aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Half Metallic Bilayer Graphene
Jie Yuan, Dong-Hui Xu, Hao Wang,
1, 3
Yi Zhou, Jin-Hua Gao,
4, 1, ∗ and Fu-Chun Zhang
1, 2, † Department of Physics, and Center of Theoretical and Computational Physics,The University of Hong Kong, Hong Kong, China Department of Physics, Zhejiang University, Hangzhou, China Department of Physics, South University of Science and Technology of China, Shenzhen, China Department of Physics, Huazhong University of Science and Technology, Wuhan, Hubei, China
Charge neutral bilayer graphene has a gapped ground state as transport experiments demonstrate.One of the plausible such ground states is layered antiferromagnetic spin density wave (LAF) state,where the spins in top and bottom layers have same magnitude with opposite directions. We proposethat lightly charged bilayer graphene in an electric field perpendicular to the graphene plane maybe a half metal as a consequence of the inversion and particle-hole symmetry broken in the LAFstate. We show this explicitly by using a mean field theory on a 2-layer Hubbard model for thebilayer graphene.
PACS numbers: 73.22.Pr, 71.30.+h, 73.21.Ac, 75.75.-c
Half metals are a class of materials in which electronswith one spin orientation are metallic and electrons withopposite spin orientation are insulating [1, 2]. In a halfmetal, electrical current can be fully spin polarized. Thisproperty is attractive in spintronics [3–5]. Possibility ofgraphene-based half metal has been interesting for its po-tential application in electronic devices. Soon after thediscovery of graphene, Son, Cohen and Louie have ap-plied first principles calculations to propose half metal-lic phase in a zigzag graphene nanoribbon with externaltransverse electric fields [6]. Due to the technical difficul-ties in the devices, the predicted half metal in grapheneribbon has not been confirmed yet in experiments. Herewe propose that bilayer graphene (BLG) in an electricfield perpendicular to the graphene plane can be a halfmetal. Our proposal is based on the recent works to sug-gest that the ground state of BLG at half filled (chargeneutral case) may be a layered antiferromagnetic spindensity wave (LAF) state with a gap of about 2 meV.In the half filled LAF state, the spins are polarized op-positely on the top and bottom layers, which break thespin SU(2) symmetry, but are invariant under a com-bined transformation of time reversal ( T ) and inversion( I ), as we can see in Fig. 1(a). The LAF state also con-serves the combined symmetry of inversion and particle-hole conjugation ( C ), where the operation of C changesan electron operator to a hole on the same site, whichalso reverses the sign of the magnetic moment of a par-ticle (or hole). The electric field breaks I and C . Theircombination I ⊗ C is conserved in the neutral graphene,but is broken in the charged graphene, under the electricfield. Therefore, an electric field applied to the chargedgraphene opens the possibility for a half metallic phasewith a net magnetization. We examine the half metallicBLG by applying a mean field theory on a 2-layered Hub-bard model. Our prediction may be tested in the BLGdevice with double gates[7].We start with a brief summary of the recent works on the BLG[7–37]. Theoretically, in the single electronband picture the BLG is a gapless semiconductor withparabolic valence and conduction bands touching at thehigh symmetry points K and K ′ . The state is unstablein the presence of electron interaction. Experimentally,there are clear evidences that the BLG at half filled hasa gapped ground state[7, 16–20]. Velasco et al. [7] haveapplied a perpendicular electric field on a high qualitysuspended BLG. The energy gap is found to decrease asthe field increases and to close at a field 15 mV nm − .Two of the most promising states consistent with thegapped ground states measured in transport experimentsare the LAF [26, 27, 33, 40, 41] and the quantum spinHall states [26, 27, 33, 38, 39]. It will be important andinteresting to explore the possible experimental conse-quences of these states and determine the true groundstate of the BLG. In the present Letter, we predict a halfmetallic phase in the LAF state of the BLG in the pres-ence of an electric field at slightly charged graphene. Ourresult may be used to resolve the controversial issue ofthe ground state in BLG, and should be of importance tothe potential application of the graphene in spintronics.We consider a BLG system in an applied perpendicularelectric field. The Hamiltonian is given by H = H + H U + H p (1)where H = H intra + H inter is the kinetic energy part, H U is the on-site Coulomb interaction, and H p describesthe electric potential due to the applied electric field. Theintra-layer hopping is given by H intra = − t X l h ij i σ [ a † lσ ( i ) b lσ ( j ) + H.c. ] + µ X liσ n lσ ( i ) (2)Here, a lσ ( b lσ ) are the electron annihilation operator onsublattice A (B), and l = 1 , σ = ↑ , ↓ and i ( j ) denotethe spin and site, respectively. hi sums over all nearestneighbor sites. Here we only consider the nearest neigh-bor hopping for simplicity and expect the small remotehopping will not change the basic physics. µ is the chem-ical potential. µ = 0 corresponds to the half filling, or thecharge neutral point. At µ >
