Domain Wall Motion in Ho doped Permalloy

rise time is short compared to the magnetization damping time, the reaction of the DW to the changing current is much higher than the reaction to the pinning field [THJ⁺07, BKM⁺09]. Therefore steep current rise times support the depin-ning and motion of the DW and hence can significantly increase the average DW velocity for short current pulses. In contrast, for the experiments using long cur-rent pulse rise times (>1µs), this dynamically enhanced part of the spin torque does not act on the DW and the magnetization is able to adiabatically adapt to the changing current leading to lower velocities and less reliable motion. This is supported by two facts: Firstly, for long current pulses with short rise times the DW velocity does not dependent on the pulse length but is constant [JKB⁺06].

Secondly, a recent study revealed that the depinning probability of a DW is re-duced at short current rise times while keeping pulse length and current density constant [BKM⁺09]. Changing the pulse length on the other hand, does not influ-ence the DW depinning. This strongly supports the interpretation of the influinflu-ence of steep rise times on the DW depinning.

Interestingly, in previous experiments it was found that for long pulses the DW velocities are reduced for more complex DW structures [KJA⁺05, KLH⁺06], whereas these results suggest no such dependence. For more complex structures the various pinning possibilities are more important and the additional torque due to the fast current rise time could be the key to moving the DWs reliably.

The results on the high current-induced DW velocity presented in this section have been published in the Applied Physics Letters [HRB⁺10].

4.4 Domain Wall Motion in Ho doped Permalloy

The β-term and its relation to the damping constant α is one of the key ques-tions for a better understanding of CIDM. The ratio β/α is predicted to control the nature of the DW motion [ZL04, TNMS05], and is the subject of much de-bate [TBB08, SSDZ07, SSDZ08, Smi08].

The special case β = α is favored by some groups, since the LLG-equation 1.32 can be rewritten in a form with Landau-Lifshitz damping without any β term occurring [TBB08, SSDZ07]. The results of the last sections show that this is not the case for permalloy. It also seems unlikely that β is always identical to α, since both are expected to depend sensitively on material and sample prop-erties, e.g. details of the band structure. However, it is predicted that α and β scale similarly with the strength of spin-dephasing processes [TBB08, GGSM09],

76 Current-Induced Domain Wall Motion

therefore the ratio β/α might be conserved. From previous experiments a wide range of estimations have been gained for permalloy ranging fromβ ≈α toβ ≈4.

But so far a systematic study of the scaling of β and α is missing. This can be achieved by doping permalloy nanowires with rare earth materials which is known to increase the damping [BKMR01, WKT⁺09]. By measuring the current-induced DW velocity vand from the scaling of the velocityv withβ/αu below the Walker breakdown [ZL04, TNMS05], information on the ration ofβ/α can be gained.

Experiment

Ho doped permalloy zig-zag wires of 1500 nm width and 20 nm thickness are fab-ricated by e-beam lithography and lift-off on Si. The experimental setup and the wire layout and thickness is identical to the section 4.3.3 (see also Fig. 4.8 for a SEM and X-PEEM image). The permalloy is co-deposited with Ho to give five sets of nanowires of different composition: pure permalloy, and permalloy doped with 1, 2, 4 and 10 at% Ho⁵. The damping constantαin the nanowires was mea-sured by ferromagnetic resonance (FMR) to be 0.01, 0.02, 0.033, 0.087 and 0.26, respectively⁶. A reduction of 5% in M_s per at% Ho is also measured [WKT⁺09].

We assume that the effect on the exchange constant A is negligible since we do not observe a significant change in the coercive field. As the effect of Ho doping on the spin polarization P is unknown, we assume the same value as for pure permalloy.

After positioning DWs at the kinks by a magnetic field pulse, 25µs long current pulses are injected and the total displacement is measured. Between 150 and 500 individual current induced DW movements are analyzed for each level of Ho doping for a range of current densities.

Results and Discussion

Fig. 4.12(a) shows the average velocity v(j)between the critical current density j_c and the Walker breakdown density j_W for wires with 0, 1, 4 and 10 at% Ho. The data for the pure permalloy wire is identical to the data shown in section 4.3.3.

