ? Vppπ sin (θ ) + Vppσ cos (θ ) cx1s′ cx0s′ +

s′

? ? + Vspσ sin2 (θ)c? z 1s′ cs0s′ + sin(θ ) cos(θ ) (Vppπ ? Vppσ ) cz 1s′ cx0s′ ? cx1s′ cz 0s′ + h.c.

(12)

where 0 and 1 denote the two atoms considered and θ is the angle between the ?xed Z axis and the direction normal to the curved surface (Fig.[2] ). The angle θ, in the limit when the radius of curvature is much longer than the interatomic spacing, a ? R, is given by θ ≈ a/R. The hopping terms induced by (intrinsic) curvature discussed here break the isotropy of the lattice and lead to an e?ective anisotropic coupling between the π and σ bands in momentum space. The previous discussion can be extended to the case of general curvature when the graphene sheet has two di?erent curvature radii, R1 and R2 corresponding to the x and y directions in the plane. In that case, the factor R?1 has to ?1 ?1 be replaced by R1 + R2 . We now expand on θ ? a/R1,2 ? 1. By projecting onto the Bloch wavefunctions of the π and σ bands at the K and K ′ points, we ?nd: HT K ≡ (Vppσ ? Vppπ ) + sin α 2 3 2 a a + R1 R2 d2 r cos α 2

? Ψ? AK ↑ (r)ψσ1BK ↑ (r) + ΨBK ↑ (r)ψσ1AK ↑ (r) +

? Ψ? AK ↑ (r)ψσ2BK ↑ (r) + ΨBK ↑ (r)ψσ2AK ↑ (r)

+ h.c.

(13)

and a similar expression for the K ′ point. The induced spin orbit coupling, however, includes only contributions from the four σ bands at K and K ′ with ?σ = V1 /2 ± (9V12 )/4 + V22 , as those are the only bands coupled to the π band by the intraatomic spin orbit term considered here, Eq.(10). We now assume that the energies of the σ bands are well separated from the energy of the π bands (?π = 0 at the K and K ′ points). Then, we can use second order perturbation theory and from Eq.(10) and Eq.(13) we obtain an e?ective hamiltonian acting on the states of the π band. HcurvKπ ≡ ?i ?(Vppσ ? Vppπ )V1 2V12 + V22 a a + R1 R2 a a + R1 R2

? d2 r Ψ? AK ↑ (r)ΨBK ↓ (r) ? ΨBK ↓ ΨAK ↑ .

HcurvK ′ π ≡ ?i

?(Vppσ ? Vppπ )V1 2V12 + V22

? ′ ′ d2 r ?Ψ? AK ′ ↓ (r)ΨBK ↑ (r) + ΨBK ′ ↑ ΨAK ↓ .

(14)

E?ect of an electric ?eld.

Now we discuss the atomic Stark e?ect due to a perpendicular electric ?eld E . In this case, we need to consider the |s orbital of the σ bands at each site, and the associated hopping terms. The hamiltonian for this case includes the couplings: HE =

? λeE c? is;s′ ciz ;s′ + ?s cis;s′ cis;s′ + h.c. + Vspσ ? a x c? 1x;s′ c0s;s′ + ay c1y ;s′ c0s;s′ + h.c. s′ =↑,↓

(15)

i=1,2;s′ =↑,↓

where λ = pz |z ?|s is a electric dipole transition which induces hybridization between the s and pz orbitals and where ax and ay are the x and y components of the vector connecting the carbon atoms 0 and 1. First, we consider

7 the situation ax = 1 and ay = 0. Again Vspσ is the hopping integral between the 2s and 2px , 2py atomic orbitals corresponding to the σ band. We can now have processes such as: |pz 0 ↑ ? → |s0 ↑

E

|pz 0 ↑ ? → |px 0 ↓ ? ? ? → |s1 ↓ ? → |pz 1 ↓

?

? ? ? → |px 1 ↑ ? → |pz 1 ↓

Vspσ E

Vspσ

?

(16)

The intermediate orbitals |s0 and |px 1 are part of the sigma bands. As before, we describe them using the analytical ?tting discussed in Appendix A. The |s0 is part of the dispersive bands, and it has zero overlap with the two non dispersive σ bands. The processes induced by the electric ?eld, in momentum space, lead ?nally to: HE K ≡ λeE 1 3 α + cos 2 d2 r sin α 2

? Ψ? r)ψσ1BK ↑ (r) + AK ↑ (r)ψσ1AK ↑ (r) + ΨBK ↑ (?

? Ψ? r)ψσ2BK ↑ (r) AK ↑ (r)ψσ2AK ↑ (r) + ΨBK ↑ (?

+ h.c.

(17)

Note that this hamiltonian mixes the states in the π band with states in the σ bands which are orthogonal to those in Eq.(10). Combining Eq.(17) and Eq.(10) we obtain, again by second order perturbation theory, an e?ective hamiltonian for the π band: √ 2 2 ?λeE V2 ? HE Kπ ≡ ?i d2 r Ψ? AK ↑ (r)ΨBK ↓ (r) ? ΨBK ↓ ΨAK ↑ . 3 2V12 + V22 √ 2 2 ?λeE V2 ≡ ?i 3 2V12 + V22

HE K ′ π

? d2 (r) ?Ψ? AK ′ ↓ (r)ΨBK ′ ↑ (r) + ΨBK ′ ↑ ΨAK ′ ↓ .

