Orbital Torque: Torque Generation by Orbital Current Injection

We propose a mechanism of torque generation by injection of an orbital current, which we call $\textit{orbital torque}$. In a magnetic bilayer consisting of a nonmagnet (NM) and a ferromagnet (FM), we consider a situation where the spin-orbit coupling (SOC) is present only in the FM. Although the SOC is absent in the NM, the orbital Hall effect can arise in the NM. When the resulting orbital Hall current is injected to the FM, the SOC of the FM converts the orbital angular momentum into spin, which exerts torque to the magnetization of the FM. Remarkably, even for small SOC strength comparable to that of $3d$ FMs, the orbital torque can be comparable to the spin torque induced by the spin Hall effect of the NM with strong SOC. This provides a way to experimentally probe the OHE and opens a new venue to achieving spin-torque devices based on light elements that exhibit gigantic orbital response. Experimental implications are discussed.

Similar to the SHE, the orbital Hall effect (OHE) allows for electrical generation of a transverse orbital current. In transition metals, for example, electron wavefunctions near atomic cores have mainly d character, and superpositions such as d zx ± id yz carry the orbital angular momentum L z = ±h. A flow of wavepackets with such superposed wavefunctions generates an orbital current. Considering that an orbital current carries the angular momentum just like a spin current does, it is reasonable to expect that injection of an orbital current (or orbital injection in short) into a FM may generate a torque on local magnetic moments of the FM. We call such torque as orbital torque (OT), which provides an experimental way to detect the OHE. Although the OHE has not yet been experimentally verified, theoretical calculations [19,20] on 4d and 5d transition metals indicate that the orbital Hall conductivities (OHCs) of these NM's are about an order of magnitude larger than the spin Hall conductivities (SHCs). Moreover, our recent theoretical analysis finds that the OHC can be gigantic σ OH ∼ 10 4 (h/2|e|)(Ω · cm) −1 even in materials with negligible SOC [1,22]. Thus the OT also provides a new venue to achieving high torque efficiency in spintronic devices.
In this Letter, we investigate the theoretical idea of the OT for a NM/FM bilayer structure (Fig. 1). When an in-plane electric field E is applied, both OHE and SHE arises in the NM in general [1,19,20,22]. In order to focus on the OT due to the orbital injection, we suppress the SHE by setting the SOC of the NM zero. Then only OHE is induced and a resulting torque in the FM can be identified unambiguously as the OT. We find that the OT indeed arises as long as the SOC of the FM is finite.
For a quantitative evaluation of the OT, we adopt the tightbinding description of the bilayer with N NM (N FM ) atomic . We assume both NM and FM to have the simple cubic structure. For the NM, we adopt the sp model that has been used previously [1] to illustrate the OHE without the SOC. In this model, each lattice site can host s, p x , p y , and p z orbitals, and the orbital hydridization, which is crucial for the emergence of the OHE [1], arises from the symmetry-allowed nearest neighbor hoppings between s and p x,y,z orbitals. For the FM, we adopt a trivial d model; each lattice site can host d xy , d yz , d zx , d z 2 , and d x 2 −y 2 orbitals with nearest neighboring hoppings allowed. This d model does not allow any orbital hybridization [23] and thus there is no OHE [1,22]. The d model is augmented by adding the SOC and the exchange coupling H FM xc = (J/h)M · S, where L is the orbital angular momentum of d character states in the FM, S is the spin, andM denotes the magnetization direction of the FM. Below, we focus on the caseM =ẑ. At the interface, the nearest neighbor hoppings exist between the sp orbitals in the NM and the d orbitals in the FM. Details of the tight-binding description are given in Ref. [24]. All parameters of the NM and FM are set to have typical energy scale of nonmagnetic and magnetic metals. In particular, we set α FM so = 100 meV, which is a typical SOC strength of 3d transition metals [22,25,26]. We emphasize that the nonzero α FM so is crucial for the OT sinceM couples only to S and there is no direct coupling betweenM and L in the Hamiltonian. Thus for the injected orbital current to generate the OT, it should be first converted to spin through H FM so and then the resulting spin can generate the torque through H FM xc (Fig. 1). Figure 2(b) shows the band structure of the NM/FM bilayer for N NM = 8 and N FM = 2, where the color represents the equilibrium expectation value of the spin-orbit correlation in the FM region L · S FM eq for each state. The correlation is negative in the lower energy range (−1.1 eV < E nk < −0.7 eV), and positive in the higher energy range (−0.3 eV < E nk < +0.2 eV). In the middle energy range (−0.7 eV < E nk < −0.3 eV), states with positive and negative correlations coexist. We later demonstrate that L · S FM eq is important for the sign of the OT.
