Anomalous Hall effect in magnetized graphene: intrinsic and extrinsic transport mechanisms approaching the quantized regime Manuel Offidani1, ∗ and Aires Ferreira1, †
arXiv:1801.07713v1 [cond-mat.mes-hall] 23 Jan 2018
1
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 [2 (spin) × 2 (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 Cr2 Ge2 Te6 [1] and CrI3 [2], while graphene and group-VI dichalcogenide monolayers acquire large exchange splitting when integrated with nanomagnets [3–9]. 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 [10–14]. The room temperature AHE observed in Gr on YIG thin films [4, 5] shows that the proximity MEC is accompanied by a sizable Rashba-Bychkov (RB) effect due to interfacial breaking of inversion symmetry [15]. Fundamentally, the RB effect entangles pseudospin (Σ) and spin (s) degrees of freedom [16, 17], endowing Bloch eigenstates with a rich spin texture in reciprocal k-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 [11, 18]. When the Fermi level is tuned inside the gap, this system is predicted to exhibit the quantum anomalous Hall effect (QAHE), with transverse conductivity σAH = 2 e2 /h [19]. However, much less is known about the non-quantized regime at finite carrier densities. The latter is the current experimental accessible regime [4, 5]. 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 [20–23]. 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 (E) and magnetic (B) 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 [4, 5], 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 interfaceinduced SOC and exchange interaction are modeled by a 2D Dirac Hamiltonian at low energies (we set ~ ≡ 1), H = v τz Σ · p + δ sz + λ τz (Σx sy − Σy sx ) + V (x) , (1) where v is the Fermi velocity of massless Dirac fermions, δ and λ are, respectively, the MEC and RB energy scale, and V (x) is a disorder potential describing impurity scattering. Here, p = −i∇ is the 2D kinematic momentum operator for states near valley K/K 0 (τz = ±1). Finally, {τ , Σ, s} are Pauli matrices acting on valley, pseudospin,
2 (a)
(e)
(b)
VH > 0
spin-flip force
VH < 0
spin-conserving
force
✏
(c) III
kx
(d)
II I QAHE
Sz
=
⌫=1 ⌫= 1
Sy
band index
=1
1
ring index
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 µ = ±1 or a ring index χ = ±1. (e) Sign of anomalous Hall conductivity σ⊥ resulting from competing effective spin Lorentz forces, with the elastic channel dominating above at energies b > a .
and spin spaces, respectively [24].PWe consider a disorder potential of the form V (x) = R2 i (u0 + ux τx ) δ(x−xi ), where {xi }i=1...N are random impurity positions and R their effective radius [25]. This allows us to interpolate between the important regimes of dominant intravalley scattering characteristic of ultra-clean samples (u0 ux ) and the “sharp defect” limit (ux ≈ u0 ) [26]. The energymomentum dispersion relation associated to clean Hamiltonian H0 = H − V (x) is easily computed as p µν (k) = µ v 2 k 2 + Mν2 (k) , (2) where k = |k| is theq wavevector measured from a Dirac p point and Mν (k) = 2λ2 + δ 2 + ν λ4 + v 2 k 2 (λ2 + δ 2 ) is a “SOC mass” (see Fig. 1). {µ, ν} = ±1 define, respectively, the carrier polarity (electron/hole) and spin chirality i.e., spin winding direction. When δ 6= 0, λ = 0 (no SOC), the Dirac cones are shifted vertically, resulting in mixed electron–hole states near the Dirac point. On the
∂t fkχ − e (E + v × B) · ∇k fkχ = 2πni
other hand, if λ 6= 0, δ = 0 (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) [16]. Finally, finite SOC and MEC opens a gap and splits the Dirac spectrum into 3 branches: regions I p 2 + δ 2 ≡ < || < = |δ| λ and III, defined by |λδ|/ I II p and || > 4λ2 + δ 2 ≡ III ; those energy regimes are characterized by a non-simply connected FS allowing for scattering between states with different Fermi momenta; and region II II < || < III with only one band intersecting the Fermi level. Hereafter, unless stated otherwise, all functions are projected onto valley τz = 1 (K point); respective expressions for K 0 are obtainable by the simple transformation v, λ → −v, −λ. The unnormalized Bloch eigenstates of the uniform part of Eq. (1) read as e−iφk
2
|ψµνk (x)i =
2 2
(µν −δ) −v k 2vkλ µν −δ vk ( −δ)2 −v 2 k2 iφk i µν 2 λ(µν +δ) e
i
ik·x , e
(3)
where φk is the wavevector polar angle. The skyrmionlike 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 z below, the k-depence of the polarization Sµνk ≡ 12 hsz iµνk 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 (S z > 0) and spin-down (S z < 0) states. For brevity, we focus on positive energies, > 0, and also λ, δ > 0, thus fixing µ = 1 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 [27]. 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
X Z Sd2 k0 fk0 0 Tk0 0 kχ − fkχ Tkχ k0 0 δ(kχ − k0 0 ) , 2 χ χ χ χ (2π) 0
(4)
χ =±1
where fkχ = fk0χ + δfkχ is the sum of the Fermi-Dirac distribution function and δfkχ , the deviation from equilibrium; e is the elementary charge and S is the area. The right-hand side is the collision term describing sin-
gle impurity scattering and ni is the impurity areal density. Subscripts χ, χ0 = ±1 are ring indices, denoting the outer and inner FS, respectively, associated with Fermi momenta k± = v −1 {2 + δ 2 ± [2 λ2 + (2 − λ2 )δ 2 ]1/2 }1/2 ;
3 see Fig. 1(d) [28]. Accounting for possible scattering resonances due to the Dirac-like spectrum [20], the quantum-mechanical transition rates are evaluated em2 0 ploying the T -matrix approach, i.e., Tkk , R 0 2= |hk |t|ki| 2 where t = V/ (1 − g0 V). Here, g0 = d k/(4π ) ( − H0k + i 0+ )−1 and V = R2 (u0 + τx ux ). We start by considering ux = 0 (no intervalley scattering). In this case, electrons undergo intra- and inter-ring scattering processes on the same valley (see SM [33] for a graphical visualization). Exploiting the isotropy of the Fermi surface, and momentarily setting B = 0, the exact solution to the linearized BTEs (∇k fkχ → ∇k fk0χ ) is δfkχ = −e
∂fk0χ ∂
!