(18)

The zero overlap between the states in the σ bands in Eq.(17) and Eq.(10) imply that only transitions between di?erent sublattices are allowed. De?ning a 4 × 4 spinor ? ? ΨA↑ (r) ? ΨA↓ (r) ? ? Ψ K (K ′ ) = ? , (19) ? ΨB ↑ (r) ? ΨB ↓ (r) K (K ′ ) it is possible to join Eqs.(14) and (18) in the following compact way: HRKπ = ?i?R d2 rΨ? σ+ s ?+ ? σ ?? s ?? ]ΨK = K [? ?R 2 ?R 2 d2 rΨ? σx s ?y + σ ?y s ?x ]ΨK K [? (20)

HRK′ π = ?i?R where ?R = ?E + ?curv ?E ?V2 = 2V12 + V22

d2 rΨ? ?+ s ?? + σ ?? s ?+ ]ΨK ′ = K ′ [?σ

d2 rΨ? σx s ?y ? σ ?y s ?x ]ΨK ′ K ′ [?

(21)

√ √ 2 2 2 2 ?λeE λeE ? 3 3 V2 a a + R1 R2 ? ?(Vppσ ? Vppπ ) V1 a a + R1 R2 V1 V2

2

?curv =

?V1 (Vppσ ? Vppπ ) 2V12 + V22

.

(22)

where the limit V1 ? V2 (widely separated σ bands) has been considered to approximate the above expressions. Eqs.(20,21,22) constitute one of the most important results of the paper. First, we recover the e?ective form for the “Rashba-type” interaction expected from group-theoretical arguments recently[24]. Even more importantly, our result shows that this e?ective spin-orbit coupling for the π bands in graphene to ?rst order in the intraatomic spin-orbit interaction ? is given by two terms:

8

111 000 000 111 000 111 000 111 000 111

SO

SO π σ

111 000 000 111 000 111 000 111 000 111

111 000 000 111 000 111 000 111 000 111

π σ

FIG. 3: “(Color online)”: Sketch of the processes leading to an e?ective intrinsic term in the π band of graphene. Transitions drawn in red (dark grey), and indicated by SO, are mediated by the intraatomic spin-orbit coupling.

? ?E : Corresponds to processes due to the intraatomic spin-orbit coupling and the intraatomic Stark e?ect between di?erent orbitals of the π and σ bands, together with hopping between neighboring atoms. The mixing between the π and σ orbitals occurs between the pz and s atomic orbitals due to the Stark e?ect λ and between the pz and px,y due to the atomic spin-orbit coupling ?. This contribution is the equivalent, for graphene, to the known Rashba spin-orbit interaction[31] and it vanishes at E =0. ? ?curv : Corresponds to processes due to the intraatomic spin-orbit coupling and the local curvature of the graphene surface which couples the π and σ bands, together with hopping between neighboring atoms. The mixing between the π and σ orbitals in this case occurs between pz and px,y atomic orbitals both due to the atomic spin-orbit coupling ? and due to the curvature. This process is very sensitive to deformations of the lattice along the bond direction between the di?erent atoms where the p part of the the sp2 orbitals is important.

Intrinsic spin orbit coupling.

We can extend the previous analysis to second order in the intraatomic spin-orbit interaction ?. We obtain e?ective couplings between electrons with parallel spin. The coupling between ?rst nearest neighbors can be written as: |pz 0 ↑ ? →

? ? ?

|pz 0 ↑ ? →

|pz 0 ↑ ? → ?1 2 |px 0 ↓ + ?1 2 |px 0

|px 0 ↓ ↓ ?

√

3 2 |py 0 √ 3 2 |py 0

σ ? → ↓ ?

σ ? ? →

V

V

? → ↓ ?

Vσ

1 2 |px 2 1 2 |px 3

↓ ? ↓ +

|px 1 ↓

√ 3 2 |py 2 √ 3 2 |py 3

↓ ? → |pz 2 ↑ ↓ ? → |pz 3 ↑

?

? → |pz 1 ↑

?

?

(23)

where the label 0 stands for the central atom. These three couplings are equal, and give a vanishing contribution at the K and K ′ points. The intrinsic spin-orbit coupling vanishes for hopping between neighbouring atoms, in agreement with group theoretical arguments[24, 29, 30]. We must therefore go to the next order in the hopping. Expanding to next nearest neighbors, we ?nd a ?nite contribution to the intrinsic spin-orbit coupling in a ?at graphene sheet, corresponding to processes shown in Fig.[3]. In this case, both the dispersive and non dispersive bands contribute to the e?ective π ? π coupling, as schematically shown by the di?erent arrows in Fig.[1]. In order to estimate quantatively the magnitude of the intrinsic coupling, we consider processes represented in Fig.[3], which are second order in ?, in momentum space, ?nally obtaining: HintK (K ′ ) = ± × 3 ?2 V14 2 2 4 V1 (V2 ? V1 )(2V12 + V22 )

? ′ ′ ′ ′ d2 rΨ? r)ΨAK (K ′ )↓ (r) ? Ψ? AK (K ′ )↑ (r)ΨAK (K )↑ (r) ? ΨAK (K ′ )↓ (? BK (K ′ )↑ (r)ΨBK (K )↑ (r) + ΨBK (K )↓ (r)ΨBK (K )↓ (r)

(24) where the ± sign corresponds to K (K ′ ) respectively. We de?ne the intrinsic spin-orbit coupling parameter ?int in the limit V1 ? V2 (widely separated σ bands) as: ?int = V14 3 ?2 3 ?2 ? 2 2 2 2 4 V1 (V2 ? V1 )(2V1 + V2 ) 4 V1 V1 V2

4

(25)

9

Intrinsic coupling: ?int Rashba coupling (electric ?eld E ≈ 50V /300nm): ?E Curvature coupling: ?curv

4 V1 3 ?2 4 V1 V2 √ 2 2 ?λeE 3 V2 ?(Vppσ ?Vppπ ) a a +R V1 R1 2

0.01K 0.07K

V1 V2 2

0.2K

TABLE I: Dependence on band structure parameters, curvature, and electric ?eld of the spin orbit couplings discussed in the text in the limit V1 ? V2 (widely separated σ bands). The parameters used are λ ≈ 0.264? A[36], E ≈ 50V/300nm[4, 24], ? = 12meV [38, 39],Vspσ ? 4.2eV, Vssσ ? ?3.63eV, Vppσ ? 5.38eV and Vppπ ? ?2.24eV [40, 41], V1 = 2.47eV, V2 = 6.33eV, a = 1.42? A, l ? 100 ? A, h ? 10 ? A and R ? 50 ? 100nm.

