Giant Extrinsic Spin Hall Effect due to Rare-Earth Impurities

We investigate the extrinsic spin Hall effect in the electron gas model due to magnetic impurities, by focusing on Ce- and Yb-impurities. In the dilute limit, the skew scattering term dominates the side jump term. For Ce-impurities, the spin Hall angle $\alpha_{\rm SH}$ due to skew scattering is given by $-8\pi\sin\delta_2/7$, where $\delta_2 (\ll 1)$ is the phase shift for$d (l=2)$ partial wave. Since $\alpha_{\rm SH}$ reaches $O(10^{-1})$ if $\delta_2 \simge 0.03$, the spin Hall effect is anticipated to be considerable in metals with rare-earth impurities. The giant extrinsic SHE originates from the large orbital angular momentum, which is also significant for the intrinsic SHE.


Introduction
The spin Hall effect (SHE) has been attracting a great deal of interest due to its fundamental as well as technological importance. The intrinsic SHE originates from the Berry phase of the multiband Bloch function, so that the spin Hall conductivity (SHC) takes a material-specific value that is independent of impurity scattering in the low resistivity regime [1,2]. In transition metals, the intrinsic SHC takes a considerably large value [3,4], because of the "effective Aharonov-Bohm (AB) phase" induced by d-orbital degrees of freedom and the atomic spin-orbit interaction (SOI) [5,6,7,8]. Intrinsic SHE in Sr 2 RuO 4 had been calculated in ref. [5] based on the t 2g tight-binding model. By the same mechanism, SHE in transition metals had been studied based on the realistic multiorbital tight-binding model in ref. [7] and the first-principles calculations [9]. The obtained SHCs agree well with recent experimental observations, in both the magnitude and sign [8].
In addition to the intrinsic mechanism, SHE is also induced by impurity scattering with the aid of SOI; this phenomenon is known as the extrinsic SHE [10,11,12]. It consists of a skew scattering term (σ ss SH ∝ ρ −1 ) and a side jump term (σ sj SH ∝ ρ 0 ); This fact suggests that considerable large SHC can be realized due to the skew scattering term in the very low resistivity metals, beyond the material-specific intrinsic SHC. For this reason, study of the extrinsic SHE is significant for realizing excellent spintronics device to produce or detect the spin current.
The extrinsic SHE due to nonmagnetic impurities or vacancies had been studied in refs. [10,11,12], by developing the theory of the extrinsic anomalous Hall effect (AHE) [13,14,15]. The obtained SHC is, however, very small unless the enhancement factor for the SOI due to the multiband effect is taken into consideration. For this reason, it is not easy to estimate the magnitudes of σ ss SH and σ sj SH due to nonmagnetic impurities. In transition metals, there is no apparent experimental evidence for a large extrinsic SHE due to natural randomness.
Dilute magnetic impurities such as Fe and Ce have been found to induce a large extrinsic AHE in simple metals [17,16]. The Hall angles in AuFe [17] and rare-earth doped metals [16] reach 0.01∼0.001 under high magnetic fields. Since the SHE has a similar mechanism to that of the AHE, one might expect that a giant SHE can be realized by magnetic impurities. However, theoretical analysis of the extrinsic SHE in a simple s-electron metal due to magnetic impurities has not been performed previously.
In this paper, we study the extrinsic SHE based on a single-impurity Anderson model for Ce and Yb atoms, both of which are typical magnetic impurities. Nonperturbative effect of the large SOI for f -electrons (∼ 3000 K) is correctly accounted for. Due to skew scattering, the obtained spin Hall angle α SH = j s h , which is the ratio between the transverse spin Hall current and the longitudinal electric current, exceeds 0.1 for δ 2 > ∼ 0.03, where δ 2 (≪ 1) is the phase shift for d (l = 2) partial wave; It is more than 10 times greater than that in nonmagnetic metals such as Pt [4]. Since the spin Hall angle represents the efficiency of the creating spin current, giant spin Hall current is expected to emerge in simple metals by introducing dilute magnetic impurities. Moreover, the extrinsic SHC can take a large value well above the Kondo temperature T K , in contrast to the anomalous Hall conductivity (AHC). The origin of the giant SHE is found to be the large SOI and angular momentum in the rare-earth impurities, which had not been treated appropriately in previous studies of the extrinsic SHE.

