SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION doi: /nPHYS147 Supplementary Materials for Bias voltage dependence of perpendicular spin-transfer torque in asymmetric MgO-based magnetic tunnel junctions Se-Chung Oh 1, Seung-Young Park, Aurélien Manchon 3, Mairbeck Chshiev 3, ae-ho Han 4, Hyun-Woo Lee 4, ang-eun Lee 1, Kyung-Tae Nam 1, Younghun o, Yo-Chan Kong 5, Bernard Dieny 3, and Kyung-in Lee 5 1 Semiconductor R&D Center, Samsung Electronics Co., Ltd., Gyeonggi-Do , Korea Nano Material Research Team, Korea Basic Science Institute, Daejeon , Korea 3 SPINTEC, UMR 8191 CEA/CNRS/UF, CEA/Grenoble, Grenoble Cedex 9, France 4 PCTP and Department of Physics, Pohang University of Science and Technology, Pohang, Kyungbuk , Korea 5 Department of Materials Science and Engineering, Korea University, Seoul , Korea Supplementary Note 1: Accuracies of fit parameters in the thermal activation model Here we discuss in more details the bias-induced heating models, and estimate possible inaccuracies caused by simplifying assumptions in the thermal activation model. Bias-induced heating: In literature on nanostructured MTs, the bias-induced heating has been estimated by several approaches. To account for the bias-induced heating effect, we apply three nature physics 1

2 supplementary information doi: /nphys147 representative approaches (case 1, & 3, see below) to the analysis of the switching phase diagrams. Case 1) T * = T I and + γ APtoP T * = T I when the system is initially in the + γ PtoAP AP configuration and P configuration, respectively [S1]. Here T is the ambient temperature, I (=V/[R(V, T, P/AP)]) is the current, R(V, T, P/AP) is the experimentally determined junction resistance, and γ PtoAP and γ APtoP are fitting parameters. The results presented in the manuscript are obtained by using this equation. Case ) T * = T + γi. Note that case is similar to case 1 except that γ is now independent of the initial magnetic configuration. Case 3) T * = T + βv where β is a fitting parameter [S, S3]. This equation is in the same form as the Holm s formula [S] for the oule heating, which is known to be exact if the transport is Ohmic and the Wiedemann-Frantz law holds. When these conditions are satisfied, β is independent of the resistance of systems. Unfortunately these conditions are not satisfied in MTs, and thus the Holm s formula does not exactly hold. Nevertheless, this form of * T still provides a useful approximation tool to take into account the bias-induced heating effect and is used for instance in [S3]. nature physics

3 doi: /nphys147 supplementary information Supplementary Table S1. Summary of the fitting parameters for case 1, case and case 3. MT type Case C 1 (Oe/V) C (Oe/V ) V C + (V) V C - (V) γ APtoP / γ PtoAP (K /A ) for case 1 γ (K /A ) for case β (K /V ) for case 3 Case x10 11 /1.3x10 11 MT1 Case x10 11 Case x10 5 Case x10 11 /0.7x10 11 MT Case x10 11 Case x10 4 Fig. S1 shows the fitting curves for the MT1 and MT using the cases and 3. The fitting parameters are summarized in Table S1. We find that the fitting curves for both MTs in Fig. S1 (case & 3) are almost identical to those in Fig. 3 (case 1) of the main text. The fitting parameters of C 1 and C for the cases & 3 are different from the + corresponding values for the case 1. Note however that for both C 1 and C (also for V C and V - C ), the difference is within 10%. Thus values of C 1 and C are weakly sensitive to details of heating models. In retrospect, this insensitivity can be understood from the following three observations: (i) the phase boundaries of the switching phase diagram in Q and 4Q are relatively insensitive to C 1 and C and thus the boundaries in 1Q and 3Q play the most important role in the determination of C 1 and C, and (ii) the effective energy barrier ± ± E ( 1 m V / ) is larger in 1Q and 3Q than in Q and 4Q. This is due to the fact that in B V C Q and 4Q, both the in-plane spin torque and the magnetic field lower the effective nature physics 3

