Anomalous Hall effect in magnetized graphene: intrinsic and extrinsic transport mechanisms approaching the quantized regime

Anomalous Hall effect in magnetized graphene: intrinsic and extrinsic transport mechanisms approaching the quantized regime

Manuel Offidani Department of Physics, University of York, York YO10 5DD, United Kingdom    Aires Ferreira Department of Physics, University of York, York YO10 5DD, United Kingdom

We present a unified theory of the anomalous Hall effect (AHE) in magnetized graphene sheets in the presence of dilute disorder. The analytical study of Berry curvature and transport coefficients in a 4-band Dirac model [(spin) (pseudo-spin)] unveils a delicate balance between intrinsic and extrinsic contributions owing to the skyrmionic spin texture of magnetic Dirac bands. We show that the anomalous Hall conductivity changes sign when the Fermi energy approaches the topological gap as result of competing spin Lorentz forces. The predicted effect foreruns the  quantum anomalous Hall regime and allows estimation of proximity spin–orbit strength directly from a Hall measurement. Our findings are relevant to outline a systematic route towards the demonstration of novel topological insulating phases in magnetized Dirac fermion systems.

Ferromagnetic order in two-dimensional (2D) crystals is of great significance for fundamental studies and applications in spintronics. Recent experiments have revealed that intrinsic ferromagnetism occurs in 2D crystals of CrGeTeCr2Ge2Te6_Gong_17 () and CrICrI3_Huang_17 (), while graphene and group-VI dichalcogenide monolayers acquire large exchange splitting when integrated with nanomagnets Ferr_G_YIG_Swartz12 (); Ferr_G_YIG_Wang15 (); AHE_G_Tang_18 (); Ferr_G_YIG_Leutnantsmeyer_17 (); Ferro_Gr_EuS_Wu17 (); Ferro_Gr_BiFeO3_Wu17 (); Ferro_vdW_mat ().

Different from bulk compounds, the magnetic proximity effect strongly impacts the electronic states of 2D crystals, opening up a completely new arena for studies of emergent spin-dependent phenomena. In this regard, graphene (Gr) with interface-induced magnetic exchange coupling (MEC) offers promising perspectives MEC_Gr_Theory_Haugen08 (); MEC_Gr_Theory_QAHE_Sun_10 (); MEC_Gr_Theory_Sun11 (); MEC_Gr_Theory_Yang13 (); MEC_Gr_Theory_Marchenko15 (). The room temperature AHE observed in Gr on YIG thin films Ferr_G_YIG_Wang15 (); AHE_G_Tang_18 () shows that the proximity MEC is accompanied by a sizable Rashba-Bychkov (RB) effect due to interfacial breaking of inversion symmetry BR_2DEG_Rahsba_Bychkov_Effect_84 (). Fundamentally, the RB effect entangles pseudospin () and spin () degrees of freedom GRashba_09 (); offidani17 (), endowing Bloch eigenstates with a rich spin texture in reciprocal -space, including skyrmions near the Dirac point (Fig. 1). Furthermore, such a sizable RB spin–orbit coupling (SOC) can drive ferromagnetic Gr through a topological phase transition to a Chern insulator MEC_Gr_Theory_QAHE_Sun_10 (); Chen_11 (). When the Fermi level is tuned inside the gap, this system is predicted to exhibit the quantum anomalous Hall effect (QAHE), with transverse conductivity QAHE_G_2 (). However, much less is known about the non-quantized regime at finite carrier densities. The latter is the current experimental accessible regime Ferr_G_YIG_Wang15 (); AHE_G_Tang_18 (). Beyond the non-quantized part of the intrinsic contribution, the presence of a Fermi surface (FS) makes the transverse (anomalous Hall) conductivity depend nontrivially on spin-dependent scattering ferreira2014 (); Tuan16 (); milletari16 (); ASP_Huang_16 (). To uncover the transverse response of magnetized Gr at finite carrier density is crucial to assess on equal footing the competition of intrinsic (Berry curvature-related) and extrinsic transport mechanisms of Dirac fermions subject to SOC and exchange interactions. Importantly, such a unified description would enlighten the crossover to the quantum regime.