Model and Hamiltonian
In this study, we use the single-impurity Anderson model for Ce and Yb atoms. In a Ce 3+ ion with a 4f 1 configuration, the J = 7/2 level is about 3000 K higher than the J = 5/2 level due to the strong atomic SOI [18]. Therefore, we consider only J = 5/2 states for the Ce-impurity. In the same way, we consider only J = 7/2 states in Yb 3+ ion with 4f 13 configuration. Here, we introduce the following single-impurity Anderson model for Ce/Yb atoms with both d-and f -orbitals [16]: where we have employed the electron (hole) picture for the Ce (Yb) impurity. c † kσ is the creation operator of a conduction electron with spin σ = ±1. f † M is the creation operator of a f -electron with total angular momentum J = 5/2 (7/2) and z-component σm is the creation operator of a d-electron with angular momentum m(−2 ≤ m ≤ 2) [19]. ε k = k 2 /2m is the energy for the conductionelectrons, and E f (E d ) is the localized f (d)-level energy. V f kM σ and V d km are the mixing potentials, which are given by where a M mσ is the Clebsch-Gordan (C-G) coefficient and Y m l (k) is the spherical harmonic function. We will show that the phase factor in Y m l (k) and a M mσ are indispensable to realize the SHE. [20,21]. Note that the SOI was neglected in the study of the AHE in ref. [16]. In the present study, we neglect the crystalline electric field of the f -orbitals, since it is small due to the small radius of the f -orbital wave function [16]. We puth = 1 hereafter.
To discuss the scattering problem, it is useful to derive an effective Hamiltonian for the conduction electrons by integrating out the f and d electrons in eq. (1). The obtained Hamiltonian is given by [16] where and [20]. According to the scaling theory [22],Ẽ f approaches the Fermi level as the temperature decreases due to the Kondo effect: |J f | is strongly enhanced near T K , and below T K , J f N(0) ≫ 1 due to strong resonant scattering, where N(0) = mk F /2π 2 is the density of state of the conduction band per spin. We also assume that

T -matrix due to c-f Resonant Scattering
In this section, we study the scattering problem. In the both models given by eq. (1) and eq. (4), the T -matrix due to the c-f resonant scattering is equivalent; it is given by where its diagrammatic expression is shown in Fig. 1 (a), N is the number of k-points, and G 0 k = (ε − ε k ) −1 . We note that the term containing both J f kσ,k ′ σ ′ and J d k,k ′ given in Fig. 1 (b) vanishes identically due to the orthogonality of spherical harmonic functions. The solution of eq. (7) for k = k ′ and σ = σ ′ is simply given by where we have used the relations is the local Green function. Assuming approximate particle-hole symmetry near µ, we put g R (0) ≡ g(+iδ) = −iπN(0). Then, the quasiparticle damping rate in the T -matrix approximation is given by where n imp is the impurity concentration. Note that eq. (9) is exact for n imp ≪ 1.

Skew Scattering Term
In this section, we study the skew scattering term using linear response theory. Initially, we consider the case N(0)J f ≪ 1 and J d ≪ J f , where the Born approximation is valid. Figure 1. In analogy to [15,16], the skew scattering term is given by where −e (e > 0) is the electron charge. Its diagrammatic expression is shown in Fig.  2 (a). Here, is the second order term of the T -matrix; the first order term in J f kσ,k ′ σ does not contribute to σ ss SH up to the first order term in J d [16]. Note that any diagram that contains the part shown in Fig. 1 (b) vanishes identically. The retarded Green function is given by G R k (0) = (µ − ε k + iγ) −1 , where γ represents the quasiparticle damping rate. Here, we put where γ 0 is the damping rate due to nonmagnetic scattering, such as c-d scattering (γ d = 5πn imp N(0)J 2 d ) and the scattering due to disorders. The charge current is given by j c ν = −e∂ε k /∂k ν , where ν = x, y. The spin current is then given by j s ν = (σ/2)∂ε k /∂k ν . First, we consider the angular integration in eq. (10), which is given by where f (k,k ′ ) ≡ m 2 k 2 ∂ε k ∂k x ∂ε k ′ ∂k ′ y = sin θ k cos φ k sin θ k ′ sin φ k ′ , and · · · Ω denotes the average over the Fermi level, which is defined as A(k 1 , · · ·,k n ) (k) appears in eq. (13). This integral is finite only when l = 2, 4. Therefore, the interference of the f (l = 3) and d (l = 2) partial waves is essential for skew scattering [16]. After performing the angular integrations, eq. (13) is given as Since γ f = 3πn imp N(0)J 2 f in the Born approximation, σ ss SH is given by Now, we derive the skew scattering term using the T -matrix approximation, which gives the exact result for n imp ≪ 1. In this case, T , where the first order term in J f kσ,k ′ σ has been dropped. The diagrammatic expression for σ ss SH is shown in Fig. 2 (b). The angular integration in the T -matrix approximation can be performed as Thus, σ ss SH in the T -matirx approximation is given by (1 + (πN(0)J f ) 2 ) −1 times eq. (14).
Therefore, the expression for the skew scattering term in eq. (15) is valid beyond the Born approximation, by considering γ f given by eq. (9). Here, we discuss the temperature dependence of σ ss SH . From eq. (15), σ ss SH is proportional to 1/γ f when γ f ≫ γ 0 . Since γ f ∝ − ln (T /W band ) for T ≫ T K [18], σ ss SH decreases as the temperature decreases. In contrast, σ ss SH ∝ γ f for γ f ≪ γ 0 , which increases at low temperatures if γ 0 is constant.