Our Hamiltonian, Eq.(24), is equivalent to the one derived in [24] d2 r?int Ψ? [? τz σ ?z s ?z ]Ψ

T

HTSO

intrinsic

=

(26)

where τ ?z = ±1 denotes the K(K ′ ) Dirac point and Ψ = (ΨK , ΨK ′ ) .

NUMERICAL ESTIMATES.

We must now estimate ?int , ?E and ?curv . We have λ = 3ao /Z ≈ 0.264? A[36], where Z = 6 for carbon and ao is the Bohr radius, E ≈ 50V/300nm[4, 24], the atomic spin-orbit splitting for carbon ? = 12meV → 1.3 × 102 K[38, 39], the energy di?erence between the π -2pz orbitals and the σ -sp2 orbitals ?π ? ?σ ? (14.26 ? 11.79)eV= 2.47eV, the energy di?erence between the 2p and the 2s atomic orbitals ?s ? ?p ? (19.20 ? 11.79)eV= 7.41eV and the hoppings between the 2s, 2px , 2py , 2pz orbitals of neighbouring atoms as Vspσ ? 4.2eV, Vssσ ? ?3.63eV and Vppσ ? 5.38eV and Vppπ ? ?2.24eV [40, 41]. We have V1 = 2.47eV and V2 = 6.33eV. We estimate ?int ? (3?2 /4V1 )(V1 /V2 )4 ? 0.1 × 10?5 eV → 0.01K. ?int is two orders of magnitude smaller than the estimate in[24]. The discrepancy seems to arise ?rst, because the intrinsic spin-orbit splitting ?int estimated here is proportional to the square of the intraatomic spin-orbit coupling ?2 , instead of being proportional to it, as roughly estimated in[24]. Besides, a detailed description of the σ bands is necessary to obtain the correct estimate. Spin-orbit splittings of order 1-2.5 K have been discussed in the literature for graphite[42]. However in graphite, the coupling between layers is important and in?uences the e?ective value of the spin-orbit splitting, typically being enhanced with respect to the single layer value[30, 43, 44]. For the other two couplings, we use the full expression obtained for√ ?E and ?curv and not the limiting form V1 ? V2 , in order to be as accurate as possible. First, we obtain ?E = 2 2/3 λeE ?V2 / 2V12 + V22 ? 0.6 × 10?5 eV → 0.07K. This estimate for ?E , depends on the external electric ?eld chosen. Our estimate, for the same value of the electric ?eld, is two orders of magnitude bigger than the estimate in[24]. So far curvature e?ects have been excluded. Curvature e?ects will increase the total value for the e?ective spin-orbit interaction in graphene. Graphene samples seem to have an undulating surface [14]. The ripples observed seem to be several ? A height and a few tens nm laterally[14]. First we consider the simplest example of a ripple being a half-sphere of radius R. The part of the sphere which intersects the plane of ?at graphene and therefore constitutes the ripple, is assumed to have a typical height h ? R so the radius R is roughly of the same order of magnitude of the lateral size in this case. It seems possible to identify ripples of lateral size ranging 50nm -100nm in [14]. Choosing a = 1.42? A and R1 ? R2 ? 50nm[14], we obtain ?curv = (2a/R) (Vppσ ? Vppπ ) ?V1 / 2V12 + V22 ? 2.45 × 10?5 eV → 0.28 K. Choosing R1 ? R2 ? 100nm we obtain ?curv ? 1.22 × 10?5 eV → 0.14 K. Now we consider a di?erent model where we assume that the sample has random corrugations of height h and length l[13]. The graphene surface presents then an undulating pattern of ripples of average radius R ? l2 /h[13]. Choosing l ? 100? A and h ? 10 ? A[13], we obtain again R ? 100nm, which leads to the same value for ?curv ? 1.22 × 10?5eV → 0.14 K. In any case, it seems clear that due to curvature e?ects, the e?ective spin-orbit coupling in graphene could be higher for curved graphene than for perfectly ?at graphene. Moreover, spin-orbit coupling in (curved) graphene would be present even for E =0. A more detailed discussion/study of the local curvature/corrugation of graphene is needed in order to obtain more accurate estimates. To conclude this section we present the e?ective hamiltonian for the π -bands of graphene including the spin-orbit interaction: ?R ?y ] + ?int [? ?x + τ τz σ ?z s ?z ] + ?z σ ?y ? [? σx s ?y + τ ?z σ ?y s ?x ] Ψ σx ? d2 rΨ? ?i vF [? 2

HT =

(27)

10 √ A being the lattice constant for graphene, γo ? 3eV the McClure intralayer coupling where vF = 3γo a/2, a ? 2.46 ? constant[30, 45], ?int ? 0.01K, ?R = ?E + ?curv the Rashba-Curvature coupling (RCC), where ?E ? 0.07K for E ≈ 50V/300nm and ?curv ? 0.2K. Table[I] summarizes the main results obtained in the paper for the e?ective spin-orbit couplings in a graphene layer.

APPLICATION TO FULLERENES, NANOTUBES AND FULLERENE CAPS. Spherical fullerenes.

