Fast current-induced domain-wall motion controlled by the ...€¦ · 1 Supplementary Information...

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Supplementary Information

Fast current-induced domain wall motion controlled by the Rashba effect

Ioan Mihai Miron 1,2*, Thomas Moore 1,3, Helga Szambolics 1, Liliana Daniela Buda-

Prejbeanu1, Stéphane Auffret 1, Bernard Rodmacq 1, Stefania Pizzini 3, Jan Vogel 3, Marlio

Bonfim4, Alain Schuhl 1,3 and Gilles Gaudin 1

1 SPINTEC, UMR-8191, CEA/CNRS/UJF/GINP, INAC, F-38054 Grenoble, France 2 Catalan Institute of Nanotechnology (ICN-CSIC), UAB Campus, E-08193 Barcelona, Spain 3 Institut Néel, CNRS/UJF, B.P. 166, F-38042 Grenoble, France 4Departamento de Engenharia Elétrica, Universidade Federal do Paraná, Curitiba, Paraná, Brazil

DW dynamics observed by micromagnetic simulations

The goal of these micromagnetic simulations is to detail the understanding of the DW

dynamics observed experimentally. Three questions will be addressed:

i) Can the terminal DW velocity be measured using ultra-short current pulses?

ii) What is the influence of HR on the DW stability and velocity?

iii) How does a single DW chirality reversal driven by HR influence the DW displacement?

The extended Landau Lifshitz Gilbert (LLG) equation

In order to model the effect of current on the magnetization1, the Landau Lifshitz

Gilbert (LLG) equation will include besides the effective Rashba field2 (HR), the spin transfer

torque (STT)3. The Extended LLG equation reads:

( )[ ] ( ) ( )[ ]MuMMuMMMHHMReff ∇⋅×+∇⋅−

∂∂×+×+=

∂∂ βαγ

SS MtMt1

0 , (1)

Here M is the local magnetization, 0γ the gyromagnetic ratio, Heff the magnetic field

including contributions from the external, anisotropic, and magnetostatic fields, HR is the

Fast current-induced domain-wall motion controlled by the Rashba effect

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effective Rashba field, α the Gilbert damping paramater, β is the ratio of the nonadiabatic to

adiabatic spin torque, and 2

Be

s

gµ PeM

=u j where je is the current density. Note that HR is

included on equal footing with the other effective fields occurring in the ferromagnet.

DW motion opposed to electron flow, in the context of STT

DW motion opposed to the carrier flow4 was also observed in (Ga,Mn)As5, where it

was attributed to negative current polarization generated by the antiferromagnetic exchange

coupling between the itinerant and localized spins, and also in the PtCoPt trilayers6, where it

was assumed to be caused by a negative current polarization. Since it is not the purpose of this

work to understand the origin of this phenomenon, we will analyze the DW dynamics within

all scenarios that could explain this observation.

In the framework of the CIDM theory1 the inverse motion could have three possible

origins:

i) Both spin-torque components are negative. Physically, in the context of the present

current induced domain wall motion (CIDM) theory this would correspond to a negative

current polarization (P<0). In this case, the DW dynamics are expected to be identical to the

ones obtained for a positive current polarization, only that the displacements occur in the

opposite direction.

ii) The non-adiabatic spin-torque is negative while the adiabatic one is positive. This

corresponds to positive current polarization (P>0), but negative β. In this case the direction of

motion is reversed only for motion occurring below the Walker breakdown (WB).

iii) The non-adiabatic component is positive while the adiabatic one is negative. In this

case both the current polarization and β are negative (P<0, β<0). With respect to the previous

scenario, the sign change of the current polarization reverses the behavior: the DW moves

against the electron flow only above the WB.

Since it is impossible to a priori distinguish between them, both physical possibilities

(negative P and negative β) explaining the direction of DW motion are analyzed by

micromagnetic simulations.

