Tight-binding theory of the spin-orbit coupling in graphene Sergej Konschuh, Martin Gmitra, and Jaroslav Fabian

Tight-binding theory of the spin-orbit coupling in graphene

Sergej Konschuh, Martin Gmitra, and Jaroslav Fabian

Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany 共Received 19 July 2010; revised manuscript received 4 October 2010; published 10 December 2010兲 The spin-orbit coupling in graphene induces spectral gaps at the high-symmetry points. The relevant gap at the⌫point is similar to the splitting of theporbitals in the carbon atom, being roughly 8.5 meV. The splitting at the K point is orders of magnitude smaller. Earlier tight-binding theories indicated the value of thisintrinsic gap of 1 ␮eV, based on the␴-␲coupling. All-electron first-principles calculations give much higher values, between 25 and 50 ␮eV, due to the presence of the orbitals of thedsymmetry in the Bloch states at K. A realistic multiband tight-binding model is presented to explain the effects thedorbitals play in the spin-orbit coupling at K. The␲-␴coupling is found irrelevant to the value of the intrinsic spin-orbit-induced gap. On the other hand, theextrinsic spin-orbit coupling共of the Bychkov-Rashba type兲, appearing in the presence of a transverse electric field, is dominated by the␲-␴hybridization, in agreement with previous theories. Tight- binding parameters are obtained by fitting to first-principles calculations, which also provide qualitative sup- port for the model when considering the trends in the spin-orbit-induced gap in graphene under strain. Finally, an effectivesingle-orbital next-nearest-neighbor hopping model accounting for the spin-orbit effects is derived.

DOI:10.1103/PhysRevB.82.245412 PACS number共s兲: 31.15.ae, 71.15.Mb, 31.15.aj, 71.90.⫹q

I. INTRODUCTION

Graphene is a two-dimensional allotrope of carbon¹ that has attracted enormous interest due to both its truly two- dimensional nature as well as due to its unique electronic properties originating in the linear energy dispersion at the Fermi level. The spectrum at the K points is akin to the Dirac cones for massless relativistic particles, causing excitement about the opportunities to test relativistic quantum mechanics in a solid-state material. This ideal picture changes qualita- tively when spin-orbit coupling共SOC兲is taken into account.

Namely, the coupling introduces a gap into the spectrum, giving a mass to the particles. The spectrum is no longer linear. The emergence of the gap moves graphene from the family of semimetals to the one of quantum Hall insulators.² It is ironic that the relativistic共spin-orbit兲effects destroy the relativistic appearance of the graphene spectrum. While for many purposes, the spectrum can be still approximated by straight lines, SOC is important when investigating such ef- fects as spin transport,^3–6 spin relaxation,^7–14 adatoms on graphene,¹⁵ or magnetoanisotropies of the predicted mag- netic edges.¹⁶

The question of how large is the spin-orbit-induced gap, the so-called intrinsic gap, at the K point has been given conflicting answers. The earliest estimates of 200 ␮^eV共Ref.

2兲were replaced by tight-binding共TB兲studies and support- ing first-principles results,^17,18predicting the gap as small as 1 ␮eV. These TB studies considered s and p orbitals. An all-electron first-principles calculation gave a much higher value, of 50 ␮^eV,19posing an interesting puzzle of what is the actual physics behind the gap. It was recently proved, again from first-principles calculations, that the gap origi- nates from the SOC of共nominally unoccupied兲dand higher orbitals.²⁰The predicted gap in that calculation is 24 ␮^{eV. A} reasonable estimate, considering the idiosyncrasies of theab initiocodes, is the value of the gap in the range between 25 and 50 ␮^eV.

The fact that one has to considerdand higher orbitals to get a sizable spin-orbit gap in graphene turns out to has al-

ready been known to Slonczewski,^21,22who devised a group theoretical argument showing in effect that the SOC of thep orbitals contributes in the second order while that of the d orbitals in the first order. Nice symmetry arguments can be found in Ref.23. The main point is that without SOC thepz

orbitals, that form the relevant states at K, do not hybridize withp_xandp_y. Their hybridization is solely via the spin-orbit interaction. On the other hand, pzdo hybridize with dxzand d_yz orbitals, forming together the ␲ band, as also proven from first-principles work.²⁰Since thedxzanddyzare split by SOC, forming “rotating” orbitals d_xz⫾id_yz, the gap of the␲ band is linearly proportional to thed splitting.

Graphene has a center of inversion symmetry, making its states doubly 共spin兲degenerate at a given momentum, even in the presence of SOC. Graphene on a substrate, or under a gate bias, loses this property and the bands further split. This splitting is termed extrinsic, and is akin to the one encoun- tered in semiconductor physics under the name of Bychkov- Rashba splitting or structure inversion asymmetry induced splitting.^24,25 Only Kramers degeneracy is left, meaning that the energies of the states of opposite spins and momenta are equal. The origin of the extrinsic splitting is the Stark effect, allowing for the ␲^-␴ hybridization, combined with the p orbital SOC. The corresponding TB theory has already been developed;¹⁷ d orbitals give negligible contribution 共of the order of 1%兲, as calculated from first principles²⁰or from our TB theory presented here. The extrinsic gap is about 10 ␮eV for the electric field of 1 V/nm. This energy scales linearly with the field. A significant enhancement of the extrinsic spin splitting has been reported for graphene placed on a substrate.^26–29 Large values of the splitting 共anything more than 1 meV should be considered as giant here兲 are likely due to charge transfer between substrate and graphene. When an impurity or an adatom is placed on graphene, the sp³ hybridization may distort graphene locally and induce split- tings comparable to the values found in zinc-blende semiconductors.¹⁵