When topological deformations in the form of pentagons are introduced in the hexagonal lattice of graphene, curved structures form. If 12 pentagons are introduced, the graphene sheet will close itself into a sphere forming the well know fullerene structure [46]. The usual hexagonal lattice “lives” now on a sphere and presents topological defects in the form of pentagons. The continuum approximation to a spherical fullerene leads to two decoupled Dirac equations in the presence of a ?ctitious monopole of charge ±3/2 in the center of the sphere which accounts for the presence of the 12 pentagons[47, 48]. The states closest to the Fermi level are four triplets at ? = 0. We consider the e?ect of the spin-orbit coupling on these triplets ?rst. Both the coupling induced by the Rashba-Curvature, Eq.(20, 21), and the intrinsic coupling, Eq.(24), are written in a local basis of wavefunctions where the spin is oriented perpendicular to the graphene sheet, |θ, φ, ⊥↑ , |θ, φ, ⊥↓ . It is useful to relate this local basis with a ?xed basis independent of the curvature. Such relation depends on the curvature of the graphene sheet considered. In the case of a fullerene the graphene sheet is in a sphere. The details are given in Appendix B. The gauge ?eld associated to the presence of ?vefold rings in the fullerene can be diagonalized using the basis: ? ? (r) = Ψ ? (r) + iΨ ′ ? (r) Ψ AKks AK ks BK ks ? ′ ? (r) = ?i Ψ ′ ? (r) + Ψ ? (? Ψ B K ks BK ks AK ks r). Equivalent transformation is obtained exchanging A ? B . In this basis, the wavefunctions of the zero energy states are[47]: |+ 1sK ≡ |0 s K ≡ |? 1sK ≡ | + 1 s K′ |0 s K′ | ? 1 s K′ ≡ ≡ ? ≡ 3 cos2 4π 3 sin 2π 3 sin2 4π 3 sin2 4π 3 sin 2π 3 cos2 4π θ 2 θ 2 θ 2 θ 2 θ 2 θ 2 eiφ cos e?iφ eiφ cos e?iφ θ 2 |AK i|BK ′ ? |s ? |s ? |s ? |s ? |s (29)

(28)

|AK i|BK ′ |AK i|BK ′ |AK ?i|BK ′ θ 2

|AK ?i|BK ′ |AK ?i|BK ′ ? |s

where |AK and |BK ′ are envelope functions associated to the K and K ′ points of the Brillouin zone and corresponding to states located at the A and B sublattices respectively. Note that, at zero energy, states at K (K ′ ) are only located at sublattice A(B ) sites. |s denotes the usual spinor part of the wave function corresponding to the electronic spin s =↑, ↓. The hamiltonian Hint couples orbitals in the same sublattice whereas HR couples orbitals in di?erent sublattices. So HR has zero matrix elements between zero energy states, as it does not induce intervalley scattering, mixing K and K ′ states [13].

11 In the {| + 1 ↑ , | + 1 ↓ , |0 ↑ , |0 ↓ , | ? 1 ↑ , | ? 1 ↓ } basis, the Hamiltonian for the K point of a fullerene looks like: ? ? ? =? ? ? ? ? ?int 0 0 0 0 √ 0 2?int 0 0 0 0 √ ??int 0 2?int 0 0 0 0 √ 0 0 0 2?int 0 √ 0 0 0 0 2?int ??int 0 0 0 0 0 0 ?int ? ? ? ? ? ? ? ? (30)

K HS ?O int

K K The Hamiltonian for K′ is HS ?O int = ?HS?O int . Diagonalizing the Hamiltonian Eq. (30), we obtain that each set of spin degenerate triplets obtained in the absence of the spin-orbit interaction split into:

′

? = +?int → Ψ? : {| + 1 ↑ , | ? 1 ↓ , ? = ?2?int → Ψ?2? : {

1 2 1 |+1 ↓ + |0 ↑ , |?1↑ + 3 3 3 2 1 2 1 |+1 ↓ ? |0 ↑ , |?1↑ ? |0 ↓ } 3 3 3 3

2 |0 ↓ } 3 (31)

Each of these solutions is doubly degenerate, corresponding to the K and K′ points. In principle, many body e?ects associated to the electrostatic interaction can be included by following the calculation discussed in[49].

Spin-orbit coupling in nanotubes.

The previous continuum analysis can be extended to nanotubes. We use cylindrical coordinates, z, φ, and, as before, de?ne the spin orientations | ↑ , | ↓ as parallel and antiparallel to the z axis. The matrix elements relevant for this geometry can be easily obtained from Eq.(55) in Appendix B, by choosing θ = π/2. The eigenstates of the nanotube can be classi?ed by longitudinal momentum, k , and by their angular momentum n, ?±,k,n = ± vF k 2 + n2 /R2 , where R is the radius of the nanotube. After integrationg over the circumference of the nanotube dφ, the Hamiltonian of a nanotube including spin-orbit interaction is: HS?O R |Aτ |Bτ = 0 vF (k + in/R) ? τ i?R π s ?z vF (k ? in/R) + τ i?R π s ?z 0 |Aτ |Bτ (32)

where the τ = ±1 corresponds to the K (K ′ ) Dirac point. Note the basis states |Aτ and |Bτ used to de?ne Eq.(32) are spinors in spin subspace where the matrix s ?z acts on (see Appendix C for details). The contribution from the intrinsic spin-orbit ?int becomes zero after integrating over the nanotube circumference (Appendix C). The spin-orbit term i?R π s ?z in Eq.(32) is equivalent to the term proportional to σ ?y obtained in Eq. (3.15, 3.16) of Ref. [37]. It is important to note that the spin orientations | ↑ , | ↓ in Eq.(32) are de?ned along the nanotube axis, whereas the spin orientations used in Eq. (3.15, 3.16) of Ref. [37] are de?ned perpendicular to the nanotube surface. On the other hand, we do not ?nd any contribution similar to the term proportional to σx (r) in Eq.(3.15, 3.16) in Ref. [37]. In any case, such contributions are not important as they vanish after integrating over the circumference of the nanotube[37]. Besides, our results are in agreement with the results obtained in [50]. The energies near the Fermi level, n = 0, are changed by the spin-orbit coupling, and we obtain: ?k = ± (π ?R ) + ( vF k )2 .