In this Letter, we present a thorough theoretical study of carrier transport in magnetized Gr systems with SOC. Specifically, starting from a generic model of magnetized 2D Dirac fermions, we evaluate the linear response of the system to external electric () and magnetic () fields, incorporating Berry phase effects and dominant disorder corrections. The calculations are carried out analytically for arbitrary strength of proximity-induced interactions and account for intervalley scattering induced by atomically sharp defects, hence applying to a range of experimental realizations and sample conditions. Focusing on the AHE, we show that the out-of-equilibrium steady state is characterized by a proliferation of scattering rates associated to magnetic Dirac states. The transport equations unveil a skew scattering contribution, strongly modulated by the magnetic spin texture and sensitive to the spin splitting of the FS. This novel effect gives rise to a change of sign in the anomalous Hall conductivity when approaches the majority spin band edge, and thus is a precursor of the QAHE. Large transferred MEC has been already achieved in Gr on YIG Ferr_G_YIG_Wang15 (); AHE_G_Tang_18 (), yet progress is needed to induce a comparable RB effect. The change of sign is a smoking gun of sizable transferred SOC, allowing for its estimation directly from Hall measurement and validating ultimately sample preparation. Finally, we calculate analytically the Berry-curvature dependent contribution in the full 4-band model, and explore the competition between extrinsic and intrinsic transport mechanisms.

Model.—The electronic properties of Gr with interface-induced SOC and exchange interaction are modeled by a 2D Dirac Hamiltonian at low energies (we set ),


where is the Fermi velocity of massless Dirac fermions, and are, respectively, the MEC and RB energy scale, and is a disorder potential describing impurity scattering. Here, is the 2D kinematic momentum operator for states near valley / (). Finally, are Pauli matrices acting on valley, pseudospin, and spin spaces, respectively CommentBasis (). We consider a disorder potential of the form , where are random impurity positions and their effective radius ResonantScatt_G (). This allows us to interpolate between the important regimes of dominant intravalley scattering characteristic of ultra-clean samples () and the “sharp defect” limit () G_Review (). The energy-momentum dispersion relation associated to clean Hamiltonian is easily computed as


where is the wavevector measured from a Dirac point and is a “SOC mass” (see Fig. 1). define, respectively, the carrier polarity (electron/hole) and spin chirality i.e., spin winding direction.

Figure 1: Band structure and spin texture with (a) only MEC (b) only RB SOC and (c) with both. Together, MEC and RB SOC open a gap and produce a non-coplanar spin texture. (d) Classification using Bloch index or a ring index . (e) Sign of anomalous Hall conductivity resulting from competing effective spin Lorentz forces, with the elastic channel dominating above at energies .

When ,  (no SOC), the Dirac cones are shifted vertically, resulting in mixed electron–hole states near the Dirac point. On the other hand, if (no MEC), the spectrum only admits a spin-gap or pseudogap region, within which the spin and momentum are locked at right angles (RB spin texture) GRashba_09 (). Finally, finite SOC and MEC opens a gap and splits the Dirac spectrum into 3 branches: regions I and III, defined by and ; those energy regimes are characterized by a non-simply connected FS allowing for scattering between states with different Fermi momenta; and region II with only one band intersecting the Fermi level. Hereafter, unless stated otherwise, all functions are projected onto valley ( point); respective expressions for are obtainable by the simple transformation . The unnormalized Bloch eigenstates of the uniform part of Eq. (1) read as


where is the wavevector polar angle. The skyrmion-like spin texture results from the combined effect of SOC and MEC: while the former favor in-plane alignment, the exchange interaction tilts the spins out of the plane, leading to a rich non-coplanar texture [Fig. 1 (c)]. As shown below, the -depence of the polarization of magnetic Dirac bands is of pivotal importance to the AHE; even if the electronic states are generally not fully polarized, it will prove useful to refer to effective spin-up () and spin-down ( states. For brevity, we focus on positive energies, , and also , thus fixing and omitting this index from the expressions.