4 supplementary information doi: /nphys147 energy barrier for the relevant switching (P-to-AP switching in Q and AP-to-P switching in 4Q) while in 1Q and 3Q, the in-plane spin torque enhances the effective barrier height for the relevant switching (P-to-AP switching in 1Q and AP-to-P switching in 3Q). (iii) Due to the enhanced height of the effective energy barrier in 1Q and 3Q, the effect of the bias-induced heating becomes less important. Instead, other factors (such as C 1 and C ) that affect the barrier height become more important. Temperature dependence of C 1 and C C 1 and C may have a slight temperature variation of the order of the temperaturedependent change of the spin polarization. The effective spin polarization P (= P P L R where P L (P R ) is the spin polarization of the left (right) electrode) is approximately given by TMR / ( + TMR). Since TMR of MT1 is 175% (117%) at 4K (300K) and TMR of MT is 195% (13%) at 4K (300K), P of MT1 (MT) at 4K/300K = 0.68/0.61 (0.70/0.6). In both cases, (P 300K -P 4K )/P 4K *100% is about 10%, which is comparable to the uncertainty of C 1 and C values originating from heating estimation methods. Moreover, the changes in critical voltage between 4K and 300K are by a factor which means that the main effect is not caused by this temperature-dependence of the spin polarization but rather by the thermal activation. To summarize this part, the fitting values of C 1 may not be quantitatively accurate because of simplifying assumptions used in the thermal activation model. However, such inaccuracies are not large enough to give a wrong sign of C 1 and thus do not alter the main conclusion of this work. 4 nature physics

5 doi: /nphys147 supplementary information Supplementary Note : Transverse switching phase diagram The switching can also occur when the magnetic field is applied along the in-plane hard axis (instead of easy-axis) of the free layer. The switching threshold field again depends on the voltage bias, which is summarized in Fig. Sa. Here we analyze this transverse switching phase diagram (SPD). The magnetization dynamics of the free layer is governed by the modified Landau- Lifshitz-Gilbert [S4] equation: d M ˆ dt F = γm ˆ F r H eff + αm ˆ F d M ˆ dt F γ ˆ τ STT (S1) where Mˆ F is the magnetization direction of the free layer, H r eff is the effective field, and τˆ STT is the torque due to spin transfer. α and γ are the Gilbert damping and the gyromagnetic ratio, respectively. The effective field arises from the combination of the anisotropy field H, the demagnetizing field H and the external field H. an d ext τˆ STT is composed of two parts ˆ τ = a M ˆ ( M ˆ M ˆ ) + b ( M ˆ M ˆ ) where Mˆ is the STT magnetization of the pinned layer, a and b are the bias-dependent amplitudes of the inplane and perpendicular torque, respectively. F P We consider a free layer lying in the x-y plane, z being perpendicular to the interfaces. The easy axis is along the x axis and we apply the external field along the y direction. F F P P Then, the effective field may be written as H ˆ eff = H a n M x x ˆ + H y y ˆ H d M z z ˆ and ˆ = x ˆ. We follow Stiles et al. [S4] to solve Eq. (S1). In the spherical coordinate M P system M ˆ = ( sinθ cosϕ,sinθ sinϕ, cosθ ), Eq. (S1) yields two differential equations: nature physics 5

6 supplementary information 1+ α & θ = h γ 1+ α γ ϕ + α h θ & ϕ sinθ = α h ϕ h θ doi: /nphys147 ( S ) ( S3 ) where h h ϕ θ = H = H y d cosϕ H an cosθ sinθ + H cosϕ sinϕ sinθ a y sinϕ cosθ + H an cosθ cosϕ b sinϕ cos ϕ cosθ sinθ a sinϕ + b cosϕ cosθ ( S4 ) ( S5) The stationary states in the absence of current are then θ = ±π / and sinϕ = ± ( H / H + ( 1 H / H ) Θ( H H ) ( S6 ) y a n y a n y a n, where Θ( x) is the step function. To determine the instability conditions at zero temperature, we study the response of the magnetization under small perturbations ( ξ ) θ = π + ξ / and ( χ ) ϕ χ ϕ = 0 +, where sin ϕ is defined by Eq. (S6) and ξ, χ << 1. In the following, we name P and AP (by 0 extension of the longitudinal case) the magnetic states for which ϕ = ϕ0 and ϕ = ϕ 0 π. Inserting the perturbed angles θ ( ξ ) and ( χ ) ϕ in Eqs. (S)-(S3), we obtain coupled differential equations. We notice that these equations are similar to the ones obtained in the longitudinal case (see for example Ref. [S4], Eqs. (4)-(44)), provided a and b are replaced by a cosϕ 0 and b cosϕ0, and H d and an are replaced by H + sin ϕ H d H an 0 and H an cos ϕ 0, respectively. In this case, assuming that b << a α and a V, we a 1 obtain the critical switching voltage at zero temperature, consistently with Morise et al. [S5]: P an 0 + H d Vc0 ± α ( S7 ) a cosϕ ( A P ) H ( 3cos ϕ 1) nature physics