2

(33)

There is an energy gap π ?R at low energies, in agreement with the results in[37, 51]. The π ?R gap originates as a consequence of the Berry phase gained by the electron and hole quasiparticles after completing a closed trajectory around the circumference of the nanotube under the e?ect of spin-orbit interaction ?R [37]. Similarly, ?R will give rise to a small spin splitting for n = 0 [37, 51] ?k = ± (π ?R ) + ( vF )2 (k 2 + (n/R)2 ) + 2(n/R) vF ?R π s ?z .

2

(34)

For a single wall nanotube of radius R1 ? 6, 12, 24? A and R2 → ∞, a ? 1.42? A and for E =0, we get ?R ? 12, 6, 3K respectively.

12

a)

b)

FIG. 4: “(Color online)”: Left: One ?fth of a fullerene cap closing an armchair nanotube. The full cap is obtained by gluing ?ve triangles like the one in the ?gure together, forming a pyramid. A pentagon is formed at the apex of the pyramid, from the ?ve triangles like the one shaded in green (grey) in the ?gure. The edges of the cap are given by the thick black line. The cap contains six pentagons and seventy hexagons, and it closes a 25 × 25 armchair nanotube. Right: Sketch of the folding procedure of a ?at honeycomb lattice needed to obtain an armchair nanotube capped by a semispherical fullerene.

Nanotube caps. Localized states at zero energy.

An armchair (5N × 5N ) nanotube can be ended by a spherical fullerene cap. The cap contains six pentagons and 5N (N + 1)/2 ? 5 hexagons. When N = 3 × M , the nanotube is metallic. An example of such a fullerene cap is given in Fig.[4]. The boundary between the semispherical fullerene and the nanotube is a circle (Fig.[5]). The solutions of the continuum equations have to be continuous accross this boundary, and they have to satisfy the Dirac equations appropriate for the sphere in the cap and for the torus in the nanotube respectively. The boundary of the nanotube in the geometry shown in Fig.[5] is a zigzag edge. Hence, zero energy states can be de?ned[52, 53], which at this boundary will have a ?nite amplitude on one sublattice and zero on the other. There is a zero energy state, |Ψn at this boundary, for each value of the angular momentum around the nanotube n. They decay towards the bulk of the nanotube as Ψn (z, φ, K ) = Ceinφ e?(nz)/R Ψn (z, φ, K ′ ) = Ceinφ e?(nz)/R n>0 n<0

(35)

where we are assuming that the nanotube is in the half space z > 0 (see Fig.[5]). A zero energy state in the whole system can be de?ned if there are states inside the gap of the nanotube π ?R , which √ can be matched to the states de?ned in Eq.(35). At the boundary we have θ = π/2, cos(θ/2) = sin(θ/2) = 1/ 2. Hence, we can combine states |l s K and |l s K′ , l = ±1 in Eq.(29) in such a way that the amplitude at the boundary on a given sublattice vanishes: 1 ≡ √ (| + 1 s K + | + 1 s K′ ) = 2 3 iφ e 8π 3 ?iφ e 8π 3 iφ e 8π 3 ?iφ e 8π |AK 0 |AK 0 0 i|BK ′ 0 i|BK ′

| +1s | ?1s | +1s |?1s

B A

A

? |s ? |s ? |s ? |s (36)

1 ≡ √ (| ? 1 s K + | ? 1 s K′ ) = 2 1 ≡ √ (| + 1 s K ? | + 1 s K′ ) = 2

B

1 ≡ √ (| ? 1 s K ? | ? 1 s K′ ) = 2

These combinations match the states decaying into the nanotube, Eq.(35). This ?xes the constant C in Eq.(35) to be C = 3/(8π ). Note that the wavefunctions with l = 0 can only be matched to states that not decay into the bulk of the nanotube, i.e. with n = 0.

13

Ψ( z ,θ,φ)

z

z

FIG. 5: “(Color online)”: (Up) Sketch of the matching scheme used to build a zero energy state at a fullerene cap. (Down)The wavefunction is one half of a zero energy state at the cap, matched to a decaying state towards the bulk of the nanotube. See text for details.

Thus, there are two states per spin s, | + 1 s , | ? 1 s , localized at the cap and with ?nite chirality, n = ±1. A sketch of this procedure is shown in Fig.[5]. This continuum approximation is in general agreement with the results in[54]. As in the case of a spherical fullerene, only the intrinsic spin-orbit coupling mixes these states. The energies of the | ± 1 s A states are not a?ected by ?int , as the contribution from | + 1 s , is cancelled by the contribution from | ? 1 s . On the other hand, the states | ± 1 s B split in energy, as the | + 1 s and | ? 1 s contributions add-up: | ? 1, s =↑, ↓ Note that each state has a ?nite chirality.

Localized states induced by the spin-orbit interaction.

| + 1, s =↑, ↓

B B

→ ?↑,↓ = ??int .