0, the chemical potentialacroses the conduction band. We consider an interlayerhopping between two sites i and i ′ on-top to each other, H inter = t ⊥ X h ii ′ i σ [ b † σ ( i ) a σ ( i ′ ) + h.c. ] . (3)The Hubbard U term is given by H U = U P li n l ↑ ( i ) n l ↓ ( i ). The effect of the external elec-tric field E is modeled by an electric potential V between the two layers, H p = X liσ V l n lσ ( i ) , (4)with V l = ( − l V /
2, and V is related to E as below.Note that the electron charge on the two layers maybe redistributed in the presence of E [13, 14]. Let E be the electric field between the two layers, and assumethe graphene sheets to be infinitely large 2-dimensionalplanes, then we have V = + ed E , with − e the electroncharge, d the interlayer distance. E is related to E by E = E − πe ( ρ − ρ ) , (5)where the electron density in layer l is given by ρ l = h P σ,i n lσ ( i ) i /S , with S the area of each layer, and h Q i isthe average value of operator Q . In the above equation,we have assumed the dielectric constant for the BLG tobe 1 as suggested in the literature[13, 42].We use a mean field approximation for the Hubbardterm, and solve the Hamiltonian self-consistently, H MF U = U X liσ h n lσ ( i ) i n l ¯ σ ( i ) (6)There are four atoms in a unit cell and eight mean fieldsin our theory, h n ηlσ i , with η = A or B indicating thesublattice. We use d = 0 .
334 nm and 2 πe d ≈ × − meV · cm At E = 0, our theory gives a gapped LAF groundstate at half filled, consistent with the experiment [7],and with previous theoretical works by using renormal-ization group theory[32, 33] and quantum Monte Carlomethod[34]. First principles calculation indicates thatsuch LAF state is stable in the presence of nonlocalCoulomb interaction and remote hopping[41]. In Fig.1(a), we illustrate the spin and charge structures of theLAF state. The charge distribution is uniform and thespins are anti-parallel to each other. There are two-fold degeneracy for the spin configurations, related tothe time reversal. The net spin polarizations in the topand bottom layers have an opposite sign, and their sumgives a null magnetization. In Fig. 2(a), we show the (a) B FIG. 1: (Color online). Schematic illustration of spin (arrows)and charge (circles) structures of the LAF state in BLG. A and B indicate sublattices, 1 and 2 are layer indices. (a) Athalf filled and E=0, the state is invariant under both com-bined transformations T ⊗ I and I ⊗ C , with T , I , and C the time reserval, inversion, and particle-hole transformationoperators, respectively. (b) At half filled and finite E, thestate is invariant under I ⊗ C . The charge structure breaks I , but the spin structure remains invariant under T ⊗ I . (c)Charged (electron) graphene in electric field: spin structurebreaks T ⊗ I and the state is ferrimagnetic. LAF energy gap as a function of U . For U ≈ . ǫ g ≈ U here for the BLG is close to the value of U ≈ . I , and alsothe particle-hole conjugation symmetry C , but conserves I ⊗ C . This can be seen from the Hamiltonian for H p . Thehalf filled LAF state in the electric field is invariant under I ⊗ C . Namely, the charge is invariant under I ⊗ C , whilethe spin remains invariant under both T ⊗ I and I ⊗ C .The system has no net magnetization. In Fig. 2(b), weplot the calculated spin dependent energy gap as a func-tion of V . As V increases, the spin-down excitation gapincreases and the spin-up excitation gap becomes nar-rowed. At a critical value V = V c ≈ . V c . At V > V c the gap increases monotonouslywith V . In Fig. 2(c,d), we show the spin polarizationand the charge transfer at the four distinct lattice sites.Our calculations confirm the symmetry analysis we havemade about the spin and charge structure. There is acharge transfer from sublattice A in layer 1 to sublat-tice B in layer 2, and a less amount of charge transferfrom sublattice B in layer 1 to sublattice A in layer 2(inter-layer nearest neighboring sites). In Fig. 2(e), weshow the screening effect and plot E as a function E ,according to Eq. (4). We remark that V c ≈ . mV cor-responds to an applied electric field E ≈ − . -505 (f)(e) (d)(c) (b) U(eV) g ( m e V ) KK’ (a) E ( m e V ) g ( m e V ) V(mV) g g P S ( X - ) V(mV) P P P P -1 -1 n ( x10 - ) V(mV) n -n n -n KK’ E ( m V n m - ) E (mV nm -1 ) non-interacting E ( m e V ) FIG. 2: (Color online). Bilayer graphene at half filled. (a)Energy gap of H in Eqn. (1) as a function of Hubbard U.Shown in inset are low energy bands (blue curves). The pa-rameters are U = 6 . t = 3 . t ⊥ = 0 .