7 doi: /nphys147 supplementary information This critical switching voltage is valid only at zero temperature and thermal activation is known to be of seminal importance in current-induced magnetization dynamics of magnetic tunnel junctions [S6]. In the longitudinal case, the thermal stability factor Λ of a ferromagnetic layer subject to both in-plane and perpendicular components of spin torque may be written as: Λ P ( A P ) KuΩ H x + b 1 k BT m H an 3 / V 1m P Vc 0 ( S8 ) ( A P ) where K is the uniaxial effective anisotropy energy, Ω is the volume of the layer, is u k B P A P the Boltzmann constant, T is the temperature and V 0 is the critical switching voltage Λ at 0 K. Using the classical Arrhenius formula for thermal activation τ = e ( τ is the thermal-activation lifetime of the magnetic state and τ 0 = 1ns is the attempt frequency of the nanomagnet), and making the replacements mentioned above, we obtain the activation formula in case of the transverse diagrams: τ ln = Λ τ 0 where Λ = K k T. 0 u Ω / B 0 c ( ) 3 / 0 c ( 1m sinϕ ) 1m 1m ( S9 ) 0 b cosϕ H an Fig. Sa shows the transverse SPDs of the MT1 measured at T=4K and 300K. Solid curves are obtained from equation (S9) using the same set of parameters as for Fig. 3b in the main text and assuming the damping constant α=0.01. We find reasonable agreement between the experimental and theoretical switching boundaries of the transverse SPD. Next we examine implicit assumptions in the above analysis of the transverse SPD. The reliability of the above analysis depends crucially on how well these assumptions are V V c0 τ 0 nature physics 7

8 supplementary information doi: /nphys147 fulfilled. Equation (S9) is obtained within two assumptions: 1) only the free layer magnetization rotates due to the hard-axis field, and ) the in-plane spin torque follows pure sinθ-dependence. These two assumptions need to be verified. Assumption 1) Magnetic configuration under a hard-axis field The layer structure of our MTs is PtMn PL Ru RL MgO FL where PL, RL and FL represent the pinned (CoFe.5nm), the reference (CoFeB nm), and the free layers, respectively. Here we estimate how much the magnetizations are tilted from the easy axis when the field is applied along the hard-axis. In order to do this, estimations of the exchange bias field (H EB ) between PtMn and PL, the RKKY interaction ( EX ) between PL and RL, and the anisotropy field (H K ) of FL are essential. Since we have already determined H K in the main text, we need to estimate H EB and EX. Due to the unpatterned synthetic antiferromagnetic pinned layer structure, we can separate the magnetization curve of the pinned layer structure from that of FL assuming that H EB and RKKY coupling field are much larger than H sh (orange-peel coupling field). In this situation, one can find H EB and RKKY coupling field in the following way. The energy per unit area of the pinned layer structure reads E M t H + H )cosθ M t H cosθ + cos( θ ), (S10) = 1 1 ( EB 1 EX 1 θ where M 1 (M ) is the saturation magnetization of PL (RL), t 1 (t ) the thickness of PL (RL), θ 1 (θ ) is the magnetization angle measured from the easy axis of PL (RL), and H is the easy axis field. Here we assume zero anisotropy fields for PL and RL, which is the worst case in the sense of possible tilting of magnetizations in the pinned layer structure. From E θ = E / θ 0, the stable θ 1 and θ are given by / 1 = 8 nature physics