→ ?↑,↓ = ±?int

(37)

The remaining states of a spherical fullerenes have multiplicity 2l + 1, where l ≥ 2 and energy ?l = ± vF /R l(l + 1) ? 2. The angular momentum of these states along a given axis, m, is ?l ≤ m ≤ l. The subbands of the nanotube with angular momentum ±m, have gaps within the energy interval ??m = ? vF |m|/R ≤ ? ≤ ?m = vF |m|/R. Thus, there is a fullerene eigenstate with l = 2 and angular momentum m = ±2 which lies at the gap edge of the nanotube subbands with the same momentum. The fullerene state is: 1 Ψl=2 m=2 (θ, φ) ≡ √ e2iφ 4 2π sin(θ) [1 + cos(θ)] |K ? [1 + cos(θ)]2 |K ′ i[1 + cos(θ)]2 |K + i sin(θ) [1 + cos(θ)] |K ′ |K ? |K ′ i|K + i|K ′ (38)

which can be matched, at θ = π/2, to the nanotube eigenstate: 1 Ψm=2 (z, φ) ≡ √ e2iφ 4 2π (39)

The spin orbit coupling acts as a position dependent potential on this state, and it shifts its energy into the m = 2 subgap, leading to the formation of another localized state near the cap. In the following, we consider only the Rashba-Curvature coupling ?R . In order to analyze the extension of the state, we assume that the localized state decays in the nanotube, z > 0 as: C Ψm=2 (z, φ) ≡ √ e2iφ eκz/R 4 2π |K ? |K ′ i|K + i|K ′ (40)

14 We match this wavefunction to that in Eq.(39) multiplied by the same normalization constant, C . We assume that the state is weakly localized below the band edge, so that the main part of the wavefunction is in the nanotube, and κ ? 1. Then, we can neglect the change in the spinorial part of the wavefunction, and we will ?x the relative components of the two spinors as in Eq.(39) and Eq.(40). The normalization of the total wavefunction implies that: C ?2 = 1 13 + 16 4κ (41)

where the ?rst term in the r.h.s. comes from the part of the wavefunction inside the cap, Eq.(39), and the second term is due to the weight of the wavefunction inside the nanotube, Eq.(40). As expected, when the state becomes delocalized, κ → 0, the main contribution to the normalization arises from the “bulk” part of the wavefunction. The value of κ is ?xed by the energy of the state: κ2 = n2 ? ?2 R 2 ( vF )2 (42)

where the energy of the state ? is inside the subgap ?m of the nanotube because of the shift induced by the spin orbit interaction ?R . We now calculate the contribution to the energy of this state from the Rashba-Curvature coupling, which is now ?nite, as this state has weight on the two sublattices: ?Rashba ≈ ±C 2 ?R 1 31 + 16κ 80 ≈± ?R 4 1? 59κ 20 (43)

The r.h.s. in Eq.(43) can be described as the sum of a bulk term, ±?R /4, and a term due to the presence of the cap, whose weight vanishes as the state becomes delocalized, κ → 0. The absolute value of the Rashba-Curvature contribution is reduced with respect to the bulk energy shift, which implies that the interaction is weaker at the cap. The bulk nanotube bands are split into two spin subbands which are shifted in opposite directions. The surface states analyzed here are shifted by a smaller amount, so that the state associated to the subband whose gap increases does not overlap with the nanotube continuum. Combining the estimate of the energy between the gap edge and the surface state in Eq.(43) and the constraint for κ in Eq.(42), we ?nd: κ≈ 59?R R 20 vF (44)

Finally, the separation between the energy of the state and the subgap edge is: ?≈ vF κ2 8R (45)

? we obtain, for E =0, a value ?R /4 ? 3K. This e?ect of the spin orbit For a C60 fullerene of radius R ? 3.55A interaction can be greatly enhanced in nanotube caps in an external electric ?eld, such as those used for ?eld emission devices[55]. In this case, spin-orbit interaction in may allow for spin dependent ?eld emission of such devices. The applied ?eld also modi?es the one electron states, and a detailed analysis of this situation lies outside the scope of this paper.

CONCLUSIONS.

We have analyzed the spin orbit interaction in graphene and similar materials, like nanotubes and fullerenes. We have extended previous approaches in order to describe the e?ect of the intraatomic spin orbit interaction on the conduction π and valence σ bands. Our scheme allows us to analyze, on the same footing, the e?ects of curvature and perpendicular applied electric ?eld. Moreover, we are able to obtain realistic estimates for the intrinsic ?int and Rashba-Curvature ?R = ?E + ?curv e?ective spin-orbit couplings in graphene. We have shown that spin-orbit coupling for ?at graphene is rather weak ?int ? 10mK and ?E ? 70mK for E =50V/300nm. Moreover curvature at the scale of the distance between neighbouring atoms increases the value of the spin orbit coupling in graphene ?curv ?E ? ?int . This is because local curvature mixes the π and σ bands. Graphene samples seem to have an undulating surface [14]. Our estimate for the typical observed ripples indicates that ?curv could be of order ? 0.2K . A more detailed study of the curvature of graphene samples is needed in order to obtain a more precise estimate.