Extrinsic contribution: Boltzmann transport equations (BTEs).—To determine the extrinsic contribution in the non-quantized regime, we solve the BTEs for a spatially homogeneous system with dilute impurities CommentFormalism (). The formalism allow for the inclusion of external magnetic field and, more importantly, for a transparent physical interpretation of the scattering processes. The BTEs read as


where is the sum of the Fermi-Dirac distribution function and , the deviation from equilibrium; is the elementary charge and is the area. The right-hand side is the collision term describing single impurity scattering and is the impurity areal density. Subscripts are ring indices, denoting the outer and inner FS, respectively, associated with Fermi momenta ; see Fig. 1(d) CommentRingIndex (). Accounting for possible scattering resonances due to the Dirac-like spectrum ferreira2014 (), the quantum-mechanical transition rates are evaluated employing the -matrix approach, i.e., , where . Here, and . We start by considering (no intervalley scattering). In this case, electrons undergo intra- and inter-ring scattering processes on the same valley (see SM supplemMat () for a graphical visualization). Exploiting the isotropy of the Fermi surface, and momentarily setting , the exact solution to the linearized BTEs () is


with . In the above, are the longitudinal () and transverse () transport times given by


where , , and similarly for . The kernels are functions of symmetric and skew cross sections , where , and . Considering the corresponding expressions for states near , the general solution involves 16 different cross sections. The exact form of the kernels is essential to correctly determine the energy dependence of the conductivity. As shown in SM supplemMat (), including a magnetic field only requires the substitution , where is the cyclotron frequency associated with the ring states. At , and accounting for the valley degeneracy, we obtain the longitudinal and transverse conductivities as


where includes the ordinary Hall background. The external field provides the magnetic thin film (e.g., YIG) with finite magnetization , hence to relate the Dirac model Eq. 1 to the experimental realizations in Refs. Ferr_G_YIG_Wang15 (); AHE_G_Tang_18 () one has to consider . At saturation the anomalous Hall (AH) signal—extracted by subtracting the Hall background to —attains its maximum value and pleataus at increasing field Ferr_G_YIG_Wang15 (); AHE_G_Tang_18 (). One can then extract by setting zero external field (no Hall background) and defining , such that .

Figure 2: Typical behavior of . (a) and (b) is the result of the competing effective spin Lorentz forces as presented in the main text; see also Fig. (1)(e). The change of sign, more evident for increasing . To plot (b) we used the relation respectively in regions I, II, and III. (c) While less evident in the unitary limit, the change of sign is robust across all scattering regimes. meV, and .

The change of sign.—Focusing on the regime , we show how, approaching low carrier density, electrons undergoing spin-conserving and spin-flip scattering processes determine a change of sign in . For the moment, we work in the weak scattering limit , and assume one can restrict the analysis to intra-ring transitions within the outer ring: (see additional discussions supplemMat ()). A first scenario for the change of sign is as follows. First, we note that as is increased from , electron states in the lower band progressively change their spin orientation from effective spin-up to -down states (see Fig. 1). Starting from , varying instead, it can be verified that the same happens within the outer ring such that by tuning one can switch between states with opposite spin polarization. As up/down states are associated with an opposite effective spin Lorentz force, this also means conducting electrons can be selectively deflected towards different boundaries of the sample. The associated AH voltage will then display the characteristic change of sign, as shown in Fig. 2(a).

A second scenario involves the spin-flip force and does not require changing the polarization of carriers. Instead, what changes when varying is the ratio of spin-flip to elastic impurity cross section. This also produces a change of sign as depicted in Fig. 1(e); the fate of the transverse conductivity will depend ultimately on the competition between the two effective spin Lorentz forces (see SM supplemMat ()). Remarkably, the change of sign in is a persistent feature as long as SOC and MEC are comparable, as shown in Fig. 2(a). In that case, in fact, the skyrmionic spin texture is well developed, such that, on one hand, it is possible to interchange between effective spin-up and -down states using a gate voltage, and, on the other, both spin-conserving and spin-flip scattering matrix elements are non-zero. Asymptotically, , the AH signal vanishes due to the opposite spin orientation of electron states belonging to bands, which produce a net zero magnetization. It is useful to note that in the strong scattering limit, , the rate of inter-ring transitions increases and the one-ring scenario presented above might break down. However, as shown in Fig. 2(c) the change of sign is still visible.