Fitting the v(j)-curves with the equationv =β/α(u−u_c) (see Eq. 4.5 in section 4.3.3), we find within the experimental error the same gradient for all datasets.

This is consistent with the assumption that althoughα increases with the amount of dopants, the ratioβ/α is not affected by the doping.

5The deposition was done by J. U. Thiele at Hitachi Global Storage Technologies Research Center in San Jose

6The measurements were done by G. Woltersdorf and C. H. Back in Regensburg

4.4 Domain Wall Motion in Ho doped Permalloy 77

Asα increases, j_c and j_W decrease, while the absolute values of v remain the same, e.g. v ≈ 0.5m/s at jW for all wires. A possible reason for the drop in jc

is the concurrent change in M_s. A reduction lowers the spin torque required for the DW motion and hence the critical current density. The reduction inM_s could also explain the drop inj_W, for similar reasons. Using Eq. 4.6, and accounting for the reduced M_s, β is determined for each wire, and is displayed in Fig. 4.12(b), alongside α, as a function of the Ho concentration. It is seen that β scales withα up to a Ho concentration of 4 at%. The ratio β/αin this region is approximately 16.

To further study the influence of the damping on the DW motion, field induced experiments are carried out⁷. For pure field-driven DW motion a simple one dimensional model predicts that v is inversely proportional to α (v = γ∆H/α), assuming a constant DW width ∆ and propagation field H[SW74]. In contrast to the current-induced motion case discussed above, the field driven DW velocity clearly decreases as α is increased[Möh10]. However, there is agreement with the 1D model only for large values of α.

For a better understanding of these discrepancies, micromagnetic simulations are carried out. A VW is placed in a 10µm-long permalloy wire with width, thickness and material parameters the same as the experiment and subjected to H = 11G. The gradient of m_x(t), the magnetization along the wire direction as a function of simulation time, which is a measure of v, tends to decrease with

7The experiments were done by P. Möhrke and T. A. Moore using a special single shot MOKE setup [Möh10].

Figure 4.12: Average DW velocity v as a function of current densityj forj < jW for a 1500 nm wide and 20 nm thick pure permalloy wire and for permalloy wires doped with Ho and hence with different dampingα. The data are fitted with Eq. 4.5. (b) Dampingα and non-adiabaticity β as a function of the Ho concentration. Scaling ofαandβ occurs up to 4 at% Ho. (From [MKH⁺09a])

78 Current-Induced Domain Wall Motion

increasing Ho content in agreement with the theory and the experiment (For details see [Möh10]).

Asα decreases, the deviation of the DW velocity from the 1D model predic-tion correlates with an increasing distorpredic-tion of the VW (although the DW width remains virtually unchanged), so that approximation of the DW as a point-like quasi-particle becomes less and less appropriate. For example, for α = 0.01, the VC oscillates perpendicularly to the wire direction with≈100nm amplitude while the DW moves forward, emitting spin waves, and this can explain the reduction of the average velocity.

In contrast to the current-induced motion where the magnitude of the DW velocity remains the same below the Walker threshold, the DW velocity for field-induced motion shows a strong dependence on the dampingα. For the field-driven DW motion the velocity is decreasing with increasingα, in qualitative agreement with micromagnetic simulations whereas for the current-driven case it is almost not affected. This demonstrates how differently the underlying mechanisms of field- and current-driven DW motion depend on the damping. It also indicates that the non-adiabatic spin torque plays a vital role in the current-induced DW propagation.

The discrepancies of the experiment concerning the thresholdβ value derived from the walker breakdown current density and the gradient ofv(j)with the theory could have multiple origins. It is not clear how applicable the simple theoretical model is for two dimensional current-induced DW motion. In addition, the effects introduced by extrinsic pinning are to the present date not well understood and more work has to be done. However, the constant gradient of v(j) below the Walker threshold for the different Ho doped samples strongly supports the fact that β directly scales with α.

The results presented in this section are part of publications in Physical Review B as well as in the Journal of Magnetism and Magnetic Materials [MKH⁺09a, MKH⁺09b].