15 We conclude that the spin-orbit coupling ?R expected from symmetry arguments[24], has an curvature-intrinsic part besides the expected Rashba coupling due to an electric ?eld ?R = ?E +?curv . Therefore ?R can be higher for curved graphene than for ?at graphene. To our knowledge, this is the ?rst time that this type of “new” spin-orbit coupling has been noticed. ?curv is in a sense a new “intrinsic/topological” type of spin-orbit interaction in graphene which would be present even if E = 0, as long as the samples present some type of corrugation. One important question now is how these ripples could a?ect macroscopic quantities. It has been already suggested that these ripples may be responsible for the lack of weak (anti)localization in graphene [13, 14]. Other interesting macroscopic quantities involving not only the pseudo spin but also the electronic spin maybe worth investigation. These issues are beyond the scope of the present paper and will be subject of future work. It is also noteworthy that our estimates ?R ? ?int , is opposite to the condition ?R ? ?int obtained by Kane and Mele[24] to achieve the quantum spin Hall e?ect in graphene. So the quantum spin Hall e?ect may be achieved in neutral graphene (E = 0) only below ? 0.01K and provided the sample is also free of ripples so the curvature spin-orbit coupling ?curv ? ?int . Further progress in sample preparation seems needed to achieve such conditions although some preliminary improvements have been recently reported[14]. Moreover, corrugations in graphene could be seen as topological disorder. It has been shown that the spin Hall e?ect survives even if the spin-orbit gap ?int is closed by disorder[26]. A detailed discussion of the e?ect of disorder on the other two spin-orbit couplings ?E , ?curv will be presented elsewhere. The continuum model derived from microscopic parameters has been applied to situations where also long range curvature e?ects can be signi?cant. We have made estimates of the e?ects of the various spin orbit terms on the low energy states of fullerenes, nanotubes, and nanotube caps. For both nanotubes and nanotube caps we ?nd that ?R ? 1K. For nanotubes we have clari?ed the existent discussion and reproduced the known appearance of a gap for the n = 0 states and spin-splitting for n = 0 states in the energy spectrum. For nanotube caps states, we obtain indications that spin-orbit coupling may lead to spin-dependent emission possibilities for ?eld-e?ect emission devices. This aspect will be investigated in the future. Note added. At the ?nal stages of the writting of the present paper, two preprints [32, 33] have appeared. In these papers similar estimates for ?int ? 10?3 meV have been obtained for the intrinsic spin-orbit coupling. Moreover similar discussion for the e?ect of an perpendicular electric ?eld has been also discussed in[33]. Our approach is similar to that followed in[33]. The two studies overlap and although the model used for the σ band di?ers somewhat, the results are quantitative in agreement ?E ? 10?2 meV The two preprints[32, 33] and our work agree in the estimation of the intrinsic coupling, which turns out to be weak at the range of temperatures of experimental interest (note, however, that we do not consider here possible renormalization e?ects of this contribution[24, 56]). On the other hand, the e?ect of local curvature ?curv on the spin-orbit coupling has not been investigated in [32, 33]. We show here that this term ?curv is as important as, or perhaps even more important than the spin-orbit coupling due to an electric ?eld ?E for the typical values of E reported.

ACKNOWLEDGEMENTS.

We thank Yu. V. Nazarov, K. Novoselov, L. Brey, G. G? omez Santos, Alberto Cortijo and Maria A. H. Vozmediano for valuable discussions. D.H-H and A.B. acknowledge funding from the Research Council of Norway, through grants no 162742/v00, 1585181/431 and 1158547/431. F. G. acknowledges funding from MEC (Spain) through grant FIS200505478-C02-01 and the European Union Contract 12881 (NEST).

16

APPENDIX A. TWO PARAMETER ANALYTICAL FIT TO THE SIGMA BANDS OF GRAPHENE.

A simple approximation to the sigma bands of graphene takes only into account the positions of the 2s and 2p atomic levels, ?s and ?p , and the interaction between nearest neighbor sp2 orbitals. The three sp2 orbitals are: √ 1 |1 ≡ √ |s + 2|px 3 √ √ 1 1 3 |2 ≡ √ |s + 2 ? |px + |py 2 2 3 √ √ 3 1 1 |3 ≡ √ |s + 2 ? |px ? |py 2 2 3 The two hopping elements considered are: V1 = i|Hatom |j |i=j = ?s ? ?p 3

(46)

V2 = i, m|Hhopping |i, n |m,n:nearest?neighbors

√ Vssσ + 2 2Vspσ + 2Vppσ = 3

(47)

where i, j = 1, 2, 3 denote “bonding” sp2 states and n, m denote atomic sites. V1 depends on the geometry/angle between the bonds at each atom and V2 depends on the coordination of nearest neighbors in the lattice. V1 and V2 therefore determine the details of band structure for the σ bands[34]. The energy associated with each “bonding” state i|Hatom |i |i=1,2,3 = (?s + 2?p ) /3 is an energy constant independent of these details and not important for our discussion here. We label a1 , a2 , a3 the amplitudes of a Bloch state on the three orbitals at a given atom, and ′ ′ ′′ ′′ ′′ { b1 , b2 , b3 } , { b′ 1 , b2 , b3 }, {b1 , b2 , b3 } the amplitudes at its three nearest neighbors. These amplitudes satisfy: ?a1 = V1 (a2 + a3 ) + V2 b1 ?a2 = V1 (a1 + a3 ) + V2 b′ 2 ?a3 = V1 (a1 + a2 ) + V2 b′′ 3 ?b1 = V1 (b2 + b3 ) + V2 a1 ′ ′ ?b′ 2 = V1 (b1 + b3 ) + V2 a2

′′ ′′ ?b′′ 3 = V1 (b1 + b2 ) + V2 a3

(48)

′′ We can de?ne two numbers, an = a1 + a2 + a3 and bn = b1 + b′ 2 + b3 associated to atom n. From Eq.(48) we obtain:

(? ? 2V1 )an = V2 bn

(? + V1 )bn = V2 an + V1

n′ ;n.?n.

an′ ,

(49)

′′ ′′ ′ ′ ′′ where n′ ;n.?n. an′ = (b1 + b2 + b3 ) + (b′ 1 + b2 + b3 ) + (b1 + b2 + b3 ) and n. ? n. denotes nearest-neighbors. This equation is equivalent to:

? ? 2V1 ?

V22 ? + V1

an =

V1 V2 ? + V1

an′

n′ ;n.?n.