9 doi: /nphys147 supplementary information RKKY sin H 1 H θ 1 = sinθ, (S11) RKKY H ( H + H ) EB RKKY RKKY 1 H1 ( H + H ) ( ) ( + ) cos, ( ) EB H H H EB θ = + (S1) RKKY RKKY H + H EB H1 H 1 H where RKKY H i = E X /( M i t i ). From -1 sin(θ1) 1 and -1 cos(θ ) 1, one finds the critical field H 1 (H ) for the deviation of θ from 0 (π). The experimental result of R-H major loop of MT1 under the easy-axis field at 300K is shown in Fig. Sb. We obtain H 1 = 140 Oe and H ~ 500 Oe. By numerically solving equations (S11) and (S1), we find H EB = 1005 Oe, H RKKY 1 = 467 Oe, and H RKKY = 875 Oe. Fig. Sb also shows the experimental and modeling results of R-H major loop under the hard-axis field. In the modeling curve, TMR is deduced by the following equation derived for symmetric MTs [S7]. Here TMR = P [1 cosθ ] 100%, 1+ P cosθ (S13) where the spin polarization P is 0.61 deduced from the TMR, and θ is the angle between PL and FL magnetizations. The modeling curve shows reasonable agreement with the experimental one at low fields (the difference in coercivity is due to the finite temperature in experiment). Modeling results of magnetization component M x along the easy axis for three layers are shown in Fig. Sc. Since both exchange-bias and RKKY coupling are sufficiently large, magnetizations in RL and PL do not tilt much, especially in the field range from 100 Oe to +100 Oe, which is the field range in the transverse SPD in Fig. Sa. Thus our approximation that only FL is tilted due to the hard-axis field is reasonably well satisfied in the field range used for the transverse SPD. nature physics 9

10 supplementary information doi: /nphys147 Assumption ) Angular dependence of the in-plane torque strength In the analysis of the transverse SPD, it is assumed that the in-plane spin torque is proportional to sinθ. For symmetric MTs, this angular dependence is supported by an ab initio calculation [S8], which takes into account realistic band structure of MT materials. Experiments in [S9, S10] also indicate that the sinθ-dependence is at least a reasonably good approximation in symmetric MTs. For asymmetric MTs, on the other hand, we are not aware of any literature dealing with this issue. The only study on this issue is our own calculation result based on the free electron model. Although this calculation result (Fig. Se) supports the sinθdependence of the in-plane spin torque in asymmetric MTs, we judge that this model calculation is not conclusive since realistic band structures of MT materials are ignored. Thus we are presently unable to completely rule out the possibility that the angular dependence of the in-plane spin torque in asymmetric MTs deviates from the simple sinθ-dependence. In Fig. Sd, we compare calculated θ s to experimental ones deduced from equation (S13). Note that equation (S13) was derived for symmetric MTs where the angular dependence of the in-plane spin torque is almost perfectly sinusoidal according to the ab initio calculation [Fig. of ref. S8]. We find that the experimental and modeling θ s deviate from each other by about 10 o in both P and AP states. To summarize, the equation (S9) used for the analysis of the transverse SPD assumes the sinθ-dependence of the in-plane spin torque. Although our model calculation supports this assumption, we are unable to rule out the possibility that realistic band structure effect modifies this angular dependence. 10 nature physics

11 doi: /nphys147 supplementary information Supplementary Note 3: Theoretical calculations of resistance and perpendicular spin torque in asymmetric MTs In the main text, we show that the asymmetric exchange splitting Δ naturally explains the observed additional linear bias dependence of b and its sign. Fig. S3a shows theoretical result on the voltage dependence of (dv/di) P for the asymmetric Δ, which is qualitatively consistent with the experiment (Fig. 1c in the main text). Here we examine the effect of asymmetric barrier height U on the linear dependence of b since Co, Fe, and B have different work functions [S11]. In the MT1 (MT), U at the interface of Free MgO is expected to be smaller (larger) than U at the interface of Reference MgO because Fe has a smaller work function than Co [S11]. Fig. S3b and S3c show theoretical results of the voltage dependence of (dv/di) P and τ using the free electron model within the Keldysh formalism for the cases of asymmetric U, respectively. MT is more conductive at V > (<) 0 when U Ref > (<) U Free, consistent with the Brinkman model [S1] and also the experimental results in Fig. 1c. The minimum of τ shifts toward a negative (positive) voltage when U Free < (>) U Ref. It is because when the free layer has a larger U, a positive bias has to be applied to equalize U Ref and U Free where τ is minimal. However, these shift directions are inconsistent with our experimental observations. It indicates that in our MTs, the asymmetry in U is less pronounced than the asymmetry in Δ. We point out that similar results are obtained when we use alternative approaches, namely the generalized version of the tight-binding model used in [S13] and the free electron model based on the Airy function calculation [S14]. Finally we comment on the nature physics 11

