Density-functional studies of spin-orbit splitting in graphene on metals
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug - Density-functional studies of spin-orbit splitting in graphene on metals
Z. Y. Li , S. Qiao , Z. Q. Yang , , ∗ and R. Q. Wu State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China Department of Chemistry, Northwestern University, Evanston, Illinois 60208, USA Department of Physics and Astronomy, University of California, Irvine, California 92697-4575, USA (Dated: September 30, 2018)Spin-orbit splitting in graphene on Ni, Au, or Ag (111) substrates was examined on the basis ofdensity-functional theory. Graphene grown on the three metals was found to have Rashba splittingof a few or several tens of meV. The strong splitting obtained on Au or Ag substrates was mainlyascribed to effective hybridization of graphene p z state with Au or Ag d z states, rather than chargetransfer as previously proposed. Our work provides theoretical understandings of the metal-inducedRashba effect in graphene. PACS numbers: 73.22.-f, 71.70.Ej, 75.75.+a
Graphene has attracted extensive attention in recentyears due to its unique and remarkable electronic prop-erties, such as gapless-semiconductor band, existenceof pseudospin, and high electronic mobility at roomtemperature.
These features are highly desirable forthe development of next-generation microelectronic andspintronic devices.
Spin currents in graphene can bemanipulated using various electronic tactics, in particularthrough exchange and spin-orbit (SO) interactions, whichare now among the most active research topics in sev-eral realms.
The intrinsic SO effect in pure graphenelayers is nevertheless very weak, 0.1 ∼ , insufficient forpractical use due to the low nuclear charge of the car-bon atom. It was hence very exciting when Dedkov et al reported an extraordinarily large Rashba SO splitting(225 meV) for the π states of epitaxial graphene layerson the Ni (111) substrate through their angle-resolvedphotoemission studies. It appears that the SO effect orhybridization in graphene can be tuned through effectiveelectric field across the interface. However, this resultwas challenged by Rader et al who found that the sumof Rashba and exchange splitting in the graphene layer oneither Ni (111) or Co(0001) is less 45 meV. They pointedout that the Rashba effect can be strongly enhanced byintercalation of one monolayer of Au between grapheneand Ni (111). Clearly, to discriminate these contradictoryexperimental results and furthermore to understand themechanism of substrate-induced SO splitting in grapheneare crucial for the progress of graphene physics.In this Letter, we report results of density-functionaltheory (DFT) calculations for the electronic and mag-netic properties of graphene on Ni, Au, or Ag (111) films.Interestingly, the Rashba splitting in graphene on Au andAg can be significantly enhanced by strong hybridiza-tion of graphene p z state and metal d z state. The SOsplitting is found to be almost independent of the chargetransfer between graphene and its substrates, contradict-ing to effective electronic field model proposed in Ref. .The electronic structures of graphene on metal (111) (II) (I) MetalC (a) -10.0-9.8-9.6-9.4-10.0-9.8-9.6-9.40.000.020.04 -20-15-10-505 (b) Gr/Ni, SO E n e r g y ( e V ) E n e r g y ( e V ) S p li tt i ng ( e V ) SO = EX+SO - EXEX+SO EX SONo SO
EX+SO EX k ( ) (c) Gr/Ni, -band FIG. 1: (Color online) (a) Two possible configurations ofgraphene adsorbed on metal (111) substrates. The rhombusgives the unit cell along the graphene plane. (b) The energybands of Gr/Ni in the configuration (I) with SO interaction.The blue arrow indicates the graphene π bands around Γ. (c)The enlarged π bands around Γ without or with SO interac-tions. The exchange and/or SO splitting of the bands are alsogiven. The k point in (c) is in the unit of the vector Γ M . substrates, abbreviated as Gr/M (M = Ni, Au, and Ag),were calculated by using the VASP code at the level of lo-cal spin-density approximation (LSDA). The projector-augmented wave (PAW) pseudopotentials were employedto describe the effect of core electrons. The equilibriumstructures were obtained through structural relaxationuntil the Hellmann-Feynman forces were less than 0.05eV/˚A. The Gr/M systems were modeled by a periodicslab geometry, with a vacuum of at least 10 ˚A betweentwo neighboring slabs. Each slab contains one graphenelayer and N atomic layers of metal. Two adsorption con-figurations of graphene on the metal substrates were con-sidered, as depicted in Fig.1(a). We found that configu-ration (I) is more stable for graphene on Ni (111), as itgives very small lattice mismatch ( ∼ . As a benchmarkcalculation for the SO effect, we first determined the SOsplitting of the surface states (SS) of the pure Au (111)film near the Fermi level (E F , set as energy zero). Ourresult, ∆ E SO ∼