This paper explains the relatively large splitting in the intrinsic graphene from using TB physics. We include the

1098-0121/2010/82共24兲/245412共11兲 245412-1 ©2010 The American Physical Society

relevants,p, andd orbitals and obtain the orbital couplings necessary to account for the splitting by fitting the TB model to first-principles calculations. Our formula for the intrinsic splitting shows that while the contribution from the SOC of the p orbitals increases with increasing lattice constant 共de- creasing hopping energy兲, the contribution from thed orbit- als decreases. This predicted trend is nicely confirmed from first-principles; as the lattice constant increases, first the splitting decreases, demonstrating the dominance of the d orbitals. After hitting a minimum the splitting increases, be- ing dominated by the SOC of the p orbitals. We also give explicit formulas for the extrinsic splitting, showing that the contributions from the dorbitals are negligible.

For many purposes, such as for investigating spin- polarized transport, magnetoelectric effects, or disorder ef- fects, it is useful to have a simple single-orbital hopping scheme. The functional form of such a hopping Hamiltonian is given by the system symmetries for the specific band region.² We derive such an effective model here by folding down our multiorbital TB scheme to the ␲ level, revealing the most relevant hopping paths共that comprise virtual hop- pings to other orbitals兲 and justifying the hopping Hamil- tonian from the conventional TB perspective. The resulting spin-dependent next-nearest-neighbor hopping model repro- duces well the spin-resolved spectrum of graphene.

The paper is organized as follows. In Sec.II, we introduce the TB model including the relevant d orbitals, and discuss their contribution to the density of states at the Fermi level.

In Sec. III A, we turn on the spin-orbit interaction and present the numerical results as well as analytical formulas for the intrinsic SOC induced gap at K. In Sec. III B, we apply a transverse electric field to the graphene sheet and obtain estimates for the ␴^-␲ ^{and p}z-d contributions to the extrinsic gap. Finally, in Sec.IVwe derive an effective next- nearest-neighbor model for the␲band in graphene including SOC.

II. TIGHT-BINDING MODEL

The hexagonal crystal structure of graphene comprises two atoms in its unit cell. Each set of atoms forms a trian- gular sublattice conventionally denoted as A and B. The Bloch states for each sublattice can be written as a linear combination of localized Wannier functions. In a tight- binding spirit, the Wannier functions are approximated by atomic directed orbitals. These orbitals point along the axes of a chosen coordinate system, in which the three nearest neighbors to an atom at origin are located at

Rជ_i=

再 ^冑

^a3

^冉

⁰¹

^冊

^,2

^冑

^a3

^冉

^{−− 1}

^冑

³

^冊

^,2a

^冑

³

^冉

^{− 1}

^冑

³

^冊 冎

^{, 共1兲}

wherea= 2.46 Å is the lattice constant. The columns are the two-dimensional coordinatesxandy, defining the coordinate system.

Considering the nearest neighbors only, the site-dependent TB Hamiltonian reads

H_␮,␯^AA共kជ兲=H_␮,␯^BB共kជ兲=␧_␮␦␮,␯,

H_␮,^AB_␯共kជ_兲=

兺

i=1 3

e^ikជRជit_␮,␯共nជi兲,

H_␮,␯^BA共kជ_兲=

兺

i=1 3

e^{−ikជRជi}t_␮,␯共−nជi兲, 共2兲

where␮^,␯are the atomic orbitals,␧_␮are their energies, and nជi=Rជ_i/兩Rជ_i兩 is the unit vector connecting the neighboring at- oms. The hopping matrix elementst_␮,␯, which depend on the bond orientation, are obtained using the two-center Slater- Koster approximation.³⁰The neighboring atomic orbitals are in general not orthogonal. This fact results in the nonzero overlap matrix elements S_␮,␯, which are usually needed to reproduce the electronic spectrum over a wide momentum range. We will give the values for the overlap matrices of the s andp orbitals by fitting to our first-principles calculation.

However, for the K points the overlaps play little role 共⬍ 10%兲, and we neglect them in our considerations for the SOC.

As the carbon atom has four electrons in the outer shell, one typically resorts to an 8⫻8 TB Hamiltonian³¹ for graphene. Without SOC the block-diagonal form of the Hamiltonian contains two different kinds of bands, the␴^and the ␲ bands. The states corresponding to the ␴ ^{bands are} responsible for the mechanical 共cohesive兲 properties of graphene while the␲bands, formed by the␲bonds between the out-of-plane pz orbitals, are responsible for the unique electronic properties. The conelike dispersion of the␲^bands in the vicinity of the inequivalent K and K⬘ points, the cor- ners of the hexagonal Brillouin zone, is described by the 2

⫻2 Dirac-type Hamiltonian,²

H₀=␧p−v_F⁰ប共␶␴x␬x+␴y␬y兲 共3兲

with the Fermi velocityv_F⁰=

冑

3aVpp␲/共2ប兲, whereVpp␲is the hopping parameter of thepzorbitals and the wave vector␬ជ ^is measured with respect to the Dirac K共K⬘^{兲 points, kជ=Kជ共Kជ}⬘^兲 +␬ជ. The Pauli matrices ␴ជ describe the pseudospin space, such that the eigenstates of␴zcorrespond to the states on the sublattices A and B, and ␶ accounts for the inequivalent Kជ

=共4␲/3a, 0兲 共␶^{= 1}兲andKជ⬘⁼共−4␲/3a, 0兲 共␶^{= −1}兲points. The eigenvalues ofH₀are given with respect to␧p, the energy of the p orbitals.

