Domain Wall Motion in Permalloy

According to the theory, vortex walls and transverse walls are affected by spin currents in a similar way and are displaced by the current in the direction of the electron flow.

An example for current-induced DW motion of transverse walls is shown in Fig. 4.2(a) and (b). The wire is imaged after the initialization with an external field in Fig. 4.2(a) with the DWs positioned at the kinks. Since the wire in (a) is narrow and the magnetic contrast is horizontal as indicated by the gray color bar, the internal structure of the DW cannot be revealed. Fig. 4.2(b) presents the configuration after a 25µs long current pulse has been injected into the wire. The two DWs in the bottom wires moved about 20µm and 30µm in the direction of the electron flow . The three DWs in the top wire did not move at all. In the other wire the center DW moved and annihilated with the right DW.

Fig. 4.2 (c) and (d) show a similar event for a 750 nm wide and 12 nm thick wire.

After the initialization by a vertical field DWs are formed at the kinks Fig. 4.2(c).

One of these DWs is a vortex wall, whereas at the other seven kinks transverse walls are formed. After the current injection only the vortex wall is displaced by the current.

Figure 4.2: (a) Wire structure after the initialization. TWs are positioned at the kinks.

(b) Configuration after a 25µs current pulse injection. (c) and (b) corresponding images from a 750 nm wide and 12 nm thick wire. The gray scale bars indicate the direction of magnetic contrast and the arrows indicate the direction of the electron flow.

4.3 Domain Wall Motion in Permalloy 59

The images suggest that the current-induced DW displacement is always in the direction of the electron flow, regardless of the DW type, but they also show, that this motion is to a certain extent stochastic.

In general, the wire dimensions only favor one DW type, either TWs or VWs.

Thus comparing the two DW types is not straight forward, since the conditions, i. e. wire dimensions, are not the same. However for certain geometries close to the phase boundary between transverse and vortex walls, both types constitute local energy minima. Both spin configurations are (meta-) stable states. As one moves away from this phase boundary, one DW type or the other becomes more stable and so prevailing vortex or transverse walls are observed and less often the energetically more unfavorable type. To a certain extent, this is the case for the wire shown in Fig. 4.2(c) and (d).

The fact that the vortex wall in Fig. 4.2(c) is displaced, but the transverse walls did not change indicates that the critical current density for vortex walls is lower than for transverse walls. This is supported by observations obtained in a 1000 nm wide and 8 nm thick wire. This geometry is also close to the phase boundary and VWs as well as TWs are observed and their critical current densities can be directly compared. The lowest critical current density at which CIDM is observed for TWs is 9·10¹¹A/m² whereas CIDM for VWs is observed already at 7.5·10¹¹A/m² [HKB⁺08b]. This explains previous results about a DW that could not be moved anymore with the same current density after it had transformed from a VW to a TW [KJA⁺05]. According to calculations [HLZ06], transverse walls experience a stronger pinning at edge irregularities due to their larger stray field, which explains the higher critical current density found for transverse walls.

4.3.1 Critical Current Densities for Vortex Wall Motion

To obtain a more comprehensive understanding of what governs the critical current density, a systematic study of the critical current densities for vortex walls in various wire geometries (12 nm to 28 nm thick and 200 nm to 1500 nm wide) has been carried out.

Experiment

The experimental setup used is the same as the one explained above.² The critical current density is determined by injecting current pulses with increasing amplitude

2The data was collected in experiments at the SLS in Switzerland, at Diamond in England and at ELETTRA in Italy.

60 Current-Induced Domain Wall Motion

till the DWs start to move. This is repeated several times to obtain reliable results concerning the threshold density. The random error for the critical current densities is estimated to be around ±5%for wires of the same thickness. On one Si chip, various wires with different width but the same thickness are studied.

Any error in the sample thickness therefore affects all results for this thickness simultaneously and adds an systematic offset to the current density for this wire thickness. Thus comparing absolute current densities for samples with different thicknesses entails larger errors. However, since we are interested in the width dependence of the critical current density at fixed thicknesses this does not pose a problem for the analysis.

Results and Discussion

For each of the different wire geometries, a series of DW motions is acquired for a range of current densities to detect the onset of DW motion. From the observed displacements and the pulse length the average DW velocities are calculated and plotted vs. the current density. An example is shown in Fig. 4.3 for the 750 nm wide and 12 nm thick wire geometry. In Fig. 4.3(a) the average DW velocities for all detected events are shown. The wires contain 8 kinks and thus 8 initial DWs.