vkχ · τχk E + τχ⊥ zˆ × E
,
(5)
with vkχ = ∇k χ (k). In the above, τχς = τχς (, λ, δ, u0 , ni ) are the longitudinal (ς =k) and transverse (ς =⊥) transport times given by −1 ˆ +Υ ˆ −1 Υ ˆ −1 1 , (6) ˆΛ ˆ ˆΛ τ k = −2 Λ 1 , τ ⊥ = τ k 1t Υ ˆ = ((Λ− , Λ+ )t , (Λ− , −Λ+ )t ), and where τ ς = (τχς , τχ¯ς )t , Λ χ χ χ ¯ χ ¯ ˆ The kernels Λ± ≡ Λ± (Γχχ , Γχχ¯ , vχ , vχ¯ ) similarly for Υ. χ χ are functions and skew cross sections R of2 symmetric t ni Γχχ0 = 2π Sd k0 Tk0 0 kχ {1, cos φ, sin φ} , where φ = χ φk0 − φk , vχ = |vkχ | and χ ¯ ≡ −χ. Considering the corresponding expressions for states near K 0 , 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 [33], including a magnetic field B = B zˆ only requires the −1 sin substitution Γsin χχ → Γχχ + ωχ , where ωχ = kχ vχ B is the cyclotron frequency associated with the ring states. At T = 0, and accounting for the valley degeneracy, we obtain the longitudinal and transverse conductivities as
σς (B, ) =
e2 X kχ () vχ () τχς (, B) , h χ=±1
(7)
where σς=⊥ includes the ordinary Hall background. The external field provides the magnetic thin film (e.g., YIG) with finite magnetization Mz (B), hence to relate the Dirac model Eq. 1 to the experimental realizations in Refs. [4, 5] one has to consider δ = δ(Mz (B)). At saturation B ≥ Bsat the anomalous Hall (AH) signal—extracted by subtracting the Hall background to σ⊥ —attains its maximum value and pleataus at increasing field [4, 5]. One can then extract σAH by setting zero external field (no Hall background) and defining δsat = δ(Mz (Bsat )), such that σ⊥ (0, )|δsat ' σAH (Bsat , )|δ . The change of sign.—Focusing on the regime λ . δ, we show how, approaching low carrier density, electrons undergoing spin-conserving and spin-flip scattering pro-
1
1 0
0.5
-1
0
−2 −3
−0.5 −1
0.1
−1.5
0
−2
−0.1
−2.5
−0.2
0
0.2
0.4
0.6
0.8
1
0
10
0
0.5
30
20
1
40
1.5
50
2
Figure 2. Typical behavior of σ⊥ . (a) σ⊥ () and (b) σ⊥ (ne ) 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 λ. To plot (b) we used the 2 2 2 2 2 relation ne = π{k+ − k− , k+ , k+ + k− } 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. δ = 30 meV, u0 = 0.36 eV, ux = 0, R = 1 nm and n0 = 1012 cm−2 .
cesses determine a change of sign in σ⊥ . For the moment, we work in the weak scattering limit |g0 u0 R2 | 1, and assume one can restrict the analysis to intra-ring tran0 (see additional sitions within the outer ring: k+ → k+ discussions [33]). A first scenario for the change of sign is as follows. First, we note that as k is increased from k = 0, electron states in the lower band ν = −1 progressively change their spin orientation from effective spin-up to -down states (see Fig. 1). Starting from I , 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 [33]). Remarkably, the change of sign in σ⊥ 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 ¯ using spin-up and -down states ”S” = ” ↑, ↓ ” = −”S” a gate voltage, and, on the other, both spin-conserving ¯ h”S”|V (x)|”S”i and spin-flip h”S”|V (x)|”S”i scattering
4 matrix elements are non-zero. Asymptotically, δ, λ, the AH signal σ⊥ vanishes due to the opposite spin orientation of electron states belonging to ν = ±1 bands, which produce a net zero magnetization. It is useful to note that in the strong scattering limit, |g0 u0 R2 | 1, 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 [29], enhancing the backscattering probability [26]. Recent studies of spin precession in graphene with interface-induced SOC showed that the in-plane spin dynamics is strongly affected by intervalley scattering [30–32]. To determine how the AHE is affected by atomically sharp potentials, we solved Eq. (4) for arbitrary ratio ux /u0 . Figure 2(c) shows the AH conductivity for selected values of ux (dashed lines). σ⊥ is reduced to nearly half of its previous value, when the intravalley potential strength achives its maximum value (ux = u0 ). However, the sign change in σ⊥ , approaching the majority spin band edge ≈ II is clearly visible. Further analysis are given in SM [33], where we also analyze the impact of thermal fluctuations, concluding the features described above are persistent up to kB T ≈ kB Troom /2 ' 12 meV for λ ≈ δ = 30 meV. A careful numerical analysis provides an estimation for 0 defined as σ⊥ (0 ) = 0, 0 = a I + b III ,
(8)
with a ' 0.3–0.4 and b ' 0.6–0.8. This relation shows that the knowledge of δ (e.g., from the Curie temperature [4]) allows to estimate the SOC strength directly from the gate voltage dependence of the AH resistance. We comment in passing on the magnitude of σ⊥ ≈ 0.1 − 1 (e2 /h) is compatible to the measurements in Refs. [4, 5], for a reasonable choice of parameters, 0 ≤ λ, δ ≤ 30 meV, ni = 1012 cm2 , R ∼ 1 nm and u0 ∼ (0, 1) eV. Note that for even cleaner samples, σ⊥ ∝ n−1 i will attain even larger values within the diffusive regime. 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 [11]—only tackled the problem numerically, also with a focus in the regime δ λ. 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 Ωnk = (∇k × Ank )z , where Ank = −ihnk|∇k |nki and n ≡ (µ, ν) is a combined band index [37]. The transverse conductivity P P isn obtained int 0 via integration of the BCs as σ = ⊥ n k Ωk fkn [35]. P n P P n 2 Note that n Ωk = 0 and k n λc ≈ δ/6 [see Fig. 2 (a)], where λc 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 = ˜0 solution of the self-consistent equation 2e2 , h (9) R b() b where Ii |a () ≡ a() dk k Ωik and θa ,b = θ (a − b ) is the Heaviside step function. In Fig. 3(b) we show the tot int total AH conductivity, given by σ⊥ = σ⊥ + σ⊥ . Remarkably, we find again ˜0 ' 0 , such that our estimate for the AHE reversal energy (0 ) 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 [11, 38, 39] 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 insuk