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Values of the physical parameters used in simulations

Most of the required values, such as the saturation magnetization magnetocristalline

anisotropy non-adiabaticity of STT and Rashba field were determined and reported

previously. The value of the Rashba field was previously measured using 100 ns current

pulses2. Since in the present work we use ultra-short pulses, we repeat the experiment to

determine the value of the effective HR for different pulse durations. Our observations are

summarized in Figure S1. As we apply shorter pulses, the current density necessary for

domain nucleations increases. This enhancement correlates with the decrease of the effective

value of HR. We believe that, this variation could be attributed to something as common as

thermal effects. The existing theoretical models predict that HR varies inversely proportional

to MS. As MS depends on the wire temperature, which in turn depends on the pulse duration

and intensity, thermal effects could account for the observed dependence of HR on pulse

duration. Note that this dependence is not central for the claims of our work. The DW motion

experiments prove that HR is sufficiently strong to prevent WB, and simulations show that its

presence doesn’t affect the velocity value of the steady motion regime. For the micromagnetic

study of the DW dynamics we use the smallest value of HR determined experimentally.

The value of the damping parameter was determined from the slope of the constant

mobility regime in the field induced DW motion. The slope can be fitted in principle with two

Figure S1. Dependence of the Rashba field intensity on pulse duration; in red the current density required for nucleation in half of the wires is plotted as a function of pulse length. In black we plot the variation of the Rashba field efficiency normalized to a current density of 1012A/m2. The increase of the current density required for nucleation correlates well with the loss of efficiency of the Rashba field.





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laws, one corresponding to the steady flow regime (α

γ Δ= 0Hv , where Δ is the wall width),

and a second corresponding to the turbulent DW motion ( 10

−+Δ

=αα

γHv ). In the previous

work by Metaxas et al.7 the DW mobility was in a range that could correspond to both these

scenarios, so that the damping value could not be uniquely determined. In our case, the

velocity can only be fitted with the law corresponding to the steady flow regime, and the

damping is uniquely determined to be approximately α ≈ 0.5 for a DW width of 4.2 nm8.

The physical parameters considered for the micromagnetic simulations are the

following: μ0Ms= 1T; AEX = 1011 J/m; KANIS = 1.19 106 J/m3 P = ±1 ; β = ±1; HR = 0.25

T/1012 A/m2; α = 0.5;

A domain wall was created in the middle of the wire by calculating the equilibrium

configuration starting from a sinusoidal configuration of the magnetization. Equation 1 was

implemented to simulate the effect of current pulses on the magnetization. The pulse shape is

similar to the measured pulses (Figure S2).

a) Measurement of the terminal DW velocity using current pulses.

The use of ultra-short current pulses, far from quasi-static conditions, requires

considering the transient regimes of DW motion.

In Figure S3 we plot an example of DW displacements observed for a 0.9 ns pulse of

6×108A/m2. Note that the DW does not actually have time to reach its terminal velocity

during the pulse (Figure S3). Moreover, its motion continues after the pulse has ended. This is

due to its structural transformation, measured by the x component of the DW’s magnetization.

Figure S2: Shape of the injected current pulses. a, used in the experiment b, used in the micromagnetic simulations





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However, the velocity determination that we use does not seem to be affected: the gradient of

displacement vs. pulse length is the same as the terminal velocity of the steady flow regime

induced by dc current. This happens because the DW distortion is “elastic” and the delay

accumulated during the pulse is compensated by DW motion occurring after the pulse.

In the inset of Figure S3 we plot the x component of the magnetization, which

measures the DW deformation. It increases during the pulse, and as soon as the pulse ends, it

starts decreasing. Together with this variation, the DW motion continues after the pulse and

only ends when the DW deformation ceases. The effect of HR is to limit the deformation of

the wall structure. Consequently the DW moves more uniformly during the pulse and less

after the end of the pulse.

b) Effect of HR on the DW stability

Figures S4 shows the DW displacement as a function of the pulse length for various

values of current densities and with and without HR. In the absence of HR, the variation of the

displacement with pulse length seems to have, besides a monotonic trend, a “random”

component. The apparent noise of this dependence is related to the DW transformations above

the WB. While in the case of steady motion the DW deformation is reversible and therefore

the two transient regimes (the beginning and the end of the pulse) compensate, above the WB

this is no longer the case. The periodic transformations of the DW structure are irreversible

and the transient regime at the end of the pulse, determined by the final DW deformation, is

dictated only by the pulse duration and amplitude. Therefore, this apparently random

Figure S3: The magnetization variation during a 0.9 ns pulse. j = 6×108A/m2. The black curve is the DW position calculated in the scenario of P < 0. In red, we plot the same variation, for the scenario that includes HR. The effect of HR is to produce a steadier motion during the pulse and to reduce the residual displacement (occurring after the end of the pulse). HR regulates the displacement by limiting the DW distortion, measured here by the MX component (along the wire).