100 meV, agrees well with data in Ref. .The band structures of Gr/Ni with N=13 are given inFig.1(b) and (c). The direction of Ni magnetization is setto be perpendicular to ΓM. Despite the strong perturba-tion from the Ni (111) substrate, one can still easily traceseveral graphene bands in Fig.1(b), e.g., the graphene σ and π states at -4 and -10 eV in the vicinity of the Γ point.Compared to the band structures of pure graphene, thesestates are spin polarized, and shifted downward in en-ergy by about 1.2 and 2.2 eV, respectively. Particularly,the feature conical points at K near E F are destroyed inFig.1(b), due to broken equivalence of A and B sublat-tices through the interaction with Ni. These results arein agreement with photoemission measurements .Now we zoom in to explore spin splitting and SO ef-fect of the π states of graphene along the - M Γ M line,following the experimental work . To separate con-tributions from different factors, we studied cases eitherwith or without the SO interaction. As illustrated inFig.1(c), bands without the SO interaction are symmetricabout the Γ point and show an induced exchange split-ting (∆ EX ) of 30 meV on the magnetic Ni substrate.After considering the SO interaction, the energy split-ting (∆ EX + SO ) contains two parts: exchange and SO(∆ SO ). The value of ∆ SO can be extracted through∆ SO = ∆ EX + SO − ∆ EX . The linear relationship of∆ SO versus k in Fig.1(c) indicates that the SO inter-action is indeed the Rashba type (∆ SO = 2 α R k , where α R is Rashba strength). The Rashba splitting ob-tained from our calculations is about 10 meV, in consis-tent with the experimental data in Ref. . In comparisonwith the SO splitting (0.37 meV) in curved graphene ,the Ni-induced SO splitting in the graphene π bands isrelatively larger.The energy bands of Gr/Au in the configuration (II),also with a fine lattice match, are given in Fig. 2, wheredifferent separation (d) between graphene and the Ausubstrate is considered. The π bands of graphene areaccompanied by a gold SS at = -7.3 eV , labeled as’Au SS’ in Fig. 2(a). This Au SS is actually localizedat the vacuum side of the Au slab and hence shows nochange for different d in Fig. 2(a). The correspondingAu SSs at the graphene side move down quickly withthe decrease of d due to the effect of graphene. At theequilibrium geometry (d eq =3.3 ˚A), the SO splitting ofgraphene π bands at 0.3 Γ M is about 21 meV, muchlarger than that on the Ni substrate. For Gr/Au inthe configuration (II), the A and B sublattices experi- -7.2-7.0-6.8 -0.60.00.6-7.2-7.0-6.8 -0.60.00.6-7.2-7.0-6.8 -0.60.00.6 E n e r g y ( e V ) x10 d = 2.9 (cid:191) d eq = 3.3 (cid:191) d = 3.7 (cid:191) d = 2.9 (cid:191) d eq = 3.3 (cid:191) d = 3.7 (cid:191) Gr (cid:83) Gr (cid:83) Gr (cid:83) Au SS Au SSAu SS
M0.40.20-0.2-0.4 K (cid:42) k ( (cid:42)(cid:3) M ) (b) Gr/Au (II)(a) Gr/Au (II) FIG. 2: (Color online) The bands of Gr/Au in the configu-ration (II) with N=9 (a) around -7.0 eV along - M Γ M and(b) near E F along Γ KM with different d values. The red linein (b) is drawn to indicate the Dirac point. The inset is theenlarged bands near the Dirac point. ence the same environment again. Thus, the graphenelayer almost restores its unique electronic structure: theDirac cone near E F . Charge transfer between grapheneand the metal and therefore the energy position of theDirac cone can be adjusted through changing d. At theequilibrium separation, the Dirac point (Fig.2(b)) is veryclose to E F , indicating almost no charge transfer betweengraphene and Au. When the separation expands/shrinks,the Dirac point moves upward/downward, revealing netcharge transfer between graphene and Au occurs. TheSO splitting of graphene, however, reduces both ways,independent of the enhancement of effective electric fieldin the interface. This contradicts to the effective electricfield model proposed in Refs. for the explanation ofenhanced SO effect in the systems. The inset in Fig.2(b)shows the SO splitting of the Dirac point. Both the elec-tron and hole bands show SO splitting of about 5 meV,close to the value (13 meV) given in recent experiment. The splitting at the Dirac point is induced by the metal,as intrinsic SO interaction at this point was predicted tobe zero. Figure 3 shows the energy bands in large energy rangefor Gr/Au with configuration (II) at equilibrium dis-tance. The two sets of parabolas at about -0.2 and -0.4eV around the Γ point are the Au (111) SSs at the twosides of the Au slab, with and without graphene, respec-tively. They are degenerate at -0.4 eV for the pure Au(111) film , but now the SS in the side with the adsorp-tion of graphene is pushed up slightly. In comparisonwith Fig. 1(b), the bulk Au 5 d bands around Γ point(about -2 ∼ -7 eV) are deeper in energy than Ni 3 d bands.This bestows the probability for Au 5 d to strongly inter-act with graphene π bands, which are in a big interstice -15-10-50 E n e r g y ( e V ) Gr/Au (II), d eq = 3.3A FIG. 3: (Color online) The energy bands of Gr/Au(II) withSO interaction in large energy range. The blue arrow indicatesthe graphene π bands. The red lines indicate the Dirac point.TABLE I: The components of wave functions for graphene π bands at Γ for the considered systems at equilibrium dis-tances. The values are in the scale of the graphene p z state.The bold expresses the more stable configuration. The SOsplitting (in meV) of graphene π bands in the configuration(I) was given at k = 0 .