The shown data is averaged across all these DWs. Not restricting the velocity measurement to one DW has the advantage that better statistics are gained and that the influence of the extrinsic pinning is averaged. In Fig. 4.3(b) the observed velocities (regardless of the current density) are shown for each of the 8 DWs. The inset explains the labeling of the DWs. Except for DW no. 3 where no motion is

Figure 4.3: (a) Shown are the observed DW displacements at various current densities in 750 nm wide and 12 nm thick wires. The critical current density is(2.1±0.1)·10¹²A/m². (b) DW displacements for the different DWs. The inset explains the DW labeling.

4.3 Domain Wall Motion in Permalloy 61

observed (most likely the DW is pinned at the kink by a defect) all DWs could be displaced by the current and show a similar stochastic behavior. The onset for the current-induced DW motion for this geometry is at (2.1±0.1)·10¹²A/m². At higher current densities the observed maximum velocity as well as the averaged DW velocity (blue line) increases. This analysis is repeated for different wire geometries and the critical current densities are extracted. In Fig. 4.4 these critical current densities are shown as a function of 1/width for different thicknesses. The data for the 25 nm thick wires was not obtained in this thesis but is taken from [JKB⁺06, Klä06]. The experimental setup, the sample layout and the pulse shape were similar, however SEMPA was used for the imaging of the wires.

The dependence of the critical current density on the wire width shows a more or less linear increase of the critical current density with the increasing inverse wire width for all thicknesses. In addition to the linear dependence, there seems to be an offset to the critical current density that depends on the wire thickness. The critical current densities for thinner wires are slightly higher than for thicker wires of the same width. Thus the critical current densityj_c might be approximated by:

j_c =f /w+j_o(t), (4.1)

with a universal constant f and an offsetj_o that depends on the wire thickness t.

The error of the critical current density is relatively high when comparing wires with different thicknesses, which makes a detailed analysis difficult. The fitted parameters for the different thicknesses are shown in Tab. 4.2. Due to the limited

Figure 4.4: Critical current density vs. 1/width for different wire thicknesses. For all wire thicknesses the critical current density increases with decreasing wire width.

62 Current-Induced Domain Wall Motion

parameter value

j₀(12nm) (1.6±0.3)·10¹²A/m² j₀(20nm) (1.0±0.5)·10¹²A/m² j0(25nm) (0.8±0.4)·10¹²A/m² j₀(28nm) (0.3± −)·10¹²A/m² f (0.5±0.1)·10¹²A/m²µm

Table 4.2: Fitted values for the linear model of the critical current density vs. wire width using Eq. 4.1.

number of data points the error for the different current offsets j₀ is rather large.

Assuming a constant slope f for the different wire thicknesses results in an error of about 20% forf.

In cooperation with Gen Tatara a simple theoretical model for the critical current density of vortex walls was developed that is able to explain the observed wire width dependence [HRC⁺09]. We start with the mechanism of the DW trans-formation based on vortex nucleations. More details on the current-induced DW transformations can also be found in the next section. This nucleation process leads to a threshold current for continuous vortex wall motion. The theory is based on a simple analytical description of the vortex neglecting thermal excita-tions and defects. It is assumed that the barrier necessary to flip the polarity and thus initiate the transformation governs the critical current density. In Fig. 4.5 simulations of the transformation process are presented with the color code repre-senting the out-of-plane component and the arrows indicating the in-plane mag-netization. The x-direction is chosen to be along the wire and y is the transverse direction so that the wire lies in the x-y plane and the VC points out-of-plane in the z-direction. Starting from a vortex wall in Fig. 4.5(a) the spin torque results in a vortex wall motion with a velocityv_s ≈u in the electron direction. The spin damping and effective force (non-adiabaticity) act as forces perpendicular to the motion with a magnitude proportional to αv_x and βj[SNT⁺06, TNMS05]. v_x is the VC velocity in thex-direction. The direction of the perpendicular VC motion is determined by the polarity of the VC [SNT⁺06, NST⁺08]. When the VC comes close to the wire edge, as shown in Fig. 4.5(b), it feels a repulsive force due to the increase in the stray field energy. This causes a deformation of the DW towards a transverse wall and will eventually stop the motion [KJA⁺05, HLZ06]. For the motion to proceed, the VC needs to flip its polarity and will then start to move towards the opposite wire edge. This DW propagation process requires a vortex–

anti-vortex pair to be nucleated both having a polarization opposite to the one

4.3 Domain Wall Motion in Permalloy 63

Figure 4.5: Micromagnetic simulation of the transformation process in a 300 nm wide and 30 nm thick permalloy wire. The color code shows the out-of-plane component (green means in-plane magnetized) and the arrows indicate the in-plane magnetization direction.