The small representation of the K point, described by the D_3hpoint group, allows some d orbitals to contribute to the

␲-band Bloch states primarily formed by thep_zstates.^21,23A natural extension of the TB model in seeking for the addi- tional contributions coming from thedorbitals, is to consider hopping between p andd orbitals. Within the Slater-Koster approximation the␲bands in graphene are formed by thep_z, dxz, and dyz orbitals, resulting in hopping matrix elements t_␮,␯^␲ 共nx,ny兲, which are given in Table I. The corresponding Slater-Koster hopping parameters are illustrated in Fig. 1.

The TB Hamiltonian, Eq. 共2兲, for the ␲ band at the K共K⬘兲 points has the form

H=

冢

^{−␧i␣000␶␣p −i␧␶␣␤i00␶␤d −−−␧i00␶␤d␣␤ −−␧i000␶␣ ␤␣p ii␧␶␣ ␣␶␤00d −i␧␶␤00d␤,}

冣

^{, 共4兲}

given in the basis of 共pz

A,d_xz^A,d_yz^A,p_z^B,d_xz^B,d_yz^B兲 orbitals. Here

␣⁼₂^3Vpd␲and␤⁼³₄共Vdd␦−Vdd␲兲.

By solving the secular equation det共H−␧I兲= 0, whereIis the identity matrix, the following two degenerate states can be identified at the Dirac points:

兩1典= 1

冑

N关兩p_z^A典+i␥共␶兩d_xz^B典+i兩d^B_yz典兲兴, 兩2典= 1

冑

N关兩pz

B典+i␥共␶兩dxz A典−i兩dyz

A典兲兴 共5兲

with the corresponding normalization N= 1 + 2␥^{2 and} ␥

⬇␣/共␧d−␧p兲, assuming␣Ⰶ共␧d−␧p兲. The presence of the d orbitals shifts the eigenenergy of the states, in Eq.共5兲, with respect to␧plevel, such that␧1,2⬇␧p− 2␥²共␧d−␧p兲. It is evi- dent that␥ and consequently Vpd␲ controls the contribution from the dorbital to the␲ ^bands.

The energy dispersion very close to the K point remains linear. Here thep-dcoupling renormalizes the Fermi velocity

according to vF=v_F⁰共1 − 2␥²兲. To obtain a quantitative esti- mate for ␥we calculate the density of states共DOS兲close to the Fermi level, which can be expressed in terms of␥^,

D共␧兲=D0共␧兲共1 + 2␥²兲², 共6兲 where D0共␧兲= 2␧/

冑

3␲^Vpp2 ␲

is the linear DOS without d or- bitals and energy ␧ is measured with respect to the Dirac point. In the analytical derivation of Eq. 共6兲 and the energy dispersion, we neglect the hopping between the d orbitals since those contribute only to the states, whose energies are much larger than the Fermi energy. By performing first- principles calculations of DOS based on the full-potential augmented plane-wave method,³² we can determine the pa- rameter␥by calculating the ratio of the DOS slopes close to the Fermi level with and without d orbitals. The ratio is 1.0306 and the slope D₀⬘共␧兲= 0.0392 共eV兲⁻². The extracted parameter ␥⬇0.0871 is justifying the assumption ␣Ⰶ共␧d

−␧p兲. The change in the Fermi velocity of only about 1.5% is neglected in further calculations. The d orbitals contribute about 3% to the DOS.

III. SPIN-ORBIT COUPLING A. Intrinsic gap

SOC connects the electron spin and orbital degrees of freedom. Its major effect comes from the orbits close to the atomic nuclei. Therefore, the crystal potential can be ap- proximated by the spherical atomic potential, which finally gives on-site contribution to the TB Hamiltonian. By averag- ing the radial degree of freedom the SOC reads

H_ᐉ;␮,␯^SO =␰_ᐉ具Lជ·sជ^典␮,␯, 共7兲 wheresជ is the vector of the Pauli matrices representing the real spin andLជ is the angular momentum operator. The ma- trix element 具¯典_␮,␯ is given in the basis of directed atomic orbitals ␮^,␯ ^and ␰_ᐉ is the angular momentum resolved atomic SOC strength with ᐉ=兵s,p,d, . . .其. The matrix ele- ments of the dimensionless SOC operator Lជ·sជ for the rel- evant orbitals in graphene are shown in Table II.

Previous TB calculations^7,17,18consideredporbitals only.

The SOC of these orbitals removes the block-diagonal form of the Hamiltonian Eq. 共2兲 by coupling the␲ ^and␴ ^bands.

This results in the intrinsic SOC splitting of the ␲ ^bands, inducing a gap at the Dirac point of only about 1 ␮eV.

However, as already discussed by Slonczewski^21,22 and proven by ab initio calculated by Gmitra et al.,²⁰ to obtain the spin-orbit splitting at K in graphene one has to considerd orbitals.

Includingdorbitals to the TB Hamiltonian共see Appendix A兲 and folding down the Hamiltonian using Löwdin trans- formation共see Appendix B兲, assuming small spin-orbit cou- pling andV_pd␲with respect to the energy difference␧d−␧p, we derive the effective 4⫻4 Hamiltonian in the vicinity of the K共K⬘^兲points,

H_eff=H₀+H_I. 共8兲 Here H_I is the intrinsic SOC Hamiltonian, having the stan- dard functional form,²³

TABLE I. Slater-Koster hopping matrix elementst_␮^␲_,␯共n_x,n_y兲of the␲bands.