25 Γ M , while in the configuration (II),at k = 0 . M due to the double lattice vectors.Ni ( I ) Au ( II ) Au (I) Ag ( II ) Ag (I)(N=13) (N=9) (N=12) (N=9) (N=12)Gr- p z state 1.0 1.0 1.0 1.0 1.0M- s state 0.8 1.3 0.1 0.5 0.7M- d z state 0.0 2.4 1.3 0.0 3.7SO splitting 6.3 32.2 89.0 1.6 36.9 formed mainly by Au 5 d states. The situation is verydifferent from that in Fig. 1(b), where the graphene π and Ni 3 d states are well separated in energy. To un-derstand the different SO splittings in Ni and Au cases,wave function compositions of the π bands at Γ are an-alyzed quantitatively. As shown in Table I, the ratio ofC 2 p z :Ni 4 s in Gr/Ni (I) ≈ d is involved.Since Ni 4 s states do not contribute to the SO effect,the enhancement of Rashba splitting in Gr/Ni only re-sults from the asymmetric potential distribution in thetwo sides of graphene. In contrast, contribution from theAu d z state is obvious for Gr/Au (II); the ratio of C2 p z :Au 6 s :Au 5 d is about 1.0:1.3:2.4. Therefore, stronghybridization of the metal d z with graphene p z is a keyfactor to produce large SO splitting in graphene π bands. -7.2-6.8-6.4-6.8-6.4-6.0-7.6-7.2-6.8-6.4 -0.4 d = 2.1A Au SS Gr Gr/Au (I) k ( )
Au SS
Au SS
Gr Gr d = 2.9A d eq = 2.5A E n e r g y ( e V ) FIG. 4: (Color online) Left: The bands of Gr/Au (I) along- M Γ M with different d values. Right: The correspondingcharge density of graphene π bands at the Γ point. The den-sity is plotted in the plane perpendicular to the interface,indicated by the dash line in Fig.1(a). The upper and loweratoms are a carbon and a gold atom, respectively. For Gr/Au, configuration (I) is less stable than config-uration (II) since the C-C bonds in graphene have to bestretched by 17% to match the structure of the Au (111)substrate. This stretching gives rise to a much shorterequilibrium distance (2.5 ˚A) in the interface, and thus al-lows us to understand the effects of adjusting the lateraland vertical distances. The whole π bands of graphene inthe configuration (I) thus disperse less; and shift upwardas well, hopefully causing more effective hybridizationbetween C 2 p z and Au 5 d states. Figure 4 gives the SOsplitting of the graphene π bands in Gr/Au (I) at d eq to be close to 100 meV, 3 times larger than that in Au(II) case. In Gr/Au (I), the Au 6 s component decreasesmeanwhile the C 2 p z and the Au 5 d z more effectively hy-bridize to each other (see Table I), as expected. Similarto the trend in Fig. 2(a) either, increasing or decreasingd cause a decrease of the SO splitting in Fig. 4. Thistrend can be rationalized by using the real-space chargedensities of the graphene π bands at Γ in the right pan-els of Fig. 4. At the equilibrium distance, a very obviousinteraction between graphene and metal and asymmetriccharge distribution above and below the graphene planeare observed, corresponding to large SO splitting. Whend becomes longer or shorter than d eq , the effective mixingbetween C p z and Au d z is weakened.Table I also contains the results of Gr/Ag (I) and(II). Again, the SO splitting in the configuration (II) ismuch less than that in (I) due to weakened hybridization. TABLE II: The SO splitting of π bands of graphene on differ-ent metal substrates with N=1 and 6, respectively. The valueis given at k = 0 .
25 or 0 . M , as stated in Table I. The boldindicates the more stable configuration.meV Ni (I) Au ( II ) Au (I) Ag ( II ) Ag (I)N = 1 6 . . . . .