k
k
(˜ 0 ) + θ˜0 ,II I1 |0+ (˜ 0 ) + θ˜0 ,III I2 |0− (˜ 0 ) = − I1 |k+ −
5 lator 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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[26]
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[27] The formalism is valid in the semiclassical limit kF l 1, where kF is the Fermi momentum and l is the mean free path, and provides the leading order extrinsic contribution σ⊥ ∝ n−1 [22]. i [28] In region I, the two states |k± i belong to the same Bloch band µ = −1, whereas in region III they respectively populate both µ = ±1 bands. In region III instead, the only-populated state χ = 1 coincide with the Bloch eigenstate |ψµ=−1k (x)i. [29] Z. H. Ni, et al. Nano Lett. 10, 3868 (2010). [30] L. A. Benítez, et al. Nat. Physics (2017). [31] A. Cummings, J. H. García, J. Fabian and S. Roche. Phys. Rev. Lett. 119, 206601 (2017). [32] T. S. Ghiasi, J. I.-Aynés, A.A. Kaverzin, B. J. van Wees. Nano Lett. 17, 7528 (2017). [33] See supplemental material appended below. [34] A. Crepieux and P. Bruno, Phys. Rev. B 64, 014416 (2001). [35] D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982); N. P. Ong and W. L. Lee, Foundations of Quantum Mechanics in the Light of New Technologies, 121 (2006); N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald and N. P. Ong, Rev. Mod. Physics 82, 1539 (2010). F. D. M. Haldane, Rev. Mod. Phys. 89, 040502 (2017). [36] P. Streda, J. Phys. C: Solid State Phys. 15 , L1299 (1982). [37] We choose to assign t he label n in descending order, starting from the band of highest energy. Therefore our convention is such that ν, µ = ±(1, 1) → n = ±2 and ν, µ = ±(1, −1) → n = ±1. [38] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988). [39] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). [40] We have exploited the fact that the bands of interest are isotropic. [41] The superscripts I, II (italic) are not to be confused with the indices I, II ( roman) indicating the different regimes in the band structure, as depicted in Fig. 1 of the main text. [42] N. A. Sinitsyn et al., Phys. Rev. B 75, 045315 (2007). [43] D. M. Basko. Phys. Rev. B 78, 115432 (2008). [44] A. Pachoud, A. Ferreira, B. Özyilmaz, A. H. Castro Neto. Phys. Rev. B 90, 035444 (2014).
6
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 kF l 1 is established. CONTENTS
References
5
I. SEMICLASSICAL THEORY A. LINEARIZED BOLTZMANN TRANSPORT EQUATIONS 1. Exact solution in zero magnetic field 2. Finite magnetic field B. TRANSVERSE SCATTERING CROSS SECTION: ELASTIC AND SPIN-FLIP CHANNELS 1. Scattering in region I and II: intra-ring transition and main contribution from χ = 1 ring 2. Spin-conserving and spin-flip Lorentz force from the collision integral C. ADDITIONAL DISCUSSIONS 1. Thermal fluctuations 2. Intervalley scattering 3. Magnetic field
7 7 7 9 11 11 11 12 12 12 13
II. SKEW SCATTERING: EQUIVALENCE BETWEEN SEMICLASSICAL AND KUBO-STREDA FORMALISM A. Linear response formalism 1. Disorder averaged Green’s function 2. Renormalized charge current vertex: solution of the BS equation and symmetry arguments
13 14 15 16
III. INTRINSIC CONTRIBUTION TO THE AHE 1. Kubo-Streda approach 2. Berry-curvature calculation
17 17 20
7 I. A.
SEMICLASSICAL THEORY
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 > 0. The scattering probability is given by 2 (1) Wkχ ,k0 0 = 2πni Tkχ k0 0 δ(kχ − k0 0 ) = 2πni hkχ0 0 |t|kχ i δ(kχ − k0 0 ) , χ
χ
χ
χ
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 (dk) = d2 k/4π 2 ,
γ0 ≡ I = τ0 Σ0 s0 ,
γKM = τ0 Σz sz ,
and γr = τz (Σ × s) · zˆ,
(2)
with wavefunction representation as in main text (A↑ K, A↓ K, B ↑ K, B ↓ K, B ↑ K 0 , B ↓ K 0 , A↑ K 0 , A↓ K 0 )t .
1.
(3)
Exact solution in zero magnetic field
Without loss of generality, we take the electric field oriented along the x ˆ direction. In the steady state of the linear response regime, the left-hand side of Eq. (4) of the main text becomes (we set e = 1) ! ! ∂fk0χ ∂fk0χ 0 − E · ∇k fkχ = −E · vkχ = −ςv |E| vkχ cos φk , (4) ∂ ∂ where vkχ is the band velocity of the χ ring and ςv is its sign, cos φk = kx /k and fk0χ = (1 + Exp[(kχ − )/kB T ])−1 [40]. To solve the BTEs, we make use of the ansatz Eq. (5) of the main text,
δfkχ = −
∂fk0χ
= −ςv E
!
∂ ∂fk0χ ∂
τχk E + τχ⊥ zˆ × E · vkχ !
vkχ τχk cos φkχ + τχ⊥ sin φkχ .
(5)
(6)
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 χ, we separate intra-ring (χχ) and inter-ring (χχ) ¯ processes S fkχ = S intra fkχ + S inter fkχ , (7) Z S intra fkχ = − (d2 k0 ) fkχ Wkχ ,k0χ − fk0χ Wk0χ ,kχ , (8) Z (9) S inter fkχ = − (d2 k0 ) fkχ Wkχ ,k0χ¯ − fk0χ¯ Wkχ¯ ,k0χ , The different scattering probabilities are Wkχ ,k0χ = 2πni δ(kχ − k0χ ) Tχχ (φ) ,
(10)
Wkχ ,k0χ¯ = 2πni δ(kχ − k0 χ¯ ) Tχχ¯ (φ) ,
(11)
Wk0χ ,kχ = 2πni δ(k0χ − kχ ) Tχχ (−φ) ,
(12)
Wk0χ¯ ,kχ = 2πni δ(k0χ¯ − kχ ) Tχχ ¯ (−φ) .
(13)
8 It will be useful in the following to work with the symmetric and antisymmetric components: s a Tχχ0 (±φ) = Tχχ 0 ± Tχχ0 .