12 supplementary information doi: /nphys147 difference between our calculation result and the corresponding result in [S14]. While our calculation indicates that τ in asymmetric MTs has its minimum value at nonzero V, [S14] reports that τ is minimized at V=0 even in asymmetric MTs. [S14] reports that the effect of the asymmetry is instead the appearance of singularity at V=0 (dτ / dv has different values for V=0+ and V=0-). We believe that this difference is due to the incorrect energy integration used in [S14] where τ is calculated from the energy window between the Fermi energy (E F ) and E F +ev, while it is known [S13] that electron states below E F also contribute to τ. When the contribution coming from the electron states below E F is also included, we find that the singularity at V=0 disappears and the minimum of τ appears at a nonzero V in asymmetric MTs. S1. Fuchs, G. D. et al., Adjustable spin torque in magnetic tunnel junctions with two fixed layers. Appl. Phys. Lett. 86, (005). S. Holm, R. Electric contacts: theory and applications 63 (Springer-Verlag, 1967);. S3. Li, Z. et al., Perpendicular spin torques in magnetic tunnel junctions. Phys. Rev. Lett. 100, 4660 (008). S4. Stiles, M. D. & Miltat,. Spin Transfer Torque and Dynamics in Spin Dynamics in Confined Magnetic Structures III: Topics in Applied Physics 101, 5-308, eds. Hillebrands, B. & Thiaville, A. (Berlin: Springer, 006). S5. Morise, H. & Nakamura, S. Stable magnetization states under a spin-polarized current and a magnetic field. Phys. Rev. B 71, (005). S6. Koch, R. H., Katine,. A. & Sun,. Z. Time-resolved reversal of spin-transfer switching in a nanomagnet. Phys. Rev. Lett. 9, (004). 1 nature physics

13 doi: /nphys147 supplementary information S7. Slonczewski,. C., Conductance and exchange coupling of two ferromagnets separated by a tunneling barrier. Phys. Rev. B 39, 6995, (1989). S8. Heiliger, C & Stiles, M. D., Ab initio studies of the spin-transfer torque in junctions. Phys. Rev. Lett. 100, (008). S9. Sankey,. C. et al., Measurements of the spin-transfer-torque vector in magnetic tunnel junctions. Nature Phys. 4, (008). S10. Kubota, H. et al. Quantitative measurement of voltage dependence of spin-transfer torque in MgO-based magnetic tunnel junctions. Nature Phys. 4, (008). S11. Michaelson, H. B. The work function of the elements and its periodicity.. Appl. Phys. 48, (1977). S1. Brinkman, W. F., Dynes, R. C. & Rowell,. M. Tunneling conductance of asymmetrical barriers.. Appl. Phys. 41, (1970). S13. Theodonis, I., Kioussis, N., Kalitsov, A., Chshiev, M. & Butler, W. H. Anomalous bias dependence of spin torque in magnetic tunnel junctions. Phys. Rev. Lett. 97, 3705 (006). S14. Wilczyński, M., Barnaś,. & Świrkowicz, R. Free-electron model of currentinduced spin-transfer torque in magnetic tunnel junctions. Phys. Rev. B 77, (008). nature physics 13

14 supplementary information doi: /nphys147 Supplementary Figure S1. Fitting of the SPDs using different heating models. a, MT1 (case ). b, MT (case ). c, MT1 (case 3). d, MT (case 3). In b and d, the fitting within the constraint C 1 =0 requires unrealistically large V C - (-3.0 V for b and -.1 V for d). 14 nature physics

15 doi: /nphys147 supplementary information Supplementary Figure S. Transverse switching phase diagrams of MT1. a, Transverse switching phase diagram measured at T = 4. K and 300 K. b, Experimental and modeling results of majors R-H loops under the easy- or hard-axis field. c, Modeling results of the easy-axis component of magnetization M x in each layer (RL = Reference Layer, PL = Pinned Layer, and FL = Free Layer). d, The angle between PL and FL magnetizations deduced from equation (S13). e. Theoretical result on the angular dependence of the in-plane spin torque in asymmetric MTs. f. Theoretical result on the angular dependence of the perpendicular spin torque in asymmetric MTs. In e and f, calculations have been done with the same parameters used for Fig. 4 in the main text and the applied voltage is 0.5 V. nature physics 15

16 supplementary information doi: /nphys147 Supplementary Figure S3. Effects of asymmetric Δ and asymmetric barrier height U on the resistance and perpendicular spin torque. a, Bias dependence of (dv/di) P for asymmetric Δ. b, Bias dependence of (dv/di) P for asymmetric U. c, Bias dependence of τ for asymmetric U. For the symmetric MT, U is 3 ev and Δ is 1 ev. For the asymmetric MTs in b and c, Δ is kept at a constant of 1 ev whereas U of a FM is 3.5 ev and U of the other FM is.5 ev. 16 nature physics