␮\␯ z xz yz

z V_pp␲ n_xV_pd␲ n_yV_pd␲

xz −n_xV_pd␲ n_x²V_dd␲+n_y²V_dd␦ n_xn_y共V_dd␲−V_dd␦兲 yz −n_yV_pd␲ n_xn_y共V_dd␲−V_dd␦兲 n_y²V_dd␲+n_x²V_dd␦

FIG. 1. 共Color online兲Sketch of the Slater-Koster hopping pa- rameters 共a兲V_pp␲,共b兲V_pd␲, 共c兲V_dd␲, and共d兲V_dd␦, needed to cal- culate the contributions of thedorbitals to the␲band.

H_I=␭I␶␴zsz. 共9兲 The intrinsic spin-orbit induced gap at the K共K⬘^{兲 points is} twice the intrinsic spin-orbit parameter,

2␭I⬇4共␧p−␧s兲

9V_sp²_␴ ␰p2+ 9V_pd² _␲

共␧d−␧p兲²␰d. 共10兲 The first term in Eq.共10兲gives a contribution to the intrinsic SOC from the p orbitals, derived previously;^7,17,18 this con- tribution is negligible, giving a gap of about 1 ␮^{eV, mainly} due to the fact that the SOC of the porbitals,␰p, appears in the second order. The second term in the above equation is due to thedorbitals and gives a gap of 23 ␮eV, as obtained from first-principles.²⁰This term dominates, mainly because the SOC of thed orbitals,␰d, appears in the first order.

To analyze in more detail the contribution from the d orbitals, let us focus on the dependence of the spin-orbit gap on the hopping parameters. In general the hopping param- eters decrease with increasing interatomic distance.³³ The contribution to the gap from the p orbitals is inversely pro- portional to the square ofVsp␴and thus should increase with increasing interatomic distance. In Fig. 2, we show the cal- culations of the intrinsic gap 2␭Ias a function of the relative lattice constant˜a/a stretching. In the absence of d orbitals, the gap increases exponentially and should approach the atomiclike splitting⌬= 3␰p⬇8.5 meV for an isolated carbon atom. Thedorbital contribution is quadratically proportional toVpd␲, thus should vanish for large˜a/a. The resulting de- crease and further increase of the gap as the function of˜a/a is an interplay between the contributions from both thepand d orbitals.

Using the results of the first-principles calculations of the gap in Eq.共10兲and the band structure of graphene, we derive the nine parameters of thes, p orbital TB model by fitting the energy spectrum at the high-symmetry points 共⌫, K兲 for different artificial lattice constants. The resulting values of the hopping parameters for graphene lattice constant 共˜a/a

= 1兲 are presented in the TableIII compared to the previous calculations.³¹ Figure 3 shows the agreement between the band structures along high-symmetry lines obtained by the

TB and first-principles calculation. The energies along the K⌫ and KM lines differs less than 5% at about 200 meV from␧F, setting the scale for the anisotropy. Taking the hop- ping parameters in Table III and results of first-principles calculations of␭I we extract the atomic SOC parameters us- ing Eq.共10兲, where␰p⬇2.8 meV. This value coincides with the literature^7,17,18 taking the different definitions of␰p into account. Thereby the numerical diagonalization shows a change in this value ⬍0.01%, that is due to the nonzero overlap parameters. The parameter ␰d⬇0.8 meV is ob- tained, in contrast, using the first-principles calculation re- sults of the DOS and spin-orbit-induced gap using Eqs. 共6兲 and 共10兲. There is no fitting of the energy spectrum like in thes,p case possible, since the needed high-energy states in the conduction bands cannot be identified. The value is lin- early sensitive to theSpd␲overlap. For a reasonable value of Spd␲= 0.1 共which is similar to Vpp␲, see Table III兲, the ex- tracted␰dis within 10% of what we get for no overlap. This sets the accuracy of the extracted parameters.

TABLE II. Matrix elements of the SOC operatorLជ·sជin the basis ofs,p, andddirected orbitals.

Orbital s p_x p_y p_z

s 0 0 0 0

p_x 0 0 −is_z is_y

p_y 0 is_z 0 −is_x

p_z 0 −is_y is_x 0

Orbital d_xy d_x2−y² d_xz d_yz d_z2

d_xy 0 2is_z −is_x is_y 0

d_x2−y² −2is_z 0 is_y is_x 0

d_xz is_x −is_y 0 −is_z i

冑

3s_y

d_yz −is_y −is_x is_z 0 −i

冑

3s_x

d_z2 0 0 −i

冑

3s_y i

冑

3s_x 0

0.8 0.9 1.0 1.1 1.2

0 10 20 30 40

˜ a/a 2 λ

I

[ µ eV ]

p orbital part

d orbital part

total (solid)

0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 0

10 20 30 40 50

p d total

FIG. 2. Results of the first-principles共circles兲, analytical共solid lines兲and numerical共squares兲tight-binding calculations of the SOC intrinsic gap in graphene as a function of the artificial lattice con- stant ratio. Those dependences originate from the hopping param- eters. The inset shows the dominance of the p orbitals for larger values of the lattices constant ratio.

B. Bychkov-Rashba effect

An external electric field, that is perpendicular to the graphene plane, breaks spatial inversion symmetry and changes the band structure of graphene.^2,20 It can originate from a gate voltage or charged impurities in the substrate.