(14)
We can now use Eq. (5) along with the trigonometric relations cos (α + β) = cos α cos β − sin α sin β ,
(15)
sin (α + β) = sin α cos β + sin β cos α ,
(16)
to write fk0χ = −ςv E vkχ
∂fk00 h χ
∂
i cos φkχ τχk cos φ + τχ⊥ sin φ + sin φkχ τχ⊥ cos φ − τχk sin φ .
(17)
The intra-ring integrals now reduce to Z intra I1 ≡ (d2 k0 )fkχ Wkχ ,k0χ Z dφ = 2πni fkχ N (kχ ) Tχχ (φ) , 2π I2intra
Z ≡
(20) Z
dφ cos φ Tχχ (−φ) + c2,kχ 2π
= 2πni Fχ vkχ N (kχ ) c1,kχ 1 k 2π vk
(19)
(d2 k0 )fk0χ Wk0χ ,kχ
where N (k ) =
(18)
Z
dφ sin φ Tχχ (−φ) , 2π
(21)
is the density of states,
{c1,kχ , c2,kχ } = {cos φkχ τχk + sin φkχ τχ⊥ , cos φkχ τχ⊥ − sin φkχ τχk } ,
and Fχ = −ςv E k
∂fk0χ ∂
.
(22)
k
The inter-ring integrals are obtained via the a similar procedure. In the following, we define (τχ , τχ¯ ) = (τ k , τ¯k ) and equally for τχ⊥ , vχ , N (kχ ), Fχ . The full scattering operator is thus Z Z Z dφ dφ dφ s s a k k ⊥ S fkχ = − 2πni Fχ N v cos φkχ τ T −τ cos φ Tχχ − τ sin φ Tχχ 2π χχ 2π 2π Z Z Z dφ dφ dφ s s a T − τ⊥ cos φTχχ + τk sin φTχχ + sin φkχ τ ⊥ 2π χχ 2π 2π Z Z Z dφ s ¯ dφ ¯ dφ inter k k ¯v a ⊥ ¯v a S fkχ = − 2πni Fχ v cos φkχ τ N T − τ¯ N cos φTχχ ¯ N sin φ Tχχ ¯ −τ ¯ 2π χχ¯ v 2π v 2π Z Z Z dφ dφ dφ s s a ¯ v¯ ¯ v¯ T − τ¯⊥ N cos φTχχ ¯k N sin φTχχ . − sin φkχ τ ⊥ N ¯ +τ ¯ 2π χχ¯ v 2π v 2π intra
(23) (24) (25) (26)
Equating the coefficients of cos φk , sin φk on the LHS and RHS of the linearized BTEs, we obtain for the steady state: v¯ 0 ⊥ sin −1 = τ k Γ0χχ − Γcos ¯k Γcos ¯⊥ χχ + Γχχ ¯ + τ Γχχ − τ ¯ +τ v χχ v¯ 0 k sin 0 = τ ⊥ Γ0χχ − Γcos ¯⊥ Γcos ¯k χχ + Γχχ ¯ − τ Γχχ − τ ¯ −τ v χχ
v¯ sin Γ¯ , v χχ v¯ sin Γ¯ , v χχ
(27) (28)
where we have defined already in the main text (0,cos,sin) Γχχ0
Z = 2πni N
χ0
dφ {1, cos φ, sin φ} Tχχ0 (φ) . 2π
(29)
9 The system of equations can now be closed considering the respective equations for the other channel χ, ¯ i.e., 0 k v cos Γ + τ⊥ −1 = τ¯k Γ0χ¯χ¯ − Γcos + τ¯⊥ Γsin χ ¯χ ¯ + Γχχ ¯ χ ¯χ ¯−τ v¯ χχ¯ 0 ⊥ v cos 0 = τ¯⊥ Γ0χ¯χ¯ − Γcos − τ¯k Γsin Γ − τk χ ¯χ ¯ + Γχχ ¯ χ ¯χ ¯−τ v¯ χχ¯
v sin Γ , v¯ χχ¯ v sin Γ . v¯ χχ¯
(30) (31)
The four equations above can be manipulated by summing and subtracting them to identify some common coefficient: ¯+ , ¯ − + τ ⊥ Υ+ + τ¯⊥ Υ −2 = τ k Γ− + τ¯k Γ ¯ + + τ ⊥ Υ− − τ¯⊥ Υ ¯− , 0 = τ k Γ+ − τ¯k Γ ¯+ , ¯ − − τ k Υ+ − τ¯k Υ 0 = τ ⊥ Γ− + τ¯⊥ Γ ¯ + − τ k Υ− + τ¯k Υ ¯− , 0 = τ ⊥ Γ+ − τ¯⊥ Γ
(32) (33) (34) (35)
or in matrix form ¯− Γ− Γ ¯+ Γ+ −Γ ¯+ −Υ+ −Υ ¯ −Υ− Υ−
¯ + τ k −2 Υ+ Υ ¯ − τ¯k 0 Υ− −Υ ¯− τ ⊥ = 0 Γ− Γ ¯+ 0 τ¯⊥ Γ+ −Γ k ˆ Υ ˆ Γ 1 τ ⇐⇒ =−2 , ˆ ˆ 0 τ⊥ −Υ Γ
(36)
(37)
where we have defined v cos 0 Γ± = Γ0χχ − Γcos , χχ + Γχχ ¯ ± Γ χχ v¯ ¯ v sin Υ± = Γsin , χχ ± Γχχ v¯ ¯
(38) (39)
and analogously for their barred version, obtained from the last two equations by replacing χ → χ. ¯ In our compact t t notation we have τ k , τ ⊥ = (τ k , τ¯k , τ ⊥ , τ¯⊥ )t , and (1, 0) = (1, 0, 0, 0)t . Together with the corresponding system at K 0 valley we thus identify 16 relaxation rates. In Fig. 1 we report a graphical visualization of the impurity scattering processes and associated rates. The formal solution of the linear system Eq. (37) gives −1 ˆ +Υ ˆ −1 Υ ˆΛ ˆ τ k = −2 Λ 1, −1 ˆ +Υ ˆ −1 Υ ˆ −1 1 , ˆΛ ˆ ˆΛ τ ⊥ = −2 Λ Υ
(40) (41)
as reported in Eq. (6) of the main text.
2.
Finite magnetic field
In the presence of a magnetic field, the LHS of the BTEs read contain the term k˙ · ∇k fkχ = − (E + vk × B) · ∇k fk0χ + δfkχ .