(a) The initial vortex wall starts to move under current injection in the direction of the electron flow (j= 2×10¹²A/m²). (b) In addition, the VC (red) also moves perpendicular to the electron flow towards one of the edges resulting in a distortion of the core (blue).

(c) If the distortion is large enough a vortex–anti-vortex pair is created follow by the immediate annihilation of the old vortex with the anti-vortex. (d) The remaining vortex has opposite polarity (blue) and starts moving towards the opposite edge of the wire.

of the original VC [WPS⁺06]. The nucleated anti-vortex will annihilate with the original vortex leaving a new vortex with opposite polarity to the original one, so that effectively the vortex polarity has switched. A snapshot of this mechanism is shown in Fig. 4.5(c). The out-of-plane component opposite to the original vortex polarity is already present in Fig. 4.5(b). It increases further until a vortex–anti-vortex pair is created followed by the immediate annihilation process [Fig. 4.5(c)].

The vortex with opposite polarity remains and starts moving in the other direction as shown in Fig. 4.5(d).

The creation energy of the vortex–anti-vortex pair is determined by the hard-axis anisotropy energy K⊥: E_K⊥ ≡2R

d³x¹₂J|∇_xS|² ∼2K⊥δ_vc² t/a³, where δ_vc is the size of the VC, ais the lattice constant, t is the thickness of the system and J is the exchange coupling. This energy has to be provided by the spin polarized current in order to secure continuous motion. Under applied current the DW gains spin transfer energy that is maximized in the limit of a transverse wall. From the Hamiltonian of the spin transfer effect [STK05]:

H_ST≡ Z ~P

2ej· ∇φ(1−cosθ)d³x, (4.2) the spin transfer energy of a transverse wall is estimated to be:

E_ST =π~tP

e jW, (4.3)

64 Current-Induced Domain Wall Motion

where we have assumed a simple two dimensional head-to-head DW profile (φ continuously changes from 0 toπandθ= ^π₂, see also Fig. 1.4(c)). ForEST> EK_⊥

a vortex–anti-vortex pair can be nucleated with the anti-vortex and the original vortex annihilating each other. The remaining vortex with the opposite polarity starts to move to the other wire edge reducing again the spin transfer energy. Thus the threshold current density given by:

j_c = 2e

This result yields a direct dependence of the critical current densityj_c on the wire width W that scales with j_c∝1/W. This contrasts with the case of the creation of a rigid transverse wall, where the threshold current is independent of the system size. In fact, when a transverse DW is created perpendicular to the current, the energy gained from the spin torque is estimated to be proportional to the wire width: E_ST ∼ ^~_e^PjW a, but the creation energy is also proportional to the width:

E_C ∼Kλ^W_a2 (K and λare the easy-axis anisotropy energy and the DW width).

This results in a size-independent threshold current density j_c∼ _~^e_a3Kλ[STK05].

Thus, the critical current density strongly depends on the DW spin structure and hence the critical current density for VC switching depends on the wire ge-ometry.

The experimental observations shown in Fig. 4.4 qualitatively agree with the theoretical model of the j_c ∝ 1/W scaling for vortex walls, where the critical current density increases with decreasing wire width for all thicknesses. To obtain a more quantitative comparison, we can estimate the magnitude of the threshold current density. Noting that the VC radius is given by the ratio of the exchange and anisotropy energy as a_v ∼p

J/K⊥, the threshold current (4.4) is written as j_c ∼ ^e_~aW¹ _a^J2. For 3d-ferromagnets generic values areJ/a² ∼10⁻¹⁹J (per site) and a∼2.5×10⁻¹⁰m. With this we obtain for the slopef =jc/w∼0.5·10¹²A/m²µm.

This result agrees with the slope of the critical current density obtained in the experiment. Given the larger number of assumptions in the theoretical model (no thermal excitations, simple analytical description of the vortex structure, no influence of defects, scaling of the vortex size with the wire thickness, etc.) the agreement between experiment and theory is quite well. The experimental results further suggest that in reality a combination of different depinning mechanisms prevail since the obtained curves are best described by a linear function with a thickness dependent offset. The slope seems to be constant and does not depend on the thickness in agreement with the theory.