The resulting extrinsic SOC leads to spin splitting of the conduction and valence bands due to the Bychkov-Rashba

effect.²⁴ We investigate this extrinsic SOC by including an electric field into the TB model in the form of the external potential eEzˆ, which represents an atomic single-particle Stark effect. Due to symmetry, this term allows for the cou- pling of the s and pz orbitals and the pz and dz² orbitals within the same atom. Nonzero matrix elements of the TB Hamiltonian共see Appendix A兲equaleEzspandeEzpd, where zsp andzpd are the expectation values具s兩zˆ兩pz典 and具pz兩zˆ兩dz²典, respectively, of the operator zˆ. The electric field leads to a shift of the electron charge density inducing a dipole moment of 0.0134 CÅ in a unit cell for a typical field of 1 V/nm.²⁰ Since this value is rather tiny we assume that the matrix elementseEz_spandeEz_pdare small with respect to␧p−␧sand

␧d−␧p, respectively. By folding down the TB Hamiltonian including the atomic SOC and external potential we derive the effective Hamiltonian in the vicinity of the K共K⬘^兲points, H_eff=H₀+H_I+H_BR 共11兲 with the Bychkov-Rashba part

H_BR=␭BR共␶␴xs_y−␴ys_x兲. 共12兲 The Bychkov-Rashba parameter is given by

␭BR⬇2eEzsp

3Vsp␴␰p+

冑

3 eEzpd

共␧d−␧p兲 3Vpd␲

共␧d−␧p兲␰d. 共13兲 The contribution of the p orbitals was already obtained in Ref. 17.

The linear dependence of the Bychkov-Rashba parameter on the electric field E is consistent with the first-principles calculations.²⁰ But unlike the intrinsic SOC, the extrinsic contribution due to thedorbitals to␭BRis rather small since the d orbitals contribution is proportional to the product of two small quantities,Vpd␲/共␧d−␧p兲and␰d/共␧d−␧p兲. Accord- ing to the first-principles calculations²⁰ ␭BR= 5 ␮^{eV for a} typical field of E= 1 V/nm. This is an order of magnitude smaller than the previous predictions by Huertas-Hernando et al.⁷of the value of 47 ␮eV and Minet al.¹⁷ of 67 ␮eV.

The contribution ofdorbitals to␭BRis about 1.5%. Compar- ing the expressions for the Bychkov-Rashba parameters in Eq. 共13兲 with first-principles calculation we obtain eEz_sp

⬇15 meV and the ratioeEzpd/共␧d−␧p兲= 0.0003, which con- firms our necessary assumption for the Löwdin transforma- tion used in the derivation of the Eqs.共11兲and共13兲. Figure4 TABLE III. Tight-binding hopping and overlap parameters. The values are obtained by fitting the band

structure to the results of the first-principles calculation as the⌫and K points, top row, compared with also presented results given in Ref.31.

Parameter ␧p−␧s V_ss␴ V_sp␴ V_pp␴ V_pp␲

Energy共eV兲 −8.370 −5.729 5.618 6.050 −3.070

Energy^a共eV兲 −8.868 −6.769 5.580 5.037 −3.033

Parameter S_ss␴ S_sp␴ S_pp␴ S_pp␲

Value 0.102 −0.171 −0.377 0.070

Value^a 0.212 −0.102 −0.146 0.129

aReference31.

-20 -15 -10 -5 0 5 10 15

K Γ M K

ε - ε

F

[e V]

-20 -15 -10 -5 0 5 10 15

K Γ M K

ε - ε

F

[e V]

-20 -15 -10 -5 0 5 10 15

K Γ M K

ε - ε

F

[e V]

s p_x+p_y p_z

-20 -15 -10 -5 0 5 10 15

K Γ M K

ε - ε

F

[e V]

-20 -15 -10 -5 0 5 10 15

K Γ M K

ε - ε

F

[e V]

-20 -15 -10 -5 0 5 10 15

K Γ M K

ε - ε

F

[e V]

d

FIG. 3. 共Color online兲 Calculated band structure of graphene obtained from first-principles calculation 共symbols兲 and tight- binding model共solid lines兲using the parameters presented in Table III. The size of the symbols reflects the contribution of the orbitals to the corresponding eigenstates.

shows the exponential increase of the Bychkov-Rashba pa- rameter ␭BR due to the realistic lattice constant stretching, which is controlled by the decay of the hopping parameter Vsp␴with increasing interatomic distance.

IV. EFFECTIVE NEXT-NEAREST-NEIGHBOR HOPPING In the phenomenological description of graphene, the in- trinsic and extrinsic SOC originate from the spin-dependent next-nearest-neighbor 共nnn兲 and spin-dependent nearest- neighbor 共nn兲 hopping, respectively.^2,34 The corresponding hopping parameters can be derived from our multiorbital TB model. Doing so, those empirical hopping parameters are expressed by TB parameters, which were introduced in the previous chapters.

The crucial point of our TB model is the absence of spin- dependent or spin-flipping direct hopping terms. The SOC acts only in the vicinity of the atom core. Hence there is only an effective spin-dependent hopping between twopzorbitals.

Such effective hoppings result from a sequence of on-site and nearest-neighbor hoppings within atomic orbitals, the so- called hopping paths. By factoring out the intermediate states 共see Appendix C兲of the real-space Hamiltonian of noninter- acting electrons,

Hˆ =

兺

i,j

兺

s,s⬘

兺

␮,␯

c_␮,s,i^† T_{␮,s,i;␯,s⬘,j}c_␯,s⬘,j, 共14兲

an effective Hamiltonian can be derived. Here the matrix elementsT_{␮,s,i;␯,s⬘,j}mark the hopping between the orbitals␮ and␯ with a spinsands⬘, respectively, which belong to the atomsiandj;c_␮,s,i^† ,c_␯,s⬘,jare the corresponding creation and annihilation operators. The matrix elements of the neighbor- ing atoms are shown in TableIVusing a specific model, that is shown in Fig. 5.