For E = E xx , we calculate the electrically generated orbital (L) and spin (S) accumulations at each atomic layer as a function of z. From the Kubo formula, one obtains the expectation value X(z) = X(z) intra + X(z) inter (X = L or S) generated by E x , where are the intraband and interband contributions, respectively.
Here, X(z) = P (z)XP (z) measures the local accumulation of X at z, where P (z) is the projection operator to the atomic layer at z. In Eq. (2), e > 0 is the unit charge,h is the Plank constant, v x is the velocity operator along thex direction, f nk is the Fermi-Dirac distribution function for a periodic part of the Bloch state |u nk with its energy eigenvalue E nk . To incorporate the effect of disorder scatterings, we phenomenologically introduce a spectral broadening Γ = 25 meV, which is a room temperature scale. Figure 3 shows the (a) y and (b) x components of the resulting L(z) and S(z) for the Fermi energy E F = −0.9 eV. We first consider a situation when the NM and the FM are disconnected (hoppings between the NM and FM turned off). In the NM (1 ≤ z ≤ 20), L y (z) [white inverted triangles in Fig. 3(a)] has nonzero values of opposite signs at the opposite edges of the NM (z = 1 and 20). This result can be interpreted as the orbital accumulation at the edges due to the OHE in the NM. The OHC in the NM is σ OH ≈ 2, 000 (h/2|e|)(Ω·cm) −1 for E F = −0.9 eV [1,24]. On the other hand, L x (z) [white inverted triangles in Fig. 3(b)] is absent in the NM. In the FM (21 ≤ z ≤ 30), both L x (z) and L y (z) are zero, confirming the absence of the OHE in the FM. The spin accumulation is zero both in the NM and the FM (not shown), which is natural since the SHE is absent in both NM and FM.
Next we connect the NM and the FM (hoppings between the NM and FM turned on). Near z = 1, which is far from the NM/FM interface, L y (z) [blue circles in Fig 3(a)] remains essentially unchanged. Near the interface (z = 20), on the other hand, L y (z) is reduced significantly since the orbital Hall current is now injected into the FM instead of getting accumulated at the interface. The injected orbital Hall current in the FM produces not only L y (z) but also S y (z) [orange squares in Fig 3(a) for 10× enlarged values] due to H FM so . Moreover once S y (z) becomes nonzero in the FM, the spin precesses aroundM due to H FM xc and produces S x (z) as well [orange squares in Fig 3(b) for 10× enlarged values]. This precession results in oscillatory profiles of S y (z) and S x (z) in the FM, which resemble oscillatory spin accumulation profiles [9] in a conventional situation, where a spin current is injected into a FM to generate the ST. The oscillatory profiles of S x (z) and S y (z) in the FM are accompanied by similar oscillatory profiles of L x (z) and L y (z) . The coexistence of the spin and orbital accumulation oscillations is due to H FM so and we note that the spin and orbital oscillations are 180 • out of phase for E F = −0.9 eV (Fig. 3), which we attribute to negative spin-orbit correlation at this energy [ Fig. 2(b)]. By the way, the spin accumulation in the NM is due to partial reflection of the orbital Hall current at the NM/FM interface.
The torque T acting on the FM can be obtained from the spin accumulation as follows, where S FM = z∈FM S(z) . When the SOC of the NM is zero, T arises from the orbital injection and thus the resulting T amounts to the OT. Analogous to the ST, the OT can be decomposed as T = τ fM ×ŷ + τ dM × (M ×ŷ), where τ f(d) refers to the field(damping)-like component. WhenM =ẑ, τ f = (J/h) S y FM and τ d = −(J/h) S x FM . We find that S y(x) (z) , which arises from the intraband(interband) contribution in Eq. (2), is even(odd) inM, thus the field(damping)like OT is odd(even) under sign reversal ofM. This is similar to the generation of the field-like and damping-like STs when a spin current polarized alongŷ direction is injected into a FM magnetized along theẑ direction [9].
Since τ d plays a more important role for the currentinduced magnetization dynamics than τ f [6,7], we focus on S x FM . A result for S y FM is given in Ref. [24]. Figure 4(a) shows that the ratio S The E F -dependence of the relative ratio sign closely resembles the energy dependence of the spin-orbit correlation L · S FM eq in Fig. 2 the sign of S x FM /E x tends to be determined by the sign of the product between the OHC of the NM and the spin-orbit correlation in the FM. Considering that S x FM determines the damping-like OT, the latter tendency may be regarded as the OT counterpart of the sign "rule" for the ST; the dampinglike ST tends to be determined by the sign of the spin Hall conductivity (SHC) in the NM [6,7,9].  