Figure 4.6: Experimental visualization of current induced vortex core gyration. a) The relative phase responseΦis plotted as a function of the excitation frequencyf. The resonance frequencyfr= 334±12 MHzis determined by fitting the phase response of a harmonic oscil-lator. The corresponding vortex core trajectory atf = 340 MHznear the resonance frequency fr, see red data point, is plotted in b). The ellipse is fitted through the data points and its eccentricity is a measure of the asymmetric pinning potential well. Differential STXM snap-shots illustrate the gyration of the vortex core under continuous ac-excitation with 340 MHz.
To remove non-magnetic contrast, two images with out-of-plane XMCD contrast recorded at 180◦ phase difference are divided by each other. A full movie of the complete image sequence can be found in appendix C, movieC.1(electronic version only).
resolved vortex core trajectory, we then determine the relative phase response Φ along the y-axis, in relation to the exciting current.
The resulting phase response Φ is plotted in figure 4.6a as a function of the excitation frequency. To determine the resonance frequency, we fit Φ(f) with the phase response of the harmonic oscillator and findfr = 334±12 MHz. The corresponding vortex core trajectory at 340 MHz, close to resonance, is plotted in figure 4.6b. We measure an eccentricity of e = qA2y −A2x/Ay = 0.77±0.04 for the elliptical vortex core trajectory. The clockwise vortex core gyration (p = −1) is visualized by the time resolved images at Φ = −90◦, Φ = 0◦ and Φ = 90◦ showing differential out-of-plane XMCD contrast.
4.4 Determination of the asymmetric potential stiffness
To determine the shape of the asymmetric pinning potential, we correlate the vortex core trajectory at resonance to the analytical model, discussed in section 4.1. The potential stiffness κx and κy can be unambiguously determined from equations (4.6) and (4.7).
The measured vortex core trajectory at 340 MHz, close to the resonance frequency fr = 334 MHz, has amplitudes Ax = 20.5±1.2 nm and Ay = 32.0±0.9 nm. With these values
we can determine the potential stiffness coefficients:
κx = 4.7±0.4 · 10−3 kg/m2, κy = 1.9±0.2 · 10−3 kg/m2. (4.8) The absolute measured values of the amplitudes Ax and Ay are significantly smaller than the amplitudes calculated using the analytical model, Ax = 42.4 nm and Ay = 66.4 nm, respectively. This can be attributed to the fact that the absolute amplitudes at the reso-nance peak are very sensitive to the excitation frequency. Furthermore, due to reflections of the microwave injection, the effective current density can be lower than the nominal injected density ofje= 9.6 · 1010A/m2.
In conclusion, we have investigated the asymmetric pinning potential for a vortex domain wall employing time-resolved STXM imaging of current-induced vortex core gyration with only minor contributions from the Oersted field. From the gyration amplitudes at the resonance frequency we determine the shape and the stiffness of the local pinning poten-tial. This technique opens a way to obtain full information about the pinning potential landscape, which is key information with regard of controlling the positions of individual domain walls on a race-track by geometric tailoring of the potential landscape. In partic-ular, is has been shown that the depinning of vortex domain walls is a resonant process [9]
and the length of the current pulse needed to move the domain wall depends on the shape of the local pinning potential.
Parts of the results shown in this chapter have been published in [16].
CHAPTER 5
Probing the non-adiabaticity by direct imaging of vortex domain wall oscillations
Spin polarized electrons traveling through a magnetic domain wall, transfer their angular momentum to the local magnetic spin texture. The challenge is to identify the torques from the injected current on the local magnetization and to understand the transport of itinerant conduction electrons in inhomogeneous magnetic textures. A prominent example for the latter is the giant magneto resistance (GMR) effect, where the electronic transport strongly depends on the alignment of two ferromagnetic layers [47]. Here, the aim is to understand the inverse effect, of the momentum transfer of the spin polarized electrons on the local magnetization. This momentum transfer consists of the adiabatic spin torque [18], arising from momentum transfer from the itinerant electrons whose spin adiabatically aligns with the direction of the local magnetization, an the non-adiabatic spin torque [19], which is due to momentum transfer of non-equilibrium conduction electrons whose spin does not point in the direction of the local spin texture.
However, the origin of the non-adiabaticity is still controversial and under debate. The non-adiabaticity can be generated by different mechanisms, such as the spin-relaxation due to spin flip scattering [19] or spin mistracking in domain walls where large magnetization gradients are present [20–22]. Recently the role of the spin diffusion as an additional mechanism contributing to the non-adiabaticity has been presented [23]. It is large when two perpendicular in-plane magnetizations gradients ∂xM and ∂yM are present, such as in the vortex core.