381 eV.The bands for U = 0 are plotted for comparison (red curves).(b) Spin-resolved energy gap as function of V. (c) spin polar-ization on four distinct lattice sites. (d) Charge transfer asfunctions of interlayer electric potential V . In (c) and (d),superindices 1(2) and A (B) are for layer indices and sublat-tices, respectively. (e) Electric field E as a function of appliedelectric field E (solid black line), also shown is E v.s. E fornon-interacting case U = 0 (dashed pink line). (f). Energybands at V = 1 . This is in good agreement with the external electric fieldat the phase transition point estimated in the transportmeasurement [7].In Fig. 2(f) we show the spin-dependent bands nearthe Fermi level for V = 1 . µ > BLG B (a) (b) ME FIG. 3: (Color online). (a) Energy bands of the BLG in an in-terlayer electric potential V = 1 . δn ≈ × cm − . Solid lines are forspin-up and dashed ones for spin-down. Model parametersare the same as in Fig. 2. (b): schematics of probing magne-tization by using torque magnetometry experiment. half metal. The spin and charge structures of this caseis illustrated in Fig. 1(c) and there is a net spin-up inthe BLG. Away from half filled, there is no particle-holesymmetry, and the system in the electric field breaks I , C , and I ⊗ C , and the spin structure of the LAF statebreaks T ⊗ I symmetry and gives a half metal with anet magnetization. The case for µ < δn ≈ × cm − . Theother parameters are the same as in Fig. 2(f). Thiselectron density is equivalent to 2 . × − electron percarbon site on the BLG. As we can see from the figure,the spin-up conduction band is partially filled but thespin-down conduction band is completely empty. Thesystem is a half metal with a full spin polarization inits carriers. The electric field induces time reversal sym-metry broken in this case. The surface magnetizationper area is M = δn × µ G = 2 × − g L µ B / ,with µ G the magnetic moment of graphene atom, and µ B the Bohr magneton and g L the Lande g-factor forthe graphene, which is about 2-2.5. The magnetizationis tiny, but possibly detectable by using torque magne-tometry experiment, as schematically illustrated in Fig.3(b). [44]We have studied µ or δn dependence of the magneti-zation and the spin polarization of the BLG away fromthe half filled. As µ increases, there is a jump in δn at acritical µ c to separate the LAF and normal states. To il-lustrate this, we consider model parameters given in Fig.4, which gives a larger LAF gap at half filled, hence amore pronounced half metallic state when it is doped.The jump in δn as a function of µ can be seen clearly inFig. 4(b). The magnetization and spin polarization areplotted in Fig. 4(c) and (d), respectively. At δn = 0, the -0.6-0.30.00.3 0.0 0.1 0.2 0.3-0.06-0.030.000.030.060.0 0.1 0.2 0.302468 0.0 0.1 0.2 0.305101520 -1 -1 (a) E -( e V ) K’ K (eV) P (d) (eV) P S (eV) (c) M ( B c m - ) (b) n ( c m - ) FIG. 4: (Color online). BLG in an interlayer electric potential V = 0 .
25 eV for model parameters U = 7 . t = 3 .
16 eV,and t ⊥ = 0 . δn ≈ × cm − . Blue solid lines are for spin-up andred dashed ones for spin-down. (b). Electron density δn vschemical potential. The jump in δn reflects first order phasetransition from LAF to normal states, and the system is phaseseparated in the ”jumped” density region. (c). Magnetizationas function of chemical potential ( g L = 2; and (d). Site spinpolarizations as functions of chemical potential. spin polarization P AS = − P BS , and P AS = − P BS . At µ >