Graphene with adatoms: tuning the magnetic moment with an applied voltage
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Graphene with adatoms: tuning the magnetic moment with an appliedvoltage
N. A. Pike a) and D. Stroud b) Department of Physics, The Ohio State University, Columbus, OH 43210 (Dated: 29 July 2018)
We show that, in graphene with a small concentration of adatoms, the total magnetic moment µ T can beswitched on and off by varying the Fermi energy E F , either by applying a gate voltage or by suitable chemicaldoping. Our calculation is carried out using a simple tight-binding model described previously, combinedwith a mean-field treatment of the electron-electron interaction on the adatom. The values of E F at whichthe moment is turned on or off are controlled by the strength of the hopping between the graphene sheet andthe adatom, the on-site energy of the adatom, and the strength of the electron-electron correlation energyU. Our result is in qualitatively consistent with recent experiments by Nair et al. [Nat. Commun. , 2010(2013)].PACS numbers: 73.20.At, 73.22.Pr, 75.70.AkThe two-dimensional structure of graphene and theDirac-like dispersion relation of its electrons are the ori-gin of many unusual properties, which may lead to novelelectronic or spintronic applications . For example,adatoms on graphene may develop magnetic momentswhich can be manipulated by an applied electric field ina manner similar to its other electric and optical proper-ties . Recent experimental work by Nair et al. showsthat both sp defects and vacancies in graphene possessmagnetic moments that can be switched on and off bychemical doping .In a recent paper, we showed, using a tight-bindingmodel, that certain non-magnetic adatoms, such as H,can create a non-zero magnetic moment on graphene .In this Letter, we extend our calculation to show thatthe magnetic moment of graphene with adatoms can beswitched on and off by varying the Fermi energy. ThisFermi energy can be controlled, in practice, by an ap-plied voltage; it can also be tuned by suitable chemicaldoping of the graphene, as in the experiments of Ref. 8.Our calculations show that the onset and turn-off of themagnetic moment depend on the parameters character-izing the adatom, such as the hopping strength betweenthe adatom and graphene, the on-site energy, and theelectron-electron correlation energy.In the following section we briefly review our model,and its solution via mean field theory. The model is ap-propriate when the adatom lies atop one of the C atomsin graphene (the so-called T site), as is the case for ad-sorbed H and several other adatom species . In sec-tion 2, we present numerical results showing the depen-dence of the magnetic moment on Fermi energy for vari-ous model parameters. In section 3 we give a concludingdiscussion. I. TIGHT-BINDING MODEL
We consider a tight-binding Hamiltonian to model thegraphene-adatom system. The graphene part of the Hamiltonian, denoted H , is written in terms of the cre-ation and annihilation operators for electrons of spin σ on a site in the n th primitive cell . Denoting the creation(annihilation) operators for the α and β sub-lattices by a † nσ ( a nσ ) and b † nσ ( b nσ ), we write H for nearest-neighborhopping on graphene as H = P k ,σ H , k ,σ , where H , k ,σ = − t ( k ) a † k ,σ b k ,σ − t ∗ ( k ) a k ,σ b † k ,σ (1)and σ = ± /