In realistic systems, structural defects and short-range impurities, such as hydrocarbons, induce scattering between inequivalent valleys Resonant_Scatt_Ni_10 (), enhancing the backscattering probability G_Review (). Recent studies of spin precession in graphene with interface-induced SOC showed that the in-plane spin dynamics is strongly affected by intervalley scattering Benitez_17 (); Cummings_17 (); Ghiasi_17 (). To determine how the AHE is affected by atomically sharp potentials, we solved Eq. (4) for arbitrary ratio . Figure 2(c) shows the AH conductivity for selected values of (dashed lines). is reduced to nearly half of its previous value, when the intravalley potential strength achives its maximum value (). However, the sign change in , approaching the majority spin band edge is clearly visible. Further analysis are given in SM supplemMat (), where we also analyze the impact of thermal fluctuations, concluding the features described above are persistent up to for  meV. A careful numerical analysis provides an estimation for defined as ,


with –0.4 and –0.8. This relation shows that the knowledge of (e.g., from the Curie temperature Ferr_G_YIG_Wang15 ()) allows to estimate the SOC strength directly from the gate voltage dependence of the AH resistance. We comment in passing on the magnitude of () is compatible to the measurements in Refs. Ferr_G_YIG_Wang15 (); AHE_G_Tang_18 (), for a reasonable choice of parameters, , , and . Note that for even cleaner samples, will attain even larger values within the diffusive regime.

Figure 3: Intrinsic contribution and total AH conductivity. (a) The BCs of hole bands . Note that develops additional “hot spots” as SOC is increased. (b) The total at selected values of ; same legend as in (a). (b) Adding the intrinsic contribution (inset) to leaves the estimate for [Eq. (8)] virtually unaffected. Parameters: nm, and .

Intrinsic contribution and total AH conductivity.—We now report our results for the intrinsic contribution. Previous studies—where the topological nature of the model was also firstly pointed out MEC_Gr_Theory_QAHE_Sun_10 ()—only tackled the problem numerically, also with a focus in the regime . We go beyond this limitation performing an analytic calculation of the intrinsic AH conductivity. Starting from the chiral eigenstates of Eq. (3), we obtain the Berry curvature (BC) of the bands along as , where and is a combined band index CommentIndex (). The transverse conductivity is obtained via integration of the BCs as TKNNFormula (). Note that and , which is the case when is tuned into the gap. The full form of is reported in Ref. supplemMat (), where we also show that the intrinsic contribution can be equivalently obtained from the clean limit of the Kubo–Streda formula. The result is plotted in the inset of Fig. 3(b), while Fig. 3(a) shows the opposite-in-sign BCs for the bands . Similarly to the situation presented for the extrinsic contribution, we find the intrinsic term also presents a peculiar change of sign under the same condition [see Fig. 2 (a)], where is a critical value for the RB strength. The effect in this case is ascribed to the profile of the BCs; in particular, in the electron sector the change of sign happens for solution of the self-consistent equation


where and is the Heaviside step function. In Fig. 3(b) we show the total AH conductivity, given by . Remarkably, we find again , such that our estimate for the AHE reversal energy () in Eq. (8) is still valid (cf. Fig. 3(b) and inset). The robust change of sign in the intrinsic contribution corroborates our previous analysis relating the typical behavior of the extrinsic AH conductivity to intrinsic properties of the model.