(42)
In the linear response regime, the contraction with the electric field only selects the equilibrium part fk0χ , as seen above. On the other hand, the contraction with the magnetic field selects the non-equilibrium part since (vk × B) · vk = 0 . It is thus convenient to use a modified ansatz from Eq. (5) δfkχ = vkχ τχk cos φk + τχ⊥ sin φk ,
(43)
(44)
10 regime II
regime I
χ=1 χ=1
inter-ring transitions
χ = −1
intra-ring transitions
=
regime III
χ=1
= χ = −1
=
−
+
=
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 Λ± χ , Υχ mentioned in the main text. Colored dots are to be identified with the indices χ. When a generic scattering amplitude (grey segment connecting yellow and green dots) is integrated over the angle, it gives rise to different components of Γχχ0 depending to which trigonometric function it is contracted with. Combinations of the various components yield the different relaxation rates.
In evaluating the term ∇k δfkχ , we use the relations between cartesian and polar derivates ∂ sin φk ∂ ∂ − = cos φk , ∂kx ∂k k ∂φk ∂ cos φk ∂ ∂ = sin φk + . ∂ky ∂k k ∂φk We thus have (omitting the index χ in the intermediate steps) h i (vk × B) · ∇k δfk = abc vka Bb ∂c vk τ k cos φk + τ ⊥ sin φk ,
(45) (46)
(47)
where abc is the Levi-Civita symbol. Let us expand the derivatives (we use ∂ki → ∂i for brevity) h i 1. ∂x vk τ k cos φk + τχ⊥ sin φk = cos2 φk ∂k vk τ k + cos φk sin φk ∂k vk τ ⊥ + + τ k vk ∂x cos φk + τ ⊥ vk ∂x sin φk , h i 2. ∂y vk τ k cos φk + τχ⊥ sin φk = sin φk cos φk ∂k vk τ k + sin2 φk ∂y vk τ ⊥ + + τ k vk ∂y cos φk + τ ⊥ vk ∂y sin φk ,
(48)
(49)
where we assumed an isotropic Fermi surface k k v. The latter equations, using Eqs. (45)-(46), can be written as τk τ⊥ 1. cos2 φk ∂k vk τ k + cos φk sin φk ∂k vk τ ⊥ + vk sin2 φk − vk sin φk cos φk , k k τk τ⊥ 2. sin φk cos φk ∂k vk τ k + sin2 φk ∂k vk τ ⊥ − vk sin φk cos φk + vk cos2 φk . k k
(50) (51)
11 Taking a perpendicular magnetic field B = B zˆ, one obtains after standard algebraic manipulations vkχ × B · ∇k δfkχ = vkχ ωBχ τχk sin φkχ − τχ⊥ cos φkχ ,
(52)
where we reinstated the index χ and defined ωBχ =
vkχ B. kχ
(53)
It is clear from Eq. (27)-(28), that the effect of the magnetic field can be reabsorbed in the definition of the skew cross sections: χ sin Γsin χχ → Γχχ + ωB ,
(54)
to which the trivial generalization of Eqs. (37) follows. Note however, due to the slight different ansatz we have used, t t the column of the know terms −2(1, 0) has to be generalized to −2Fχ (1, 0) .
B.
TRANSVERSE SCATTERING CROSS SECTION: ELASTIC AND SPIN-FLIP CHANNELS
We discuss the effective “spin Lorentz forces” presented in the main text (spin-conserving and spin-flip), which are responsible for the change of sign of the extrinsic transverse response. We also justify how the outer ring χ = 1 yields the dominant contribution to the Hall conductivity in the weak scattering limit.
1.
Scattering in region I and II: intra-ring transition and main contribution from χ = 1 ring
The starting point is the expression for the single-valley Hall conductivity at T = 0, σ⊥ =
e2 X kχ vχ τχ⊥ . 2h χ=±1
(55)
It is obvious that the change of sign is due to the behavior of the transverse scattering times τχ⊥ . In Fig. 2 we show a ⊥ ⊥ |. comparison between the two τχ⊥ where inter-ring transition are neglected. Both τχ⊥ change sign, although |τ+ | > |τ− −1 The outer ring is also associated with a larger density of states N+ > N− , where Nχ = kχ |2πvkχ | , as displayed in ⊥ ⊥ Fig. 2(b). The larger τ+ and N+ as compared to their χ = −1 counterparts motivate our discussion in the main text concerning the change of sign of σ⊥ focused on k+ → k+ transitions. Supporting also this choice is the fact that in the χ = 1 ring the out-of-plane spin polarization S z can change sign, as displayed in Fig. 2(c) and (d).
2.
Spin-conserving and spin-flip Lorentz force from the collision integral
Having demonstrated the important role of intra-ring χ=1 processes, let us get a physical picture behind the change of sign in σ⊥ approaching the spin majority band. The transverse relaxation time, in the absence of a magnetic field, is given by ⊥ τ+ =
(Γ0++
−
Γsin ++ cos Γ++ )2
2 + (Γsin ++ )
.
(56)
⊥ 0 and hence the change of sign of τ+ is controlled by the antisymmetric part of T++ (φ) = |hk+ |t|k+ i|2 . The T matrix can be decomposed according to the following form
t = t0 γ0 + tKM γKM + tz sz + tm Σz + tr γr .
(57)
In Eq. (57) all the terms are associated with diagonal matrices in spin space, except the Rashba term Tr γr . The latter is indeed what connects orthogonal states in spin space. The resulting terms in Γsin ++ give rise to the effective spin-conserving and spin-flip Lorentz forces, as described in the main text. In Fig. 3 we plot the modulus square of
12 0.6
Inner ring
0.5
Outer ring 0.4 0.3 0.2 0.1 0
10
20
30
40
50
0.5 Outer ring
Inner ring
-0.5 -1
0.5
Lorem ipsum
Figure 2. (a) Transverse scattering time in two cases of interest λ ≷ δ, where the larger and smaller energy scales are, ⊥ respectively, 10 and 30 meV. Despite both displaying a change of sign, τ+ is dominant. (b) Also the density of states of the outer ring is generally larger (here λ > δ). (c) Finally it is proven the change of sign of the out-of-plane spin polarization S z happens within the ring χ = 1. Parameters: u0 = 0.36 eV, R = 1 nm and ni = 1012 cm−2 .
the scattering amplitude T++ (φ) as a function of the scattering angle and for different values of the Fermi energy lying in region I or II.
C.
ADDITIONAL DISCUSSIONS 1.