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component is deterministic and does not cancel by averaging. The fact that we do not observe

such variations experimentally further supports the conclusion that DW motion is steady, not

turbulent.

When HR is included, the displacements lose their “noisy” character as the DW

transformations no longer occur. Once again we find a “clean” linear dependence and the DW

mobility obtained from the linear regression of the displacement vs. pulse length corresponds

to the terminal velocity values expected from the steady motion regime (Figure S4).

Figure S4: DW displacement vs. pulse duration. a) and b) the effect of HR is not included. c) and d) once HR is included the dependence becomes perfectly linear and the difference between the two scenarios disappears. The slope of the gray dotted line on the four graphs is the theoretically8 predicted (v =β/α·u) DW velocity for a current density of 250·108 A/m2. Note that the inclination of the gray line is the same as the slope of the DW displacement vs. pulse length (for 250·108 A/m2) only in the presence of HR. Otherwise the DW displacement is much smaller.





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c) DW chirality reversal by HR

Figure 4c of the main text shows the DW displacement obtained for the two possible

orientations of the DW chirality. First, it is set along the Rashba field, and after opposed. The

experimental observation of the decrease in the displacement produced by the chirality

reversal is reproduced by simulations. Once again, we find not only good qualitative

agreement, but also good quantitative similarity. Note that the two scenarios of negative β and

negative P give identical results.

Figure S6: Distribution of the measured DW displacements for a current density of 2×1012A/m2; a, 0.98 ns pulse; b, 1.18 ns; and c, 1.3 ns; Note that changes of the average displacement don’t influence much the dispersion. This indicates that the distribution is associated to the rise and fall time of the pulse, which is the same for all pulse lengths.

Figure S5: Delay time dependence on the injected current density. The delay time is defined as the intercept of the linear interpolation of displacement vs. pulse length with the abscissa (Figure 4a). As the current density is increased, the delay time decreases, and saturates for values close to the rise and fall time of the pulse (0.3 ns).





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Supplementary References:

1 Thiaville, A. Nakatani, Y. Miltat, J. & Suzuki, Y. Micromagnetic understanding of current-

driven domain wall motion in patterned nanowires. Europhys. Lett. 69, 990 (2005).

2 Miron, I.M. Gaudin, G. Auffret, S. Rodmacq, B. Schuhl, A. Pizzini, S. Vogel, J. &

Gambardella, P. Current-driven spin torque induced by the Rashba effect in a ferromagnetic

metal layer. Nat. Mater. 9, 230 - 234 (2010).

3 Szambolics H., Toussaint J.C., Marty A., Miron I.M., & Buda-Prejbeanu L.D. Domain wall

motion in ferromagnetic systems with perpendicular magnetization JMMM 321, 1912-1918

(2009)

4 Moore, T.A. et al. High domain wall velocities induced by current in ultrathin Pt/Co/AlOx

wires with perpendicular magnetic anisotropy. Appl. Phys. Lett. 93, 262504 (2008); ibid. 95,

179902 (2009).

5 Yamanouchi, M. Chiba, D. Matsukura, F. Dietl, T. & Ohno, H. Velocity of domain-wall

motion induced by electrical current in the ferromagnetic semiconductor (Ga,Mn)As. Phys.

Rev. Lett. 96, 096601 (2006).

6 Lee, J.C. Kim, K.J. Ryu, J. Moon, K.W. Yun, S.J. Gim, G.H. Lee, K.S. Shin, K.H. Lee,

H.W. Choe, S.B. Roles of adiabatic and nonadiabatic spin transfer torques on magnetic

domain wall motion arXiv:1006.1216v1

7 Metaxas, P.J. et al. Creep and flow regimes of magnetic domain-wall motion in ultrathin

Pt/Co/Pt films with perpendicular anisotropy. Phys. Rev. Lett. 99, 217208 (2007).

8 Miron, I.M. Zermatten, P.-J. Gaudin, G. Auffret, S. Rodmacq, B. & Schuhl, A. Domain wall

spin torquemeter. Phys. Rev. Lett. 102, 137202 (2009).





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