The predictions of topological insulating phases in 2D Dirac systems Haldane_88 (); Kane_05 (); MEC_Gr_Theory_QAHE_Sun_10 () have triggered numerous efforts aimed to realize topological fermions in ultra-clean graphene heterostructures. To guide such efforts, it is essential to characterize the response of magnetized Dirac systems to external fields in the presence of disorder corrections, as the current work shows. In this respect, our unified theory—incorporating extrinsic and intrinsic contributions to the anomalous Hall effect on equal footing, and unveiling a skew-scattering mechanism modulated by the rich spin texture of magnetic Dirac states—constitutes a major step towards the understanding of transport phenomena in graphene–magnetic insulator heterostructures.

Acknowledgments.—The authors are grateful to Denis Kochan, Chunli Huang and Mirco Milletarì for useful discussions. We thank Stuart Cavill and Roberto Raimondi for critically reading the manuscript and for helpful comments. A.F. gratefully acknowledges the financial support from the Royal Society (U.K.) through a Royal Society University Research Fellowship. M.O. and A.F. acknowledge funding from EPSRC (Grant Ref: EP/N004817/1).


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Supplementary Information

In the Supplementary Information, we provide additional details on the anomalous Hall conductivity, including a detailed analysis of scattering mechanisms and intrinsic contribution. The equivalence between semiclassical transport theory and Kubo–Streda linear response theory in the diffusive regime is established.

Appendix A Semiclassical Theory

a.1 Linearized Boltzmann Transport Equations

We present the solution of linearized BTEs [Eq. (4) in main text] for intravalley scattering potentials. For brevity, we work at fixed Fermi energy . The scattering probability is given by


where all the quantities appearing in the last equation are defined in the main text. Throughout this supplemental material, we also employ the following definitions


with wavefunction representation as in main text


a.1.1 Exact solution in zero magnetic field

Without loss of generality, we take the electric field oriented along the direction. In the steady state of the linear response regime, the left-hand side of Eq. (4) of the main text becomes (we set )


where is the band velocity of the ring and is its sign, and Comment1 (). To solve the BTEs, we make use of the ansatz Eq. (5) of the main text,


Note that in regime II, only intra-ring processes are allowed, whereas in regime I and III, one needs to take into account inter-ring transitions. For fixed index , we separate intra-ring and inter-ring processes


The different scattering probabilities are


It will be useful in the following to work with the symmetric and antisymmetric components:


We can now use Eq. (5) along with the trigonometric relations


to write


The intra-ring integrals now reduce to


where is the density of states,


The inter-ring integrals are obtained via the a similar procedure. In the following, we define and equally for . The full scattering operator is thus


Equating the coefficients of on the LHS and RHS of the linearized BTEs, we obtain for the steady state:


where we have defined already in the main text


The system of equations can now be closed considering the respective equations for the other channel , i.e.,


The four equations above can be manipulated by summing and subtracting them to identify some common coefficient:


or in matrix form


where we have defined


and analogously for their barred version, obtained from the last two equations by replacing . In our compact notation we have and . Together with the corresponding system at valley we thus identify 16 relaxation rates. In Fig. 1 we report a graphical visualization of the impurity scattering processes and associated rates.

Figure 1: Graphic visualization of the different impurity scattering processes in this model for the different Fermi energy regimes (I, II and III; see main text). Also we report graphically the different relaxation rates mentioned in the main text. Colored dots are to be identified with the indices . When a generic scattering amplitude (grey segment connecting yellow and green dots) is integrated over the angle, it gives rise to different components of depending to which trigonometric function it is contracted with. Combinations of the various components yield the different relaxation rates.

The formal solution of the linear system Eq. (37) gives


as reported in Eq. (6) of the main text.

a.1.2 Finite magnetic field

In the presence of a magnetic field, the LHS of the BTEs read contain the term


In the linear response regime, the contraction with the electric field only selects the equilibrium part , as seen above. On the other hand, the contraction with the magnetic field selects the non-equilibrium part since


It is thus convenient to use a modified ansatz from Eq. (5)


In evaluating the term , we use the relations between cartesian and polar derivates


We thus have (omitting the index in the intermediate steps)


where is the Levi-Civita symbol. Let us expand the derivatives (we use for brevity)