Thermal fluctuations
The finite temperature response is obtained from Z +∞ 0 ∂f (, µ, T ) T T =0 σ⊥ (µ) = − d σ⊥ () , ∂ −∞
(58) −1
T =0 where σ⊥ () is the zero-temperature response and f 0 (, µ, T ) = (1 + Exp [(0 − ) /kB T ]) . Fig. (4) shows the temperature dependence of the anomalous Hall conductivity for typical parameters. The characteristic change of sign is visible up to temperatures T ≈ Troom /2.
2.
Intervalley scattering
To account for intervalley scattering, we use the following simplified model for the disorder potential with site C6v symmetry V (x) =
Ni X i=1
R2 (u0 γ0 + ux τx )δ(x − xi ) ,
(59)
13
0.15 3 0.1 2 0.05 1 0 0 −0.05 0
0.2
0.4
0.6
0.8
0
1
0.2
0.4
0.6
0.8
0.4
0.6
0.8
1
0.6
0.05 0.4
0
0.2
0
− 0.05 0
0.2
0.4
0.6
0.8
1
0
0.2
1
Figure 3. Effective spin-conserving and the spin-flip Lorentz forces. Parameters: u0 = 36 meV, R = 1 nm and ni = 1012 cm−2 .
which interpolates between a pure intravalley potential (ux = 0) and a atomically-sharp potential with ux = u0 (e.g., a resonant adatom) leading to strong intervalley scattering [43, 44]. The intervalley term (ux ) introduces new rates Z sin Γχζ = Γ0χζ , Γcos , Γ = 2πn (d2 k0 ) {1, cos φ, sin φ} Tχζ (φ) , (60) i χζ χζ where ζ = ±1 is an index associated with outer and inner ring respectively for states at K 0 valley, obtainable from Eq. (3) of the main text by performing the substitution λ, v → −λ, −v. We plot the result in Fig. 4. The intervalley scattering results in a reduction of the total transverse relaxation time.
3.
Magnetic field
The formalism at finite magnetic field [Sec. I A 2] allows us to model various quantities of interest in graphene/thin film heterostructures. Fig. 5 shows the anomalous Hall resistivity ρAH as function of the applied magnetic field. To reproduce a typical hysteresis loop for magnetized graphene, the effective exchange coupling is chosen as δeff = δ tanh (B/Bsat ), where δ = 30 meV, and with the saturation field Bsat = 1.5 × 103 Gauss [4, 5].
II.
SKEW SCATTERING: EQUIVALENCE BETWEEN SEMICLASSICAL AND KUBO-STREDA FORMALISM
We evaluate the disorder correction to the anomalous Hall conductivity by means of linear response theory. The results are shown to agree with the BTEs to numerical accuracy.
14 2 0.1 1 0
0
−1
−0.1
−2
−0.2 10
20
30
40
50
10
60
20
30
40
50
Figure 4. Effect of thermal fluctuations and intervalley scattering. (a) Change of sign in σ⊥ is distinguishable for T . Troom /2, where Troom = 300 K. (b) In the less favorable scenario (ux = u0 ), the transport times τ k,⊥ () are reduced by a factor ≈ 2, while keeping their basic functional dependence with . Parameters: u0 = 3.6 eV, R = 1 nm and n = 1012 cm−2 .
200 100 0 −100 −200 −10
−5
0
5
10
Figure 5. ρAH as a function of the magnetic field for different Fermi energies . Parameters: λ = 10 meV, u0 = 1 eV, R = 1 nm, and ni = 1012 . Also δeff = δ tanh (B/Bsat ), δ = 30 meV, Bsat = 1.5 × 103 Gauss [5].
A.
Linear response formalism
The longitudinal (Drude) and transverse (Hall) responses are denotes as σxx and σyx , respectively. The transverse I II () , where response is calculated using the Kubo-Streda formalism [41], σyx () = σyx () + σyx Z I σyx () = (dk) tr vy GkR ()˜ vx ()GkA () , (61) Z Z dG R () dG R () II σyx () = − dω (dk) Re tr vx k vy GkR () − vx GkR ()vy k , (62) d d −∞ where R/A
Gk
−1 () = − H0k − ΣR/A ,
(63)
are disorder averaged Green’s functions associated with the Hamiltonian Eq. (1) of the main text and ΣR/A = ni tR/A is the self energy. Here, tr is the trace over internal degrees of freedom τ, Σ, s and vx,y = ∂kx,y H0k = τz σx,y are the bare current operators. v˜x is the renormalized current vertex, obtained from the Bethe-Salpeter (BS) equation X v˜x () = vx + ni tr tR () GkR () v˜x () tA () GkA () . (64) k
15 I II σyx , σyx are the so-called “Fermi surface” and “Fermi sea” contribution to the Hall response. The semiclassical (skew scattering) contribution originates from the leading disorder correction to the former, scaling as σ⊥ ∝ τ ⊥ ∝ n−1 [22]. i
1.
Disorder averaged Green’s function
We provide the explicit form of the disorder averaged Green’s function for intravalley scattering potentials (ux = 0). To obtain those, one first needs to calculate the disorder averaged T matrix R/A
tR/A = hV (1 − g0
V )−1 idis ,
(65)
R R/A R/A where is the momentum-integrated bare Green’s function g0 = (d2 k) G0k . Explicit calculation yields R,A 1 1 R,A R,A 2 2 2 2 2 2 2 2 ( γ0 − δ sz ) L+ + λ + δ γ0 − δ − λ sz + δ λ γm + λ γKM − λ − δ γr L− . g0 = − 2 D (66) with D =
p
2 δ 2 + λ2 (2 − δ 2 ) . Also we have defined Lp± = Lp1 ± Lp2 (p = R/A = ±) with (67) Lp1,2
1 = 2 2 p , v k − z1,2
p z1,2 = 2 + δ 2 ± 2 D + i p 0 + .
(68)
From Eq.(66) we see that beyond the identity and the matrix structures γr , sz , already present in the bare Hamiltonian, we obtain two additional terms γm = Σz , γKM = Σz sz , which will appear in the self-energy and thus in the disorder averaged Green’s functions. R/A
To obtain an analytical expression for Gk R/A appearing in g0 , namely
(), we consider an effective model containing all the matrix structures
Heff = H0 + m γKM + λm γm .