2. We note that here and in subsequentequations the notation of Ref. 9 is used.In Eq. (1), t ( k ) = t h (cid:0) ik x a (cid:1) cos (cid:16) √ k y a (cid:17)i ,where a = 1 . A is the nearest-neighbor bond length forgraphene, and t is the hopping energy between nearestneighbor carbon atoms (for graphene t = 2 . eV ) .The extra part of the Hamiltonian due to an adatom ata T site may be written in real space as H I = ǫ X σ h † ,σ h ,σ − t ′ X σ (cid:16) h † ,σ a ,σ + h ,σ a † ,σ (cid:17) , (2)where h † ,σ and h ,σ are creation and annihilation opera-tors for an electron of spin σ at the site of the adatom, ǫ is the on-site energy of an electron on that site (relativeto the Dirac point of the pure graphene band structure),and t ′ > α sub-lattice. In terms of Bloch eigenstates of H , H I = ǫ X σ h † ,σ h ,σ (3) − t ′ √ N X σ " h † ,σ X k e − iφ k ( γ k ,σ, + γ k ,σ, ) + h.c. . raphene with adatoms: tuning the magnetic moment 2Here γ k ,σ,i (i=1,2) is the annihilation operator for a Bloch electron of wave vector k , spin σ , and within the i th band. φ k is a phase angle given in Ref. 9 and for a hydrogen adatom we take ǫ = 0 . eV and t ′ = 5 . eV .To calculate the magnetic moment we add a Hubbard term H U , which acts only on the adatom. Within themean-field approximation H U is given by H U ∼ U h h † ↑ h ↑ h n ↓ i + h † ↓ h ↓ h n ↑ i − h n ↑ ih n ↓ i i . (4)where h n ,σ i is the average number of electrons with spin σ on a T site of the α sub-lattice. For a hydrogen adatomwe take U to be the difference between the ionization potential and the electron affinity, which gives U ∼ . eV =4 . t .The total mean-field Hamiltonian consists of the sum of Eqs. (1), (3), and (4), which is quadratic in the electroncreation and annihilation operators and therefore readily diagonalized. The total spin-dependent densities of states ρ tot,σ ( E ) can then be calculated, given the values of h n ,σ i . The result is ρ tot,σ ( E ) = N ρ ( E ) − π (cid:20) Im (cid:18) ddz ln[ z − ǫ − U h n , − σ i − t ′ N G ( z )] (cid:19)(cid:21) z = E + i + , (5)where ρ ( E ) is the density of states per spin and per primitive cell of pure graphene, N is the number of primitive cells,and G ( z ) the corresponding Green’s function, as defined in Ref. 9. The quantities h n ,σ i are obtained by integratingthe local spin-dependent density of states on the adatom, ρ ,σ ( E ), from the bottom of the valence band up to theFermi energy. ρ ,σ ( E ) is, in turn, given by ρ ,σ ( E ) = − π Im z − ǫ − U h n , − σ i − t ′ , N G ( z ) ! z = E + i + . (6)The total magnetic moment µ T can be calculated as afunction of the Fermi energy E F from the expression µ T ( E F ) = µ B Z E F − t [ ρ tot, ↑ ( E ) − ρ tot, ↓ ( E )] dE, (7)where µ B is the Bohr magneton.Using these results, we can numerically calculate µ T ( E F ) for various choices of tight binding parameters.This is done iteratively, for a given choice of E F , as fol-lows. First, we make initial guesses for h n ,σ i . Next, weintegrate the spin-dependent local density of states up tothe Fermi energy using Eq. (6) and the initial guesses for h n ,σ i . This gives the next iteration of h n ,σ i . We iterateuntil the changes in h n ,σ i in two successive cycles areless than 0 . II. NUMERICAL RESULTS
We have carried out these calculations as a function of E F for a variety of values of ǫ , t ′ , and U . In every case,we find that the total magnetic moment is non-zero onlywhen E F lies within a limited range. We write this con-dition as E ℓ < E F < E u where E ℓ and E u are the lower a) Electronic mail: [email protected] b) Electronic mail: [email protected] and upper energies within which µ T = 0. Within thisrange, the magnitude of µ T is controlled by varying theFermi energy E F , typically by applying a gate voltageto the sample which interacts with the conduction elec-trons via the electric field effect . If the graphene-adatomsystem is neutral and no voltage is applied, E F will beconstrained to have a particular value controlled by thecharge neutrality condition. Introducing a gate voltagewill shift E F from this neutral value (denoted E F ) andhence change the magnetic moment.As an illustration of this