The Hilbert space of the effective Hamilton operator is reduced to that ofpzorbitals; we omit these labels in further equations. As predicted, distinct terms, the intrinsic and ex-

trinsic SOC, appear in the Hamilton operator. The intrinsic SOC operator

Hˆ

SO=it_SO

兺

具具j,l典典

兺

s,s⬘␨j,lc_s,j^† szcs⬘^,l 共15兲 contains the nnn hopping with the indexl going through all the six next nearest neighbors of the atom j. The parameter

␨j,l= 1共−1兲 for anticlockwise 共clockwise兲 hopping if j is in sublatticeA; vice versa ifjis inB. The extrinsic SOC Hamil- tonian

Hˆ

BR=it_BR具

兺

j,m典

兺

s,s⬘

c_s,j^† 共sជ⫻nជm兲zc_s⬘,m, 共16兲

appears as hopping within nearest neighbors connected by unit vector nជm. The strengths of the effective hopping are given by the parameters

TABLE IV. Real-lattice hopping matrix T within the nearest- neighbor approximation. The geometrical arrangement is defined in Fig.5, where the atom labeled by共p,q兲belongs to the sublatticeA.

The index pcounts the zigzag chains and the indexq atoms in a chain. The matrix elements t_␮,␯共nជ兲 共Ref.30兲 contain the nearest- neighbor hopping parameters and the on-site matrix elements␧␮,␯, the energies of the orbitals and the atomic SOC parameters Eqs.

共A3兲and共A7兲.

p,q− 1 p,q p,q+ 1

p,q− 1 ␧␮,␯ t_␮,␯共−nជ2兲 0

p,q t_␮,␯共nជ2兲 ␧␮,␯ t_␮,␯共nជ3兲 p,q+ 1 0 t_␮,␯共−nជ3兲 ␧␮^,␯

p− 1 ,q− 1 p+ 1 ,q p− 1 ,q+ 1

p,q− 1 t_␮,␯共−nជ1兲 0 0

p,q 0 t_␮,␯共−nជ1兲 0

p,q+ 1 0 0 t_␮,␯共−nជ1兲

0.8 0.9 1.0 1.1 1.2

8 10 12 14

First-principles calc.

Numerical TB Analytical TB

˜ a/a 2λBR[µeV]

FIG. 4.共Color online兲Calculated Bychkov-Rashba constant as a function of the artificial lattice constant ratio: first-principles calcu- lations共circles兲, numerical diagonalization of thep-orbital part of TB Hamiltonian including overlap共squares兲, and the analytical cal- culations 共solid line兲. Those dependences on the lattice constant arise from the hopping parameterV_sp␴.

FIG. 5.共Color online兲The zig-zag chains in the graphene lattice are defined in the way that the corresponding hopping path goes horizontally, alternating in the direction of the unit vectorsnជ2and nជ3. The hopping between the chains is possible only in the direction of the unit vectornជ1.

t_SO= ␭I

3

冑

3, t_BR=␭BR, 共17兲 which correspond to the SOC parameters given by Eqs.共10兲 and共13兲. The effective TB Hamilton matrix Eq.共11兲is ob- tained by Fourier transforming into the k space of the cre- ation and annihilation operators.³⁴The validity of this effec- tive Hamiltonian is again limited, because its eigenenergies␧ are constrained by兩␧−␧p兩Ⰶ兩␧s−␧p兩and兩␧−␧p兩Ⰶ兩␧p−␧d兩but also 兵␰p,eEzsp其Ⰶ兩␧s−␧p兩 and␰dⰆ兩␧p−␧d兩. These conditions limit the wave vector kto the vicinity of the K共K⬘^{兲 points;}

the effective hopping Hamiltonian is valid roughly for ␬

⬍2/a. Figure6shows the band structure of graphene around K共K⬘^兲 points for artificially large valuest_BR= 500 ␮^{eV and} eEzsp= 1.5 eV. These values are at the limit for the validity of the effective hopping Hamiltonian since higher electric fields would break the linear relation betweent_BRandeEz_sp. The spin-orbit interaction respects the symmetries of the crystal so no spin-dependent band splitting occurs. Hence it does not couple states of opposite spins, for the spin quan- tized perpendicular to the sheet. There are two leading hop- ping paths between two p_z orbitals having the same spin, which results in the intrinsic SOC term shown in the Eq.

共10兲. The paths through the states of the␴bands, which are constructed by thes,px, andpyorbitals, are shown in Fig.7.

They include two SOC-induced on-site hoppings with the strength ␰p, because of the spin flipping. Hence this effect appears in the second order in SOC and the corresponding effective hopping parameter should be negligible small. In Fig. 8, we illustrate the paths through the unfilled ␲^-band states, thed_xzandd_yzorbitals. These paths include only one SOC induced on-site spin-conserving hopping with the strength ␰d. Therefore this path of hopping is the most re- sponsible for the gap opening, even if the hopping parameter V_pd␲is small compared to the parameterV_pp␲.

The Bychkov-Rashba SOC effect originates from the ef- fective nn hopping of two p_z orbitals of the opposite spin, appearing in the presence of an applied transverse electric field. The corresponding hopping paths are shown in Fig.9.