(69)
The disorder averaged Green’s function of the original Hamiltonian Eq. (1) in the main text for ux = 0 is obtained by taking the bare Green’s function of Eq. (69) and performing the analytical continuations AR/A : → − ni ( ± iη0 ) ,
(70)
λ → λ + ni (∆λ ± iηr ) , δ → δ + ni (∆δ ± iηz ) ,
(71)
m0 → ni (∆m ± iηKM ) , λm → ni (∆λm ± iηm ) ,
(72)
where λi , ηi denote respectively the real and imaginary parts of one component of the T matrix associated with the matrix γi . We then find X Gp = (M0,ξ + Mφk ,ξ ) Lξ , (73) p ξ=± A
meaning that the analytical continuation Ap has to be done to find the p = R/A sectors. Above, expressing the ˜ ,M ˜ jk = 1/4 tr Mσ ˜ j sk , we have explicitly matrices in the notation M = τi ⊗ M
16
M0,+
M0,−
τ0 0 =− 2 0 λm
0 0 0 0
−δ 0 , 0 m
0 0 0 0
(74)
λ2 + δ 2 + m λm δ − λ2 τ0 0 =− ˜ 0 2Γ λ2 (λm + δ) − m ( δ + m λm ) i δ h 2 2 − (λm + δ) − ( − m) γr , ˜ 2Γ
0 0 0 0
0 δ 2 (λm + δ) − (m λm + δ) 0 0 0 0 2 2 2 0 λ − δ λm − m λ + λm (75)
and
0 0 0 0 τz vk cos φk λ sin 2φk −λ cos 2φk 0 Mφk ,+ = − 2 vk sin φk −λ cos 2φk −λ sin 2φk 0 0 0 0 0 0 vk ( − m) λ sin φk −vk ( − m) λ cos φk 0 τ0 0 0 0 0 Mφk ,− = − ˜ 0 0 0 0 2Γ 0 vk λ (λm + δ) sin φk −vk λ (λm + δ) cos φk 0 0 0 0 0 2 2 2 2 2 2 2 2 τz cos 2φ −vk (mλ sin 2φ −λ + δ − m − λ 0 λ + δ − m − λ k m + δ) cos φk k m m − 2 2 2 2 2 2 2 2 ˜ sin 2φ −vk (mλ cos 2φ −λ + δ − m − λ 0 −λ + δ − m − λ k m + δ) sin φk k 4Γ m m 0 0 0 0
(76)
(77)
with ˜= Γ
r
h i 2 2 2 δ 2 (m − ) − (λm + δ) + (m λm + δ)
˜ L± = v 2 k 2 − 2 + δ 2 − m2 − λ2m ± 2Γ
2.
−1
.
(78) (79)
Renormalized charge current vertex: solution of the BS equation and symmetry arguments
The recursive BS equation Eq. (64) for the charge vertex produces new matrix structures, which are associated to observables that can be “excited” upon the application of external fields. We show now how these observables can be predicted by simple symmetry arguments. The continuum model Eq. (1) of the main text is invariant under the following operation C2 : τx sz eiπlz
(80)
Λz : τz
(81)
Λx : τx Σz
(82)
Λy : τy Σz .
(83)
C2 is the rotation of π around the zˆ-axis, and lz = −i∂φ the generator of the rotation in coordinate space. Reflection around x ˆ: Rx : Σx sy rx , with rx : (x, y) → (x, −y) , and time reversal operation Θ, present in the continuum limit of bare graphene, are broken symmetries in this model; we can refer to the transformations above as pseudosymmetries Θ−1 H (δ) Θ = H (−δ) Rx−1
H (δ) Rx = H (−δ) .
(84) (85)
17 To constraint the number of observables with finite (nonzero) expectation value, we examine the form of a generic response function A Rα = Tr γα GR (86) 0 vx G 0 , for an observable associated with the operator γα . Above Tr includes the trace in momentum space. Exploring any of the symmetries S listed in Eqs. (80)-(83) S : GR,A → GR,A . 0 0
(87)
Eq. (86) can be manipulated as follows † † Sγα S † S GR Svx S † S GA 0S 0S = x α Tr γα GR vx GA ,
Rα = Tr
(88) (89)
where = ±1 is the parity of some operator under S. From the last equation, we see that a non-zero response requires the operator γα to have the same parity of the current vertex vx under the action of S. We have C2 : vx → −vx Λx,y,z : vx → +vx .
(90) (91)
By spanning the 64-dimensional algebra τi σj sk , we conclude that only eight responses are allowed τ0 Σ0 sx,y : spin-x,y (spin-Galvanic) magnetisation
(92)
τ0 Σz sx,y : staggered spin-x,y polarization
(93)
τz Σx, s0,z : x-charge current (Drude) and spin-z x-current
(94)
τz Σy s0,z : charge y-current (Hall) and z-spin y-current (spin Hall) .
(95)
We verified this is confirmed by explicit diagrammatic calculation. Solution of Eq. (64) then requires the inversion of a 8 × 8 matrix. The solution can be written in the form X v˜x = vxi γi , (96) i
where γi as given above. By inserting the renormalized vertex in Eq. (96) into Eq. (61), one can finally find the skewscattering contribution to the Hall response. Similarly one can calculate the Drude response by replacing vy → vx in Eq. (61). Figure 6 shows a comparison between the Drude and transverse response obtained using the BTEs and the present formalism.
III.
INTRINSIC CONTRIBUTION TO THE AHE
In this section we give details about the calculation of the intrinsic contribution to the AHE. This can be done in two equivalent ways: via a direct Berry-curvature calculation or using the clean limit of the Kubo-Streda formula. We show they provide the same result.
1.
Kubo-Streda approach
We start here by computing the intrinsic contribution within the Kubo-Streda formalism. In practice, one needs to evaluate Eqs. (61)-(62) in the clean limit, which corresponds to the substitution for the Green’s functions Gka → Ga0k .
18
1250 0.5 1000 750
0
500
−0.5
250 −1 0
10
20
30
40
50
60
10
20
30
40
50
60
70
Figure 6. Comparison between Kubo–Streda (diagrammatic) and Boltzmann calculation for both the longitudinal and transverse response in a representative case u0 = 0.36 eV, R = 1 nm. Also ni = 1012 cm−2 . It can be seen the agreement between the two formalisms is excellent.
Using the expression presented above for the latter, we can extract the (single valley) σ I contribution as 2(3λ2 δ 2 +δ 4 −22 (λ2 +δ 2 )) √ region I 4 2 2 (λ +3λ δ +δ 4 −2 (λ2 +δ 2 )) 2 (λ2 +δ 2 )−λ2 δ 2 2 λ λ δ 1 2+ √ I σyx () = − 2 (λ2 +δ 2 )−λ2 δ 2 2 √2 2 2 2 2 region II 2 +δ 2 + λ (λ +δ )−λ δ 2(2λ2 +δ 2 ) region III λ4 +3λ2 δ 2 +δ 4 −2 (λ2 +δ 2 ) .