(50)

Hence, the amplitudes an satisfy an equation formally identical to the tight binding equations for a single orbital model with nearest neighbor hoppings in the honeycomb lattice. In momentum space, we can write: ?k ? ? 2V1 ? where: fk ≡ 3 + 2 cos(ka1 ) + 2 cos(k? a2 ) + 2 cos[k(a1 ? a2 )] (52) V22 ?k ? + V1 =± V1 V2 fk ?k ? + V1 (51)

17 and a1 , a2 are the unit vectors of the honeycomb lattice. The derivation of equation(51) assumes that an = 0. There are also solutions to eqs.(48) for which an = 0 at all sites. These solutions, and Eq.(51) lead to: ?k ? = ?k ? V1 9V12 ± + V22 ± V1 V2 fk 2 4 = ?V1 ± V2

(53)

The knowledge of the eigenstates, Eq.(50) allows us to obtain also the eigenvalues of Eq.(54). The spin orbit coupling induces transitions from the |K, A, ↑ state to the sigma bands with energies V1 ± V2 and spin down, and from the |K, A, ↓ state to the sigma bands with energies V1 /2 ± (9V12 )/4 + V22 and spin up. The inverse processes are induced for Bloch states localized at sublattice B . In the limit V1 ? V2 , the σ bands lie at energies ±V2 , with corrections associated to V1 . The spin orbit coupling induces transitions to the upper and lower bands, which tend to cancel. In addition, the net e?ective intrinsic spin orbit coupling is the di?erence between the corrections to the up spin bands minus the those for the down spin bands. The ?nal e?ect is that the strength of the intrinsic spin orbit coupling scales as (?2 /V1 )(V1 /V2 )4 in the limit V1 ? V2 .

APPENDIX B. MATRIX ELEMENTS OF THE SPIN ORBIT INTERACTION IN A SPHERE.

These equations give the six σ bands used in the main text. In order to calculate the e?ects of transitions between the π band and the σ band on the spin orbit coupling, we also need the matrix elements of the spin orbit interaction at the points K and K ′ . At the K point, for instance, the hamiltonian for the σ band is: ? ? 0 V1 V1 V2 0 0 ? ? V1 0 V1 0 V2 e2πi/3 0 ? ? 4πi/3 ? ? V1 V1 0 0 0 V2 e ? (54) HσK ≡ ? ? ? V2 0 0 0 V1 V1 ? ? ? ? 0 V2 e4πi/3 0 V1 0 V1 2πi/3 0 0 V2 e V1 V1 0

Both the coupling induced by the curvature, eqs.(20,21), and the intrinsic coupling, Eq.(24), can be written, in a simple form, in a local basis of wavefunctions where the spin is oriented perpendicular to the graphene sheet, |θ, φ, ⊥↑ , |θ, φ, ⊥↓ . Using spherical coordinates, θ and φ, the basis where the spins are oriented parallel to the z axis, | ↑ , | ↓ , can be written as: | ↑ ≡ cos | ↓ ≡ sin θ θ eiφ/2 |θ, φ, ⊥↑ ? sin e+iφ/2 |θ, φ, ⊥↓ 2 2 θ θ e?iφ/2 |θ, φ, ⊥↑ + cos e?iφ/2 |θ, φ, ⊥↓ 2 2

(55)

and for K′

where the states |+ and |? are de?ned in terms of some ?xed frame of reference. From this expression, we ?nd in the basis {|A ↑ , |A ↓ , |B ↑ , |B ↓ , basis for K: ? ? θ θ θ cos 2 ?i?R cos2 2 e?iφ ?int cos(θ) ?int sin(θ)e?iφ +i?R sin 2 θ θ ? ? e+iφ ?i?R sin 2 cos θ ?int sin(θ)eiφ ??int cos(θ) +i?R sin2 2 K 2 ? ? (56) HS 2 θ θ θ ?O = ? ?iφ ?iφ ? ?i?R sin 2 cos 2 ?i?R sin 2 e ??int cos(θ) ??int sin(θ)e θ θ iφ +i?R sin 2 cos 2 ??int sin(θ)eiφ ?int cos(θ) +i?R cos2 θ 2 e ??int cos(θ) ? ??int sin(θ)eiφ =? ? +i?R sin θ cos θ 2 2 θ eiφ +i?R sin2 2 ? ? θ ?iφ ?i?R sin2 θ ??int sin(θ)e?iφ ?i?R sin θ 2 cos 2 2 e θ θ ? ?int cos(θ) +i?R cos2 2 e+iφ +i?R sin θ 2 cos 2 ? ?iφ ? ?i?R cos2 θ ?int cos(θ) ?int sin(θ)e?iφ 2 e θ θ iφ ?i?R sin 2 cos 2 ?int sin(θ)e ??int cos(θ)

K HS ?O

′

(57)

18

APPENDIX C. MATRIX ELEMENTS OF THE SPIN ORBIT INTERACTION IN A CYLINDER.

The previous continuum analysis can be extended to nanotubes. We use cylindrical coordinates, z, φ, and, as before, de?ne the spin orientations | ↑ , | ↓ as parallel and antiparallel to the z axis. The matrix elements can be obtained in a similar way to Eq.(57) by choosing θ = π/2. ? 0 ? ?int e?iφ +i?R /2 ?i?R /2e?iφ 0 +i?R /2e+iφ ?i?R /2 ? ? ?iφ ?i?R /2e 0 ??int e?iφ ? +i?R /2 ??int eiφ 0 ? ??int e?iφ ?i?R /2 ?i?R /2e?iφ 0 +i?R /2e+iφ +i?R /2 ? ? ?iφ ?i?R /2e 0 ?int e?iφ ? ?i?R /2 ?int eiφ 0 dφ the Hamiltonian above becomes:

and for K′

? ?int eiφ K ? HS ?O = ? ? i ? / 2 R +i?R /2eiφ ? 0

(58)

After integrating over the nanotube circunference ?

? ??int eiφ K′ ? HS ?O = ? + i ? / 2 R +i?R /2eiφ

(59)

K HS ?O

and for K′

K HS ?O

′

? 0 0 +i?R π 0 ? 0 0 0 ?i?R π ? ? =? ? ? ?i?R π 0 0 0 0 +i?R π 0 0 ? 0 0 ?i?R π 0 ? 0 0 0 +i?R π ? ? =? ? ? +i?R π 0 0 0 0 ?i?R π 0 0 ?

(60)

(61)

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