picture, we show in Fig. 1 thetotal magnetic moment µ T ( E F ) under various conditions.In each case, we assume all parameters but one are thosethought to describe H on graphene, and vary the remain-ing parameter . We assume a single H adatomis placed on a graphene sheet containing N = 500 car-bon primitive cells, giving 1 H atom per 1000 C atoms.In Fig. 1(a), we assume that t and t ′ are those of theH-graphene system, while each curve represents a differ-ent value of the on-site energy ǫ . We find that, when E F > ǫ and we allow ǫ to become much less then E F that the onset energy E ℓ → E u shows only a minimal dependence on E F .In Fig. 1(b) we plot µ T ( E F ) versus E F for severalvalues of t ′ , with other parameters the same as the H-graphene system. As t ′ is reduced, the upper energy cut-off, E u , also decreases, whereas the lower energy onsetremains approximately unchanged at E ℓ = 0 . eV , inde-pendent of t ′ . In Fig. 1(c), we plot µ T ( E F ) versus E F forseveral values of the electron-electron energy U , assumingraphene with adatoms: tuning the magnetic moment 3 µ t / µ m a x E F (eV) ε =0.4 ε =-0.4 ε =-1.0(a) µ t / µ m a x E F (eV) t’=5.0t’=5.8t’=6.0(b)0.00.51.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 µ t / µ m a x E F (eV)U=4.59t U=10.0t U=12.0t(c) FIG. 1. (Color online) Total magnetic moment verse Fermienergy for an adaom on graphene calculated using Eq. (7). Ineach case, all the parameters but one are appropriate to H ongraphene, as given in the text. In (a) each curve represents adifferent on-site energy ǫ (given in eV in the legend). In (b),each curve represents to a different hopping energy t ′ (in eV),and in (c) each curve represents a different Hubbard energy U (in eV). The maximum calculated magnetic moment in eachFigure is noted within the text. the other parameters same as the H-graphene system. As U increases, so do both E ℓ and E u . As U → ∞ , E u → ∞ as well, i. e., µ T persists no matter how large E F in thiscase.For all parameters we have considered, we find thatthe magnitude of the magnetic moment µ T < µ B . InFig. 1(a). the maximum value of µ T ≈ . µ B , corre-sponding to ǫ = − . eV and − . eV . In Fig. 1(b)the maximum value of µ T ≈ . µ B for t ′ = 5 . eV ,while in Fig. 1c a maximum value of µ T ≈ . µ B for U = 12 . t . In previous work , it has been shown that U → ∞ , µ T → µ B . III. DISCUSSION AND CONCLUSIONS
The numerical results of Fig. 1 can be qualitativelyunderstood as follows. If the system of graphene plusadatom has a net magnetic moment, then the partialdensities of states ρ ↑ ( E ) and ρ ↓ ( E ) will be different. Thetotal magnetic moment is then obtained by integratingthese two densities of states up to E F . If E F lies belowthe bottom edge of the lower band, there will be no netmagnetic moment. The moment becomes non-zero at theenergy E F = E ℓ when E F moves above the bottom of thelower band. It reaches its maximum when E F lies some-where between the peaks in ρ ↑ ( E ) and ρ ↓ ( E ) (assumingthe two bands overlap), and thereafter decreases until itbecomes zero at E F = E u , the energy above which bothsub-bands are filled. Thus, for any choice of the param-eters ǫ , t ′ , and U , there should be a finite range of E F ,within which µ T = 0.Of course, this description is an oversimplification be-cause the self-consistently determined ρ ↑ ( E ) and ρ ↓ ( E )depend on the quantities h n ,σ i ( σ = ± / µ T . However, even with the oversimpli-fication, the qualitative description remains correct. Themaximum value of µ T depends, in part, on how much thetwo sub-bands overlap. If the overlap is large, maximumof µ T will be small, everything else being equal, while asmall overlap will tend to produce a larger µ T . Anotherreason why µ T is reduced below µ B is