Side Jump Term
In this section, we briefly discuss the side jump term σ sj SH . In the T -matrix approximation, σ sj SH is given by [15] σ sj whose diagrammatic expression is shown in Fig. 3. In contrast to the skew scattering term, J d is not necessary for the side jump term. As reported in ref. [21], the large anomalous velocity v a kM σ , which is not perpendicular to the Fermi surface, arises from the k-derivative of the phase factor in the c-f mixing potential as follows: The final expression for the side jump term in the T -matrix approximation is given by where we have used the relations M σ Mσ|V f kM σ | 2 = 3|V f | 2 (1 − sin 2 θ) and M σ σ 2 |V f kM σ | 2 = 6|V f | 2 . By using eqs. (15) and (19), the ratio σ sj SH /σ ss SH is given by 28 3π γ J d n . Therefore, σ sj SH will exceed σ ss SH in dirty metals. However, the skew scattering contribution will be dominant in simple metals with magnetic impurities for n imp < ∼ 0.01 (ρ 0 < ∼ 10µΩcm) [16].

Origin of the Skew Scattering Mechanism
In section 3.2, we have studied the SHC due to skew scattering by using the Green function method. In this section, we discuss the origin of the skew scattering mechanism based on the Boltzmann transport theory. For this purpose, we study a simplified two-orbital model with M = ±5/2, assuming the strong crystalline electric field. In this model, (k) ∝ e −3iσφ k approximately, the second order term of T f is simply given as T f (2)R ∼ −iN(0)e −3iσ(φ k −φ k ′ ) . In the Boltzmann transport theory, the spin Hall resistivity due to skew scattering is ρ ss w represents the scattering probability, which is proportional to due to Fermi's golden rule in the present model. According to ref. [16], skew scattering occurs when the scattering probability includes an assymetric component In the present model, w ss arises from the interference of the f and d scattering channel, w ss ∈ T . In summary, a conduction electron with σ hybridizes with l z = −3σ state due to the strong SOI, and therefore the spin-dependent skew scattering probability w ss (kσ → k ′ σ) ∝ Ime iσ(φ k −φ k ′ ) arises from the interference of the f and d angular momenta. Thus, the origin of the SHE due to skew scattering mechanism is well understood based on the simplified two-orbital model.

Estimations of the Spin Hall angle and the Skew Scattering Term
First, we estimate the magnitude of the spin Hall angle tan α SH ≡ σ ss SH σ xx 2ē h . The longitudinal conductivity is given by σ xx = e 2 n 2mγ , where n = k 3 F /3π 2 is the density of the conduction electrons. Then, the spin Hall angle is given by Since tan δ 2 = −πN(0)J d [18], eq. (20) can be rewritten as tan α SH = − 8π 7 In the case where γ f ≫ γ 0 , α SH reaches ∼ 0.3 for δ 2 ∼ 0.1 [24]. Therefore, giant SHE will be realized in metals with rare-earth impurities.
Recently, giant SHE was observed in FePt/Au devices [23]. It was reported that the large spin Hall angle in Au is ∼ 0.1. Since the intrinsic SHE in Au is small due to the Fermi surface being mainly composed of s-electrons [7], the observed giant SHE may be explained by the skew scattering mechanism. Very recently, the extrinsic SHE in AuFe metal has been studied with the first-principles band structure calculations independently of our study [25].
Second, we estimate the magnitude of the skew scattering term for δ 2 = 0.1, where the spin Hall angle is 0.3. Then, σ ss SH = (h/2e)0.3 × σ xx from the above discussion. In the case where the resistivity is 10µΩcm, we obtain σ ss SH = 15000(h/e)Ω −1 cm −1 if we put the length of unit cell a = 4Å, which is the case for Pt and Au. The obtained skew scattering term is about 60 times larger than the SHC observed in Pt: 240(h/e)Ω −1 cm −1 . Since the resistivity in Pt and Au is ρ ∼ 10µΩcm and ρ ∼ 1µΩcm, respectively, the skew scattering term due to magnetic impurities should dominate the intrinsic term. In contrast, the skew scattering due to "natural randomness" has not been observed in the AHE in transition metals, even in ρ ∼ 10µΩcm [26]. Therefore, the orbital degrees of freedom of the impurity is indispensable for the giant extrinsic SHE.

Summary
In summary, we have studied the extrinsic SHE due to magnetic impurities, where Ce and Yb impurities have been discussed as a typical case. The obtained expression for the skew scattering term shows that the spin Hall angle reaches O(10 −1 ), which is more than 10 times larger than that in Pt. Therefore, we propose that the extrinsic SHE due to rare-earth impurities is useful as a method for creating large spin current efficiently. The present study also suggests that the giant SHE will arise from other rare-earth atoms as well as 4d and 5d atoms that possesses strong SOI. As in the case for the intrinsic SHE, orbital degrees of freedom are essential for generating a huge extrinsic SHE. According to eq. (20), giant spin Hall angle always appears when γ f ≫ γ 0 . Thus, the Kondo resonance, N(0)J f ≫ 1, is not a necessary condition for the giant SHE. In contrast, the AHC (= σ xy /H z ) is enhanced by the Kondo effect since it is proportional to the uniform susceptibility.