Experimentally, it has been reported that the strength of the non-adiabaticityβdepends on the domain wall spin structure, i.e. transverse and vortex domain walls, or Bloch domain
61
a
90°-Néel walls Half-antivortices
b
Figure 5.1: Schematic Illustration of the vortex domain wall spin structure a) The vortex domain wall consists of a centered vortex encompassed by two half-antivortices with opposite polarity and90◦-Néel walls. b) Micromagnetic simulation of a vortex domain wall in a500 nmwide and30 nm thick permalloy wire. The color code indicates the magnetization component along the horizontal axis, blue (red) corresponds to magnetization pointing to the left (right). Clearly visible is the counter-clockwise vortex structure surrounded by two anti diametrically arranged transverse wall spin structures.
walls in nanowires with perpendicular magnetic anisotropy. In vortex cores, where large magnetization gradients are present, the non-adiabaticity has been precisely determined β = 0.15± 0.02 by the static and dynamic response of a vortex core to spin-polarized currents [44,115] in permalloy nanostructures. This is a relatively high value compared to the non-adiabaticity found in transverse domain walls β = 0.010±0.004 [116], where only small magnetizations gradients occur. However in very sharp domain walls in FePt nanowires with perpendicular magnetization anisotropy, a non-adiabaticity β ≈ α was measured [117]. The experimental observations indicate that the strength of the non-adiabaticity not only depends on the magnetization gradient, but also on the dimensionality of the domain wall spin structure and the material [23].
An ideal candidate to test the theoretical models on a single domain wall spin structure is the vortex domain wall. The vortex domain wall is composed of a vortex, where the magnetization curls in the plane of the wire around the central vortex core, where the magnetization turns in- or out-of-plane, two anti diametrically arranged half-antivortices and 90◦-Néel walls forming the domain wall boundaries [58], as illustrated in figure 5.1.
In the vortex core and at the half-antivortices large magnetization gradients are present, compared to the magnetization gradient at the 90◦-Néel wall boundaries. Furthermore, the spin structure of the vortex core has a two fold dimensionality, with strong magnetization gradients along the horizontal ∂xM and vertical ∂yM axis.
The aim of this work is to spatially resolve the strength of the non-adiabaticity in a vortex domain wall, i.e. at the vortex core and at the domain wall boundaries, and to correlate these findings to the mechanisms that lead to non-adiabatic transport. Following analytical
63 calculations and micromagnetic simulations, we demonstrate that the phase response is a direct measure of the strength of the non-adiabaticity. By direct imaging of current induced vortex domain wall oscillations, the phase response of the vortex core and the domain wall boundaries is measured and compared to micromagnetic simulations, in order to determine the strength of the non-adiabaticity.
The different mechanisms that lead to non-adiabatic transport are discussed in section5.1.
The role of the non-adiabaticity in the current induced steady-state vortex core gyration is then studied analytically in the framework introduced by Thiele [57,62], extended to include the adiabatic and non-adiabatic spin transfer torque [45,58,108], see section 5.2.
In particular, we are interested in the effect of the asymmetry of the pinning potential on the phase response of current induced vortex core gyration. This is important, because in the vortex domain wall, the potential landscape of the vortex core is asymmetric and tilted.
Employing time-resolved STXM, we imaged resonant oscillations of a vortex domain wall induced by alternating currents injected into the nanowire. Not only we determined the phase response of the vortex core gyration and the oscillations of the domain wall bound-aries, but also the vortex core trajectory at resonance. This allows us to precisely determine the pinning potential landscape of the vortex domain wall [16]. The experimental results are presented in section 5.3, including the description of the the experimental set-up.
The phase response of the vortex core gyration is a direct measure of the direction of the driving force, which is determined by the relation of the strengths of the adiabatic and non-adiabatic spin transfer torque. To relate the measured phase response of the vortex domain wall spin structure oscillations to the strength of the non-adiabaticity quantitatively, we performed micromagnetic simulations, see section 5.4. For free vortex domain walls, we find a small intrinsic domain wall resonance at frequencies much higher than the gyrotropic resonance frequency. Resonant oscillations of the vortex domain wall only occur in the presence of a confining pinning potential, which has to be introduced artificially in the micromagnetic simulations. The phase response of the vortex domain wall oscillations is determined by the shape of the pinning potential, which, on the other hand, depends on the shape of the geometric confinement. Therefore, to corroborate our experimental findings it is important to model the exact same shape of the pinning potential, namely the asymmetry and the orientation with respect to the injected current.
In order to determine the strength of the non-adiabatic spin transfer torque, we correlate the experimentally measured phase response to the analytical model and micromagnetic simulations in section 5.5. The best fit is obtained for β = 8.5±0.8α, which is in good quantitative agreement with the non-adiabaticity found previously [115,116,118]. Addi-tionally, we can determine a higher limit for β < ∼13α from the qualitative shape of the experimentally observed phase response. The conclusion can be found in the last section 5.6 of this chapter.