(97)
II we need instead to calculate the derivative of the Green’s functions with respect to the For the Fermi sea term, σyx energy variable. By doing that, performing the trace over internal indices and angular integration we arrive at
II σyx () = 64 λ2 δ lim Im η→0
Z
Z dω
−∞
0
∞
v2 k2 − ω2 dk k 2 2 2π (v k − z1R )2 (v 2 k 2 − z2R )2
Starting form Eq. (98) we want to perform integration over energies of the integrand g(k, ω) =
R ω
.
(98)
first. Doing partial fraction decomposition
64 λ2 δ (v 2 k 2 − ω 2 ) , (v 2 k 2 − z+ )2 (v 2 k 2 − z− )2
we look for the poles in ω. Obviously we get the eigenvalues q p µν (k) = ν v 2 k 2 + 2λ2 + δ 2 + µ v 2 k 2 (λ2 + δ 2 ) + λ4 ,
(99)
(100)
displaced by the small imaginary part iη. Note the following relations hold for the eigenvalues ++ = −+− ,
(101)
−− = −−+ .
(102)
19
k+
k
k+
region III
k
region II
k+
region I
Figure 7. Schematic of band structure in the hole sector and different regimes. Depending on the position of the Fermi level the integration over momentum variables has to be performed between different boundaries as reported in details in Table I.
Let us write the partial fraction decomposition in the form g(k, ω) =
Aω + B (ω − −1 + iη)
2
+
Cω + D 2
(ω − −2 + iη)
+
Eω + F (ω + −2 + iη)
2
+
Gω + H (ω + −1 + iη)
2
,
(103)
where we have labeled the eigenvalues according to the prescription for the contracted index n adopted in the main text −1 = +−
(104)
−2 = −− ,
(105)
which are respectively the lower and the upper branches of the spectrum on the hole side (see Fig. 7). The eigenvalues appear as poles of second order. As no small η is left in the denominator of the ω-independent coefficients A, B, C, ..., we can focus on the imaginary part of the ω integration. We basically need to solve two classes of integrals Z (1, ω) Ja,b = dω (106) 2 . (ω − n + iη) −∞ sWe have −1 −1 P Ja = = =− + iπδ ( − n ) , ω − n + iη −∞ − n + iη − n Z − n ω − n + iη − iπθ ( − n ) + n Ja , Jb = dω 2 + n Ja = log Λ (ω − n + iη) −Λ
(107) (108)
where in the last integral we have considered a regularization UV cutoff Λ. When taking the imaginary part we have Ja = iπ
X δ (k − kχ ) , vχ χ=±1
X δ (k − kχ ) Jb = −iπ θ ( − n ) − vχ χ=±1
(109) ! ,
(110)
where we have used vkχ ≡ vχ . The structure of the integrals Eq. (107)-(108) is important and allows to read already at this stage the quantization of the transverse conductivity. To illustrate that, let us take negative energies, where electronic states can populate bands labeled with −1, −2. We can see now how the terms proportional to θ, δ in Eqs. (107)-(108) will contribute differently depending on the position of . In particular we have the following table
20 I θ ( − −1 ) = R∞ 0
dk k g(k)
θ ( − −2 ) =
R∞ − Rk∞
k+
II
dk k g(k) dk k g(k)
δ ( − −1 ) = k− g(k− )/v− δ ( − −2 ) = k+ g(k+ )
R∞ R0∞ k+
III R∞
dk k g(k) dk k g(k)
gap R∞
dk k g(k) dk k g(k) R∞ R0∞ dk k g(k) + k+ dk k g(k) 0 dk k g(k) 0
R k− 0
0 k+ g(k+ )
0 k− g(k− ) + k− g(k− )
0 0
Table I. Table summarizing the momentum integrals in the different regimes.
The relevant observation is that inside the gap the δ-parts do not contribute, as does not intersect any band. Only θ-parts are left, with the integration extending from 0 to ∞. According to the partial fraction decomposition of Eq. (103), we see, by using Eqs. (98), (108), we are left with need to calculate Z ∞ 1 II σyx dk k (A + C) . (111) (|gap ) = 2π 0 In this respect we can identify A, C ≡ Ak , Ck as the Berry curvatures of the lower and upper band respectively. We discuss below how they exactly match a direct calculation of the respective Berry curvatures. The analytic expression for the sum A + C is quite complicated. However, explicit calculation shows (re-establishing explicitly the units and considering the contribution of the K 0 valley) II σyx (|gap ) =
2e2 . h
(112)
II acquires a energy dependence due to non-quantized θ-parts and If the Fermi level lies instead outside the gap, σyx a finite contribution of the δ-parts. However, we verified the δ-parts cancel exactly opposite in sign contributions in I (). A similar cancelation has been reported for a massive Dirac band model in Ref. [42]. We conclude the intrinsic σyx contribution is a result of the θ-parts only. Finally note that for positive Fermi we can simply extract the result from int int what we have obtained in the hole sector; we have in fact σyx () = σyx (−), which is a direct consequence of the Berry curvatures of the bands summing to zero.
2.
Berry-curvature calculation
We provide details now on a direct calculation of the intrinsic contribution via the well-known TKNN formula XX int σyx () = Ωnk n0 (n (k)) , (113) n
k
where the band-BC is Ωnk = (∇k × Ank ) · zˆ stemming from the Berry-connection Ank = −ihnk|∇k |nki. Let us focus again on the two bands in the hole sector. Performing the calculation in polar coordinates, we find the Berry-connection only contains the angular components: λ2 δ δ 4 + 2λ2 (2v 2 k 2 + 3S + 3λ2 ) + δ 2 (3v 2 k 2 + 3S + 4λ2 ) −1 Ak φ = , (114) 3/2 (S 2 (v 2 k 2 + 2S + δ 2 + 2λ2 )) λ2 δ δ 4 + 2λ2 (2v 2 k 2 − 3S + 3λ2 ) + δ 2 (3v 2 k 2 − 3S + 4λ2 ) −2 Ak φ = − . (115) 3/2 (S 2 (v 2 k 2 + 2S + δ 2 + 2λ2 )) We verified the latter expressions exactly agree respectively with the coefficients A, C of the previous section. It follows the results match those from the Kubo-Streda formalism presented earlier.