Fig. 4(a) (denoted by the yellow star and the red cross). For these favorable choices of E F , values of τ d /E x are −0.08 ea and +0.05 ea for α FM so = 100 meV, which is SOC energy scale for 3d FMs. Here a is the lattice constant. By increasing α FM so , they reach up to −0.22 ea and +0.09 ea for α FM so = 200 meV, which is SOC energy scale for 4d transition metals. Note that these values τ d /E x ∼ 0.1 ea for α FM so = 100, 200 meV are not negligible compared to the corresponding value ∼ 0.5 ea for the damping-like torque calculated for the Pt/Co bilayer [18,27] with the SOC strength of 500 meV for Pt. Then, considering that the OHC in real materials such as V is gigantic σ OH ∼ 12, 000 (h/2|e|)(Ω · cm) −1 , which is about 6 times larger than the OHC of the sp model used in our calculation, τ d /E x for real NMs may be proportionally larger and comparable to the corresponding ST value for the Pt/Co bilayer. Although quantitative predictions on τ d /E x require realistic calculations that take material details into account, we argue it is still reasonable to expect that the OT may be sizable for a FM with weak SOC, thus providing an alternative route to enhancing the torque efficiency.
So far we have assumed the SOC is absent in the NM. Now we consider a situation where not only the FM but also the NM have the SOC. Thus the sp model Hamiltonian for the NM now includes where α NM so is the SOC parameter in the NM. Since s character states do not carry the orbital angular momentum, L in Eq. (S23) acts only on p character states. Due to H NM so , the NM exhibits SHE, as well as the OHE. Thus, on top of the OT, injection of the spin Hall current into the FM generates the ST. It is known that OHE and SHE occur in the same(opposite) direction if L · S NM eq is positive(negative) at E F [1,19,20]. Thus, when L · S FM eq > 0 at E F , which is a case for Fe, Co, and Ni, the OT and ST add up if L · S NM eq > 0 and cancel each other if L·S NM eq < 0. This situation becomes the opposite when L · S FM eq < 0. The result is summarized in Table I, which is supported by our numerical calculation [24].
Lastly we discuss experimental implications. In a magnetic bilayer consisting of only light elements the OHC is much larger than the SHC, thus the total torque is dominated by the OT as in Fig. 1, which is advantageous for unambiguously quantifying the OT. Several groups have reported nonnegligible effective spin Hall angles for bilayers that consist only of light elements with weak SOC such as V [28], Cr [29,30], Py [31], CuO x [32,33], and AlO x [34]. These experimental results may be related to the OT.
When the NM in a magnetic bilayer consists of heavy elements, the OT and ST coexist (Table. I). Unfortunately, since the OT and ST have same symmetry, one should go beyond symmetry analysis for their separation. We expect that the orbital Hall current injection into a FM can be sizable only when atomic ordering near the NM/FM interface is sufficiently good, since even spin-conserving scatterings can suppress the orbital angular momentum [24]. We thus expect that the orbital information carried by the orbital Hall current is likely to be more easily destroyed by interface disorder than the spin information carried by the spin Hall current is. This may provide a possible experimental method to differentiate the OT and the ST. For instance, in bilayers with the same sign of the OT and the ST, the NM/FM interface quality enhancement will increase the OT and the total torque. This implies that the total torque may go even beyond the level expected from the theoretically predicted SHC of a NM in favorable situations. Sizable increase of the total torque has been reported for Pt/Co [35,36] upon the interface quality enhancement. Values above theoretically predicted SHCs are also reported for Pt/Co [37] and Au 0.25 Pt 0.75 /Co [38]. Since L · S NM eq > 0 and L·S FM eq > 0 in these bilayers and the OHC is larger than the SHC in Pt [19,22], we suspect that these experimental re-L · S FM eq > 0 L · S FM eq < 0 L · S NM eq > 0 same sign opposite signs L · S NM eq < 0 opposite signs same sign sults may also be related to the OT. Meanwhile, it has also been suggested that the orbital degree of freedom in Co plays an important role for the increased torque in Pt/Co [39]. On the other hand, in bilayers with opposite signs of the OT and the ST, the interface quality enhancement will increase the OT and reduce the total torque. We expect that Ta/Co and W/Co may exhibit this type of behavior. The OT may be probed also through material variation of a FM. This is in clear contrast to present experimental efforts focused on the material variation of a NM while the material choice for a FM is mostly fixed to 3d FMs with weak SOC. According to Fig. 4(b), the OT is enhanced in FMs with strong SOC (Gd for instance). Then in bilayers with opposite signs of the OT and the ST, the total torque may even have a sign that is opposite to the sign expected from the SHC sign of a NM.  References S10