The effective nn spin-flipping hopping originates from the coupling to the␴-band states: a spin-conserving on-site hop- ping to thesord_z² orbitals and an on-sites-p coupling共via SOC兲 which flips the spin. The nn hopping takes place be- tween the s,p orbitals and p,d orbitals. In contrast to the intrinsic SOC, the effect of thedorbitals is negligible. Hence the Stark effect coupling of the p_z and d_z2 orbitals is tiny compared to the coupling ofpzto thesorbitals. In addition, the SOC coupling of thedorbitals appears an order of mag- nitude smaller then that of theporbitals. Finally, the hopping between the p and d orbitals is also smaller then the one between p and s orbitals. Thus the contribution of the se- quence of the hopping paths in which thedorbitals take part is quite limited.

Finally we point out that the presented arguments go in hand with the results of our first-principles calculations.

V. CONCLUSIONS

We have presented a multiorbital extended TB model for graphene considerings,p, anddorbitals relevant when SOC

10⁴ K 10⁴

2 1 0 1 2

κ [nm

⁻¹

]

ε [meV ]

FIG. 6. Calculated spectrum in the vicinity of the K point, plot- ted along the high-symmetry lines in the⌫-K-M direction. Circles correspond to the numerical diagonalization of the multiorbital TB Hamiltonian 共neglecting thedorbitals which do not play any role here兲and solid lines give the eigenenergies of the effective hopping Hamiltonian. The artificially large spin-orbit parameters for this plot are given in the text. The effective hopping model describes well the spin-orbit physics in graphene at the K point.

(a)

−ξp

√3 2 Vspσ

1 2Vspσ

−iξp

pz

py

s

pz

px

(b)

iξp

−Vspσ

−^√2³Vspσ

−ξp

pz

px

s pz

py

FIG. 7. 共Color online兲 Two of the possible nnn hopping paths through thes,porbitals,共black兲arrows, with a corresponding spin, shown by共yellow兲arrows on the orbitals. The opposite sign for the clock-wise共a兲and the anti-clock-wise共b兲effective hoppings is de- termined by the signs of the two SOC of theporbitals.

(a)

−^√₂³Vpdπ

−iξ_d

−₂¹V_pdπ p_z dxz

dyz

p_z

(b)

V_pdπ

iξ_d

√3 2V_pdπ

p_z dyz dxz

pz

FIG. 8. 共Color online兲 Two of the possible nnn hopping paths throughdorbitals,共black兲arrows. Their spin is shown by共yellow兲 arrows on the orbitals. The opposite sign for clock-wise 共a兲 and anti-clock-wise 共b兲 hopping is given by the opposite sign in the SOC of thedorbitals.

is present. Sincedorbitals contribute to the␲bands without SOC, their共atomic兲spin-orbit splitting determines the value of the band gap at the K共K⬘^兲 points. This fact is indepen- dently confirmed by performing first-principles calculations of the dependence of the SOC induced gap on the lattice constant. We have also derived an effective single-orbital hopping Hamiltonian that captures all the essential spin-orbit physics of itinerant electrons in graphene. With the param- eters derived from the multiorbital theory and the insight given by showing the relevant effective hopping paths, such a model should be useful for spin-polarized transport inves- tigations, within the limitations restricting its use close to the K共K⬘^{兲 points.}

ACKNOWLEDGMENT

This work was supported by the DFG SPP 1285 and SFB 689.

APPENDIX A

The extended TB Hamilton matrix can be divided into four blocks,

H=

冉

^HH␲s† H^H_␴^s

冊

^{. 共A1兲}

The diagonal blocks describe the ␴ ^bands 共H_␴兲 and the ␲ bands共H_␲兲, which are coupled by SOC, leading to nonzero off-diagonal blocks of the TB Hamiltonian共H_s兲. The␴^-band block is divided into site-dependent parts due to the two sublattices 共A and B兲 of graphene, which are given in the basis of the兵s,px,py,dxy,dx²−y²,dz²其atomic orbitals,

H_␴=

冉

^HH␴AA_␴^{BA HH␴AB}_␴^BB

冊

^{, 共A2兲}

whereH_␴^AA=H_␴^BBand

H_␴^AA=

冢

^{␧00000s is␧0z000p␰p −is␧0000pz␰p − 2is␧0000dz␰d 2is␧0000zd␰d ␧00000d}

冣

^._共A3兲

Here␧_ᐉare the energies of the atomic orbitals,␰_ᐉ^{the atomic} SOC strength of the orbitals with ᐉ=兵s,p,d, . . .其 and the Pauli matrices s_i withi=兵x,y,z其. The off-diagonal matrices 共H_␴^AB兲^†=H_␴^BA are presented here at the K 共␶= 1兲 and the K⬘ 共␶= −1兲points

H_␴^BA=

冢

^{−i−␶␣␣i00␶␣␣sdds −−ii␶␣␶␣␤i␥i␶␤␶␥−p−spp−− −−−−−−ii␶␤␶␥␤␥␣␣p−−sp−p− i−i−ii␶␣␶␥␶␤␶␦␥␦−−−−dd −i−i␶␥␶␦␥␦␣␤−−−−dd −i−␶␤i␣00␶␣␤pddp}

冣

_共A4兲^,

end at the⌫point

H_␴^BA=

冢

^−3V

^冑

^{200003ss␣␴d 220␤000␥+p+ 2200␤00␥+p+ − 220000␦␥++ − 220000␦␥++ −}

^冑

^{␦2000030␣d}

冣

_共A5兲

defining the parameters

␣s=3

2Vsp␴, ␣p=3

4Vpd␴, ␣d=3 4

冑

3Vsd␴,

␤p⫾=3

4共Vpp␲⫾Vpp␴兲, ␤d=3

8

冑

3共Vdd␦−Vdd␴兲,

␥^⫾=3

8共2V_pd␲⫾

冑

3V_pd␴兲, ␦⁰⁼³

4共3V_dd␦+V_dd␴兲,

␦^{⫾= 3}

16共Vdd␦⫾4Vdd␲+ 3Vdd␴兲, 共A6兲 which are given by the Slater-Koster hopping parameters V_ᐉᐉ⬘␣共Ref.30兲and␣⁼兵␴^,␲^,␦其.