7N = 6 6 . . . . . While the graphene p z state has strong interaction withthe Ag d z state in the configuration (I), only Ag s stateis involved in the more stable Gr/Ag (II), similar to Ni.Therefore, heavy metals may not always produce largeSO splitting in graphene.Finally, we also explored the effect of the thickness ofmetal films. The SO splitting of graphene π bands forGr/Au (I) and (II) and Gr/Ag (I) with N=6 in TableII are almost the same as the values listed in Table I,respectively. For the rest two cases: Gr/Ni (I) and Gr/Ag(II), only one monolayer of metal substrate is enough togive a saturated SO splitting of graphene. Since Au SSsusually extend several atomic layers into the bulk , a fewlayers of metals are needed to obtain the saturated SOsplitting. Nevertheless, the SO splitting for graphene on Au mono- and bi-layer films is already large, explainingwhy one Au atomic layer intercalated between grapheneand Ni (111) can cause a substantial Rashba effect in theexperiment. In conclusion, we investigated what determines the SOsplitting of graphene bands on Ni, Au, or Ag (111) sub-strates through first-principles calculations. While theRashba splitting for Gr/Au is sizeable, the effect of Niis very limited. The hybridization between graphene p z and metal d z states is identified as the chief factor forthe enhancement of the SO effect in graphene. This re-quires not only large SO strength from metal atoms, butalso effective overlap of metal d and graphene states inenergy. A few atomic layers of metals are sufficient toproduce saturated strong Rashba splitting in graphene.Our findings point out a direction for the manipulationof SO strength in graphene that is needed for the devel-opment of spintronic materials and devices.The work was supported by National Natural Sci-ence Foundation of China (No.10674027), 973 project(No.2006CB921300), Fudan High-end Computing Cen-ter, and Chemistry and Materials Research Division(MRSEC program) of NSF in USA. Work at UCI wassupported by DOE grant DE-FG02-05ER46237 and com-puting time at NERSC. ∗ Electronic address: [email protected] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov and A. K. Geim, Rev. Mod. Phys. , 109(2009). K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M.I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A.Firsov, Nature (London) , 197 (2005). Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature(London) , 201 (2005). N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, andB. J. van Wees, Nature (London) , 571 (2007). Y. G. Semenov, K. W. Kim, and J. M. Zavada, Appl. Phys.Lett. , 153105 (2007). S. Datta and B. Das, Appl. Phys. Lett. , 665 (1990). D. D. Awschalom and M. E. Flatt´e, Nature Phys. , 153(2007). C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 226801(2005). F. Kuemmeth, S. Ilani, D. C. Ralph, and P. L. McEuen,Nature (London) , 448 (2008). Y. Yao, F. Ye, X.-L. Qi, S.-C. Zhang, and Z. Fang, Phys.Rev. B , 041401(R) (2007). E. I. Rashba, Sov. Phys. Solid State , 1109 (1960); Yu.A. Bychkov and E. I. Rashba, JETP Lett. , 78 (1984). Yu. S. Dedkov, M. Fonin, U. R¨udiger, and C. Laubschat,Phys. Rev. Lett. , 107602 (2008). O. Rader, A. Varykhalov, J. S´anchez-Barriga, D.Marchenko, A. Rybkin, and A. M. Shikin, Phys. Rev. Lett. , 057602 (2009). G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169 (1996); G. Kresse and J. Hafner, Phys. Rev. B , 14251(1994). J. P. Perdew and Y. Wang, Phys. Rev. B , 13244 (1992). E. Durgun, R. T. Senger, H. Sevin¸cli, H. Mehrez, and S.Ciraci, Phys. Rev. B , 235413 (2006). V. M. Karpan, G. Giovannetti, P. A. Khomyakov, M. Ta-lanana, A. A. Starikov, M. Zwierzycki, J. van den Brink,G. Brocks, and P. J. Kelly, Phys. Rev. Lett. , 176602(2007). G. Giovannetti, P. A. Khomyakov, G. Brocks, V. M.Karpan, J. van den Brink, and P. J. Kelly, Phys. Rev.Lett. , 026803 (2008). G. Nicolay, F. Reinert, S. H¨ufner, and P. Blaha, Phys. Rev.B , 033407 (2001). G. Bertoni, L. Calmels, A. Altibelli, and V. Serin, Phys.Rev. B , 075402 (2005). A. Nagashima, N. Tejima, and C. Oshima, Phys. Rev. B , 17487 (1994). I. Gierz, J. H. Dil, F. Meier, B. Slomski, J. Oster-walder, J. Henk, R. Winkler, C. R. Ast, and K. Kern,arXiv:1004.1573v1. S. H. Liu, C. Hinnen, C. Nguyen Van Huong, N. R. DeTacconi, and K. M. Ho, J. Electroanal. Chem. , 325(1984). A. Varykhalov, J. S´anchez-Barriga, A. M. Shikin, C.Biswas, E. Vescovo, A. Rybkin, D. Marchenko, and O.Rader, Phys. Rev. Lett. , 157601 (2008). A. Yamakage, K.-I. Imura, J. Cayssol, and Y. Kuramoto,Europhys. Lett.87