A. Tight-Binding Model
The tight-binding model for a magnetic bilayer presented in the Letter is composed of a nonmagnet (NM) and a ferromagnet (FM). The numbers of layers for the NM and the FM are N NM and N FM , respectively. We assume the simple cubic structure for both NM and FM with only nearest neighbor hoppings allowed. We also assume that the layer is periodic in x and y directions, and the layers are stacked along z direction. Thus, the NM is located from z = 1 to z = N NM and the FM is located from z = N NM + 1 to z = N NM + N FM (in unit of the lattice spacing a), and we use the Bloch theorem for x and y directions by introducing the crystal momentum k = (k x , k y ). The total Hamiltonian is formally written as where H 2d NM(FM) (k) is the Hamiltonian for a two-dimensional NM(FM) layer, T NM(FM) is the hopping between nearest NM(FM) layers, and T int is the interface hopping between the last NM layer (z = N NM ) and the first FM layer (z = N NM + 1).

NM
We assume the NM hosts sp α (α = x, y, z) orbitals at each site, which was introduced in Ref.
[S1]. Writing the Hamiltonian in a finite film structure is straightforward as follows. The Hamiltonian within each two-dimensional NM layer consists of the kinetic energy and spin-orbit coupling (SOC) parts:

S2
First, the kinetic part is where and I 2×2 is an identity operator in the spin space. Here, the basis states are where |φ lσR is a Wannier function localized at the Bravais lattice R = (R x , R y ) with its orbital character l = s, p x , p y , p z and spin σ, which is defined in a layer located at z. For the Wannier states, E s , E pα are onsite energies for s and p α orbitals, and t s , t pσ(π) , γ sp are the nearest hopping amplitudes between s orbitals, between p orbitals via σ(π) bonding, and between s and p orbitals, respectively. Second, the SOC part is where S is the spin operator and L (p) is the orbital angular momentum (OAM) operator in p orbital space. Here, α NM so > 0 is the strength of the SOC in the NM. The OAM operator is explicitly expressed in a matrix representation with p x , p y , and p z orbital Wannier functions. Finally, the interlayer coupling between neighboring NM layers is described as where the basis states for the row and column are ϕ l ′ σ ′ k , respectively, for z = 1, · · · , N NM − 1.

FM
In the FM, we assume there are d β (β = xy, yz, zx, x 2 − y 2 , z 2 ) orbitals at each site. The Hamiltonian within each twodimensional layer is where each term describes kinetic energy, SOC, and exchange interaction with magnetization, respectively. The kinetic energy term is

S3
where E dxy (k) = E dxy + 2t dπ cos(k x a) + 2t dπ cos(k y a), (S11a) E dyz (k) = E dyz − 2t dδ cos(k x a) + 2t dπ cos(k y a), (S11b) E dzx (k) = E dzx + 2t dπ cos(k x a) − 2t dδ cos(k y a), (S11c) Here, E d β is the onsite energy of the d β orbital, and t dσ , t dπ , t dδ are nearest neighbor hoppings between d orbitals via σ, π, δ bondings, respectively. The basis states are defined similarly as Eq. (S5) but for d β orbital Wannier functions. The SOC term is where α FM so > 0 is the SOC strength. Here, L (d) is the OAM operator in d orbital space, whose matrix representation is written as where the basis states are d xy , d yz , d zx , d x 2 −y 2 , d z 2 orbital Wannier functions. The exchange interaction is where J > 0 is the strength of the exchange interaction, andM is the direction of the magnetization. We assumeM =ẑ in the calculation. The interlayer coupling between neighboring FM layers is where the basis for the row and column are ϕ l ′ σ ′ k , respectively, for z = N NM + 1, · · · , N NM + N FM − 1.