that there is gen-erally a large electron transfer from the adatom onto thegraphene sheet . We also note that the energy range, E u − E ℓ , where µ T = 0 is approximately equal to thewidth of the extra density of states due to the adatom.The Fermi energy E F can be controlled experimentallyin several ways. One is to apply a suitable gate voltage V , which raises or lowers E F by an amount eV , where e is the magnitude of the electronic charge. Another is bychemical doping: if one adds or subtracts charge carriersto the graphene-adatom system by doping with suitablemolecules, this will also raise or lower E F as was done inRef. 8. One could also add a small number of vacanciesin the graphene, as also done in Ref. 8. This will reducethe number of charge carriers and hence lower E F . Ofcourse, vacancies would also change the graphene densityof states; so the present calculations would have to bemodified to treat this situation.The model used here treats only the effects of adatomson graphene, but it does qualitatively reproduce the up-per energy cutoff found in the experiments of Ref. 8. Forexample, Nair et al. found an upper cutoff of around E u = 0 . eV , which we can approximately obtain by as-suming an on-site energy ǫ = 0 . eV , t ′ = 5 . eV and U = 4 . t . However, our model does not account for theonset energy found in Ref. of E F ∼ E ℓ > eV .In summary, by using a simple tight-binding model ofadatoms on graphene we are able to calculate the totalmagnetic moment of graphene with a small concentra-tion adatoms as a function of E F . The model is ex-raphene with adatoms: tuning the magnetic moment 4pected to apply to the case of H adatoms, but could alsobe applicable to other adatom species, characterized bydifferent model parameters. The Fermi energy E F canbe controlled experimentally by a suitable gate voltage.Our results show that, for realistic tight-binding param-eters ( ǫ = 0 . eV, t ′ = 5 . eV, U = 4 . t ), the magneticmoment can be switched off at a relatively low voltage( eV ∼ . eV ), in rough agreement with the experimentsof Ref. 8. These results are potentially of much interestsince they suggest that the magnetic moment of graphenewith adatoms can be electrically controlled. IV. ACKNOWLEDGMENTS
This work was supported by the Center for EmergingMaterials at The Ohio State University, an NSF MRSEC(Grant No. DMR0820414). We thank R. K. Kawakamifor helpful discussions. D. Pesin and A. H. MacDonald, Nat. Maters. N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J.van Wees, Nature , 06037 (2007). K. M. McCreary, A. G. Swartz, W. Han, J. Fabian, and R. K.Kawakami, Phys. Rev. Lett. , 186604 (2012). M. Garnica, D. Stradi, S. Barja, F. Calleja, C. Diaz, M. Alcami,N. Martin, A. L. Vazquez de Parga, F. Martin, R. Miranda. Nat.Phys. A. K. Geim. Science F. M. Hu, T. Ma, H. Lin,and J. E. Gubernatis. Phys. Rev. B. , 075414 (2011). K-H. Yun, M. Lee, and Y-C. Chung. J. Magn. Magn. Mater. R. R Nair, I-L Tsai, M. Sepioni, O. Lehtinen, J. Keinonen, A.VKrasheninnikov, A. H Castro Neto, M. I Katsnelson, A. K Geim,I. V Grigorieva, Nat. Comm. N. A. Pike and D. Stroud Phys. Rev. B O. V. Yazyev and L. Helm, Phys. Rev. B , 125408 (2007). A. L. Rakhmanov, A. V. Rozhkov, A. O. Sboychakov, and F.Nori. Phys. Rev.
B 85 , 035408 (2012). K. Nakada and A. Ishii, Solid State Commun. , 13 (2011). K. T. Chan, J. B. Neaton, and M. L. Cohen, Phys. Rev. B ,235430 (2008). P. W. Anderson, Phys. Rev. , 41 (1961). R. Pariser and R. Parr, J. Chem. Phys. , 767-775 (1953). K. R. Lykke, K. K. Murray, and W. C. Lineberger, Phys. Rev.
A 43 , 6104-6107 (1991). T. O. Wehling, M. I. Katsnelson, and A. I. Lichtenstein, Chem.Phys. Lett.
125 (2009). J. Hobson and W. A. Nierenberg, Phys. Rev. , 662 (1953). S. Yuan, H. De Raedt, and M. I. Katsnelson, Phys. Rev.