(a) (b)

eEzsp

Vspσ

−iξp

eEzpd

i√ 3ξd

Vpdπ

pz

s

py

pz

pz

d_z2

dyz

pz

FIG. 9. 共Color online兲 A representative leading hopping path, 共black兲arrows, which is responsible for the Bychkov-Rashba SOC effect, by coupling states of different spins, illustrated by共yellow兲 arrows on the orbitals. The effective hopping is between nearest neighbors. 共a兲 The dominantporbital contribution.共b兲The negli- gible d orbital contribution. For clarity the orbitals of the same atoms are separated vertically, according to their contribution either to the␴bands共bottom兲or to the␲bands共top兲.

The ␲-band block is given in the basis of the 兵pz

A,p_z^B,d_xz^A,d_yz^A,d_xz^B,d_yz^B其atomic orbitals. At the K共K⬘^兲points, it reads as

H_␲=

冢

^{−␧000i␣␶␣p −−␧i000␶␣p␣ −isi␤i␶␣␧0z␶␤d␰−d− −−−−isi␧0␶␤␤␣dz␰−−d isii␶␤␤␧␶␣0zd␰−−d −−i␶␤␧is␣0␤dz,−␰−d}

冣

_共A7兲^,

and at the⌫point

H_␲=

冢

^{3V␧0000ppp␲ 3V␧0000ppp␲ is␤␧00z0d␰+d −is␤␧000d+z␰d is␤␧000z+d␰d −␧␤is000d+z,␰d}

冣

_共A8兲^,

where

␣=3

2Vpd␲, ␤^⫾=3

4共Vdd␦⫾Vdd␲兲, 共A9兲 such that the off-diagonal blocks共separated by lines兲contain the hoppings between the p andd orbitals.

Finally the SOC part of the TB Hamiltonian reads as

Hs=

冢

^{hhh000xzyzz hhh000xzyzz}

冣

^{, 共A10兲}

where

h_z=共eEzsp, −isy␰p, isx␰p, 0, 0, eEzpd兲,

hxz=共0, 0, 0, isx␰d, −isy␰d, i

冑

3sy␰d兲, h_yz=共0, 0, 0, −isy␰d, −isx␰d, −i

冑

3sx␰d兲

共A11兲 with nonzero Stark-effect matrix elements eEzsp andeEzpd. Hereeis the electron charge,Ethe electric field andzsp,zpd

are the expectation values of the zˆ operator within corre- sponding orbitals.

APPENDIX B

To be self-contained, we also present briefly the Löwdin partitioning. The scheme makes use of a unitary anti-

Hermitian operatorS, such that the transformed Hamiltonian

H˜ =e^−SHe^S⬇H+关H,S兴+1

2关关H,S兴,S兴 共B1兲

has block-diagonal form, with

H=

冉

^HT†0 ⌬^T

冊

^{, S=}

冉

^{−0M† M0}

冊

^{, 共B2兲}

whereM can be obtained iteratively from the equation T+H₀M−M⌬= 0. 共B3兲 Keeping only second-order terms in⌬⁻¹, the matrixM reads as

M⬇T⌬⁻¹+H₀T⌬⁻². 共B4兲 Inserting this expression into the Eq. 共B1兲, the first element includes the effective Hamiltonian

H_eff⬇H₀−T⌬⁻¹T^†−1

2兵T⌬⁻²T^†,H₀其, 共B5兲 where higher-order terms in⌬⁻¹are neglected.

APPENDIX C

To derive the effective hopping energies, Eq.共17兲, of the next-nearest-neighbor hopping model, with the microscopic SOC parameters specified by Eqs.共10兲and共13兲, and to iden- tify the contributing virtual hopping paths, we start with the secular equation for the multiorbital TB Hamiltonian,

T_{␮,s,i;␯,s⬘,j}c_␯,s⬘,j=␧c_␮,s,i, 共C1兲 where ␮^,␯ count the orbitals, s,s⬘ the spins, and i,j the atoms. In the nearest-neighbor approximation, the corre- sponding secular equations of the TB Hamiltonian of the␲ bands, comprising pz,dxz, and dyzorbitals, read as

共␧−␧p兲cz,i=

兺

_m,␯^{tz,␯共nជm兲c␯,m, 共C2兲}

共␧−␧d兲cxz,i= −isz␰dc_yz,i+

兺

m,␯

t_xz,␯共nជm兲c_␯,m, 共C3兲

共␧−␧d兲cyz,i=isz␰dc_xz,i+

兺

m,␯

t_yz,␯共nជ^m兲c␯,m. 共C4兲

The spin index is omitted as there are no spin flips; the trace of the spin is kept bysz. The labelmcounts the three nearest neighbors to the atom i and ␯⁼兵z,xz,yz其 enumerates the

␲-band orbitals. The strategy is to eliminate thedorbitals by expressing them via the neighboringp_zones,

共␧p−␧d兲cxz,m=

兺

m⬘

关txz,z共−nជ^m_⬘^兲−isztyz,z共−nជ^m_⬘^兲␷d兴cz,m⬘,

共C5兲