Interface
At the interface, there are hoppings between the last NM layer (z = N NM ) and the first FM layer (z = N NM + 1), which are expressed in where the basis for the row and column are ϕ (NNM+1) lσk | and |ϕ (NNM) l ′ σ ′ k , respectively. Here, γ pdσ(π) is the nearest neighbor hopping between p and d orbitals via σ(π) hoppings. We neglect the hopping from a s orbital in the NM to d orbitals in the FM, since the s orbital does not carry the OAM, thus not affecting the orbital injection.

Parameter Setting
For the tight-binding model defined above, parameters which we used for the calculation in Figs. 2 and 3 of the Letter are set as for the NM, for the FM, and for the interface. All parameters are expressed in unit of eV.

B. Spatial Profiles of the Orbital and Spin Hall Currents
Linear responses of the orbital and spin Hall currents are evaluated using the Kubo formula as follows: where P (z) is the projection operator to a layer at z and is the orbital(spin) current operator for X = L(S). The velocity operator along the z direction is defined as (S22) Figure S1 shows spatial profiles of the orbital and spin Hall currents obtained from the tight-binding model introduced in Sec. . The parameters are set as Eqs. (S17)-(S19), and the numbers of the NM and FM layers are N NM = 20 and N FM = 10. For this calculation, we set the Fermi energy as E F = −0.9 eV. We find that the orbital Hall conductivity in the NM region is more than ≈ 2, 000 (h/2|e|)(Ωcm) −1 . In the FM region, part of the orbital Hall current is injected, which is converted to the spin current by the SOC of the FM [Eq (S12)]. We also find the spin current in the NM region, which is decaying from the interface. This is because reflected current from the interface becomes spin-polarized. The decay is due to finite spectral broadening Γ = 25 meV in Eq. (S20).
. The E F -dependence of the sign of the relative ratio between L y FM and S y FM strikingly resembles the energy dependence of the spin-orbit correlation L · S FM eq in Fig. 2(b). However, the variation of X y FM /E x [ Fig. S2(a)] differs from the variation of X x FM /E x [ Fig. 4(a)]. A reason for such difference is due to the fact that while X y FM arises from the intraband contribution for the states at the Fermi surface [Eq. 2(a)] X x FM arises from the interband contribution for the states in the Fermi sea [Eq. 2(b)]. However, for each state in the band structure, they have strong correlations. To demonstrate this point, we present in Fig. S2(b) a plot of ∂{ X x FM /E x }/∂E F , which corresponds to the contribution within the energy slice near E F . By comparing this with X y FM /E x [ Fig. S2(a)], we find strong resemblance for both orbital and spin over the whole range of E F , except that their relative signs are opposite. Figure S3 shows Fermi energy dependences of (a) L y Fig. S3(a)]. This is expected because

D. Spin-Orbit Coupling Dependence
Thus, L x FM /E x also increases monotonically with increasing α FM so [ Fig. S3(c)]. In Fig. 4(b), the SOC dependence of τ d FM /E x , which is proportional to S x FM /E x , is shown for fixed Fermi energies

NM SOC included
When the SOC in the NM is nonzero, the spin Hall effect (SHE) follows the OHE [S1] thus orbital torque (OT) and spin torque (ST) coexist. Relative sign of the OT and ST is determined by the spin-orbit correlations L · S NM eq in the NM and L · S FM eq in the FM as summarized in Table I of the Letter. In this section, we demonstrate this point from the numerical calculation by setting finite SOC strength in the NM: Except for α NM so , the rest of the parameters are set equal to Eqs. (S17), (S18), and (S19). Figure  Letter] by setting SOC strength parameters in the NM and FM by (i) α NM so = 0.2 eV, α FM so = 0.1 eV, and (ii) α NM so = 0.2 eV, α FM so = −0.1 eV. Note that the sign of α FM so is reversed in (ii). This reversal is motivated by the fact that it reverses the sign of the OT while it does not affect the sign of the ST. By this way, the band structure is barely affected and also the sign of the L · S FM eq for the case (ii) becomes opposite to that for the case (i). Thus, we define the OT and ST contributions as such that S(z) (i) = S(z) OT + S(z) ST . Therefore, we calculate S(z) (i) and S(z) (i) from the Kubo formula in Eq. (2) of the Letter, and extract the OT and ST contributions by Eq. (S24) for E F = −0.9 eV, where L · S NM eq < 0 and L · S FM eq < 0 [ Fig. S4]. Thus, the OT and ST contributions are expected to have the same sign.
In Fig. S5(a), spatial profile of S y (z) /E x is shown in the NM region (1 ≤ z ≤ 20) and the FM region (21 ≤ z ≤ 30). The OT contribution (orange squares) is similar to the result of S y (z) /E x when α NM so = 0 [ Fig. 3(a) of the Letter]. On the other hand, the ST contribution (cyan diamonds) exhibits a standard behavior of the SHE; the spin is accumulated at the boundary. Near z = 1, sign of S y (z) /E x is negative, which implies that the OHE and SHE occurs in the opposite directions. In the absence of the FM, S y (z) /E x is positive near z = 20 (not sown). However, due to presence of the FM attached, S y (z) /E x is reduced near z = 20, which is injected to the FM. We find that the signs of the OT and ST contributions are same in the FM region. This is expected from the spin-orbit correlations of states near E = −0.9 eV and from the fact that S y (z) /E x results from the intraband contribution at the Fermi surface [Eq. (2a)]. When S y is injected to the FM, it precesses along the magnetization by the exchange interaction, regardless of whether it is the OT or ST contribution. In the Kubo formula calculation, S x (z) /E x is captured by the interband contribution in the Fermi sea [Eq. (2b) of the Letter]. Nevertheless, we find that the signs of the OT and ST contributions are same in the FM region [ Fig. S5(b)], which is because all the states below E F = −0.9 eV satisfy L · S NM eq < 0 and L · S FM eq < 0.