4.3 Domain Wall Motion in Permalloy 65

The non-zero offset j_o could be explained by VC pinning that contributes to the critical current density. The relative strength of surface defects that could pin the VC is reduced at higher wire thicknesses. The pinning and hence the critical current densities therefore should be lower for thicker samples in agreement with the observations.

In conclusion, the dependence of the threshold current density for continuous vortex wall motion has been determined experimentally and theoretically. The-oretically, a scaling of j_c ∝ 1/width is found. This size-dependence arises from the fact that the vortex–anti-vortex creation energy is independent of the wire width, while the energy gain depends on the wall size, which scales with the wire width. In comparison, the threshold current density for rigid transverse walls is independent of the geometry.

Experimentally, the observed scaling of the critical current densities for vortex DWs is in good agreement with the theoretical model of periodic DW transfor-mations. An additional thickness dependent contribution of the critical current density is linked to the extrinsic pinning of the VC.

Parts of the results shown in this section have been published in Physics Review B [HRC⁺09].

4.3.2 Domain Wall Transformations

The DW transformation process introduced in the last section was discussed as a possible explanation for the observed scaling of the critical current density with the wire width. In this section, the process of current-induced DW transformations is discussed in more detail.

Simulation

To better understand the influence of spin-polarized current on the internal DW structure, micromagnetic simulations are carried out. To reduce the simulation time, a 300 nm wide and 15 nm thick permalloy wire is studied. A DW in the wire is created, current is injected and the time evolution of the system is studied. A sequence of images from this process is shown in Fig. 4.6. The time step between adjacent images is 1.5 ns. The injected current density is 3.3·10¹²A/m² and the corresponding spin torque velocity is u = 100m/s. The damping constant is α= 0.01 and the non-adiabaticity parameter is β = 0.06.

The image sequence in Fig. 4.6(a) reveals the transformation process. Starting from a vortex wall (VW1) the DW transforms after 6 ns to a transverse wall

66 Current-Induced Domain Wall Motion

(TW1). During this time the DW has moved by 2µm. A new vortex is nucleated with opposite polarity and the VC now moves towards the opposite wire edge. In Fig. 4.6(b) the DW displacement vs. time is shown (blue), calculated from the total magnetization component along the wire (mx). The corresponding DW velocity is shown by the red curve. The green curve belongs to the total magnetization component perpendicular to the current (m_y). For a symmetric VW the total magnetization along the y-direction is zero, whereas for TWs with magnetization pointing upwards (downwards) m_y is positive (negative).

The transformation process strongly effects the velocity. This is also visible in Fig. 4.6(c), where the trajectory of the VC inside the wire is shown. Starting in the center of the wire the VC moves to the lower edge, this transformation sequence yields the highest velocities of up to 300 m/s. When the DW transforms to a TW (TW1) the velocity drops to zero. A new VC is nucleated as indicated by the color change of the VC trajectory in Fig 4.6(c). The new VC starts moving to the opposite wire edge. Initially, its motion is roughly perpendicular to the electron flow and the DW velocity is low. Only after the VC has reached the wire center (VW2) the DW gains speed till the VC reaches the other wire edge and the velocity again drops to zero. The complete transformation process (VW1→VW3)

Figure 4.6: (a) Image sequence of one current-induced DW transformation period in a 300 nm wide and 15 nm thick permalloy wire. The time step between adjacent images is 1.5 ns. (b) DW displacement as a function of simulation time (blue) and the corresponding DW velocity (red) deduced from the total magnetization component along the wire mx. The total magnetization componentmycorresponds to the green curve. The gray shaded area corresponds to the first transformation cycle shown in (a). (c) Position of the VC in the wire during the current-induced DW motion. Red and blue corresponds to the VC polarity

4.3 Domain Wall Motion in Permalloy 67

yields an averaged DW velocity of 180 m/s with its peak velocity of 320 m/s. It should be noted that the DW velocity during DW transformations can be defined by different methods. For instance looking at the component of the VC velocity along the wire direction, results in a different velocity distribution, whereas the average velocity remains unchanged.

The ratio of average DWv andu is aboutv/u= 1.8. According to the theory the average DW velocity should be betweenu andβ/αu= 6u, since we are above the Walker breakdown, in agreement with the simulations.

Experiment

The experimental setup for the study of the DW transformations is similar to the previous ones, but with one small difference. Since special considerations are put

The experimental setup for the study of the DW transformations is similar to the previous ones, but with one small difference. Since special considerations are put

Im Dokument Manipulation of Magnetic Domain Walls and Vortices by Current Injection (Seite 70-87)