NM SOC included, FM onsite changed
The result shown in the previous subsection considers the case when the sign of the OT and ST is same. In this subsection, we present a result in another regime where the OT and ST have the opposite signs. To achieve this, we shift the onsite energy of d orbitals in the FM by setting E F = −0.9 eV, L · S NM eq < 0 but L · S FM eq > 0, thus we expect the opposite signs for the OT and ST. From the same method [Eq. (S24)], we calculate the electric field responses of the OT and ST contributions. Figures S7(a) and S7(b) show the results for S y (z) /E x and S x (z) /E x , respectively. We find the signs of the OT and ST contributions are opposite in both cases. The result for S x (z) [ Fig. S7(b)], which is the interband contribution from the Fermi sea, is understood as the following. Although there are many FM bands with L · S FM eq < 0 below E F = −0.9 eV, the hotspots for the OHE and SHE in the NM is concentrated in the energy range −1.0 eV < ∼ E nk < ∼ 0.0 eV [S1] thus the torque contribution from the FM bands with L · S FM eq < 0 is negligible and the major contribution is from the states in an energy range −1.0 eV < ∼ E nk < ∼ −0.9 eV.

F. Role of the Interface Hoppings for the Orbital Torque
In this section, we demonstrate crucial role of the interface hoppings for the OT. For the orbital injection from the NM to the FM in the tight-binding model presented in the Letter, the orbital information in the p orbitals in the NM should be transferred to the d orbitals in the FM. At the interface, two types of hoppings are crucial for this: Once a state carrying finite OAM, |L (p) y = ±1 = |p z ± i |p x for example, is induced in the NM, the interface hoppings in Eq. (S26) can generate a state |L (d) that also carries net OAM. Thus, the relative sign of γ pdσ and γ pdπ is crucial, by which the OT changes the sign. In the tight-binding model used in the Letter, we assume the same sign for γ pdσ and γ pdπ [Eq. (S19)]. In order to demonstrate this effect, we present calculation results for X y (z) /E x and X x (z) /E x in Figs. S8(a) and S8(b), respectively, by assuming γ pdσ = −0.4, γ pdπ = 0.1, which is to be compared with Fig. 3 of the Letter. We find that L y (z) /E x is unchanged near z = 1, which is away from the interface. However, near the interface (z = 20) and in the FM region (21 ≤ z ≤ 30), the sign of the L y (z) /E x in Fig. S8(a) is opposite to that in Fig. 3(a) of the Letter. As a consequence, S y (z) /E x , which is converted from the injected orbital angular momentum, also changes the sign [ Fig. S8(a)]. Since S x (z) /E x precesses along the magnetization by the exchange interaction and L x (z) /E x follows by the SOC in the FM, the signs of L x (z) /E x and S x (z) /E x in Fig. S8(b) are flipped compared to Fig. 3(b) of the Letter. Therefore, the interface crystallinity is crucial for the generation of the OT. In dirty interface, the interface hoppings, such as Eq. (S26), are randomized, and this reduces the magnitude of the OT. On the other hand, the spin injection is not affected by the relative sign of the interface hoppings, thus the ST is less susceptible to the interface crystallinity.