5.1 Origin of the non-adiabaticity of the spin transfer torque
The aim of this section is to give an overview over the recent theoretical work on the origin of the non-adiabatic transport, starting with the most widely accepted form of the spin transfer torque [19,45]
T=bJ(u·∇)M−β bJ
MSM×(u·∇)M, (5.1)
where bJ = P jeµB/eMS is the adiabatic spin torque and β describes the strength of the non-adiabaticity of the spin torque. The strength of the spin torque is proportional to the spin drift velocityu. The first term describes the momentum transfer from electrons whose spin adapts adiabatically the local magnetization direction. The second term describes the momentum transfer of electrons whose spin does not point in the direction of the local magnetization, and therefore has a component perpendicular toM. Such non-adiabaticity can occur due to various mechanisms, that depend on the material properties and on the local spin texture.
The essence of all the theoretical models is to distinguish between two types of electrons;
the spin dependent transport is provided by the conduction electrons near the Fermi level, and the local magnetization, which is defined by the localized d-electrons below the Fermi sea [19]. The interaction between these electrons is generally modeled by the s-d Hamilto-nian, but in a "real" ferromagnet it is impossible to unambiguously separate the itinerant conduction electrons from the localized electrons of the magnetization,
Hsd = SJex
Ms s·M(r, t). (5.2)
Jex is the strength of the exchange coupling and s are the spins of the itinerant spin polarized s-electrons. The spins of the localized d-electrons are substituted by the classical magnetizationS/S =−M/MS. This semi classical approximation holds, because generally these magnetization dynamics are much slower than the spin dynamics of the itinerant electrons.
Following this semi classical treatment, various models of non-adiabatic transport have been derived recently [18–20,22,23]:
Non-adiabatic transport from spin relaxation due to spin flip scattering
By introducing a relaxation time model for the spin relaxation, <Γre(s)>=δm/τsf into the spin continuity equation for the spin densitym=<s>, Zhang and Li [19] derived an equation for the spin torque T, which the spin density exerts on the magnetization:
T= 1 1 +ξ2
"
− n0 MS
∂M
∂t +ξn0
MS2M× ∂M
∂t − 1
MS2M×[M×(u·∇)M]− ξ
MSM×(u·∇)M
# , (5.3)
5.1. Origin of the non-adiabaticity of the spin transfer torque 65 where the strength of the non-adiabatic spin torque ξ = τex/τsf depends on the ratio between the spin precession time τex (≈ 10−14s) and the spin flip relaxation time τsf (≈10−12 s), which describes the spin relaxation due to scattering with electrons, phonons and material impurities etc.
The first two terms are independent of the current and only evolve from time varying magnetization. These two terms can be integrated in the standard Landau-Lifshitz-Gilbert (LLG) equation by renormalizing the gyromagnetic ratioγ and the phenomenological dam-ping term α,
γ0 =γ(1 +η)−1, γ0α0 =γ(α+ξη), (5.4) whereη = (n0/MS)/(1 +ξ2) andn0 is the local equilibrium spin densitys whose direction is parallel to the magnetization M. The second two terms represent current induced magnetization dynamics and scale with the current density je.
In permalloy bothη≈ξ≈10−2are relatively small numbers and thus only slightly modifies the gyromagnetic ratio γ0 and the damping α0 [19]. Furthermore the strength βsr ≈ ξ of the non-adiabaticity only depends on material properties such as the exchange couplingJex and the spin-flip relaxation time τsf and this theory does not yield weather α > β,α = β orα < β, which is crucial for applications of current induced domain wall motion.
Non-adiabatic transport in narrow domain walls
Tatara and Kohno [18] reformulated the problem of domain wall dynamics in the presence of electric current introduced by Berger [74]. Following the collective coordinates model of magnetic domain walls, the dynamics of the domain wall are described by its position X and the polar angle φ,
~N S
∆ φ˙+αX˙
∆
!
=Fpin+Fel
~N S
∆
X˙ −α∆ ˙φ= K
2 sin(2φ) +Tel,
(5.5)
where ∆ is the domain wall width, Fpin is the pinning force andFel represents force from the momentum transfer from the electrical electrical current to the domain wall, Fel is proportional to the domain wall resistance. Tel is the spin transfer torque from the spin angular momentum transfer of the conduction electrons to the domain wall.
The electrical current affects the domain wall motion in two ways. First, a force acts on the domain wall position X due to reflection of conduction electrons on the domain wall, this effect is proportional to the domain wall resistance. Second, spin angular momentum is transferred from the conduction electrons to the domain wall, acting as a torque on the polar angleφ. In the case of wide domain walls (adiabatic limit), the domain wall resistance is negligible and thus the spin-transfer effect due to spin current dominates Fel = 0, i.e.
only adiabatic spin transfer torque. In contrast, the resistance of a narrow domain wall
Figure 5.2: Strength of the non-adiabaticity in narrow domain walls a) The non-adiabaticity a function of the wall width, scaled by the characteristic length L (from [20] FIG. 7). Note the logarithmic scale. b) The non-adiabatic spin torque as a function of the wall widthλ(from [22] FIG.
2). The non-adiabaticity is oscillatory and negative for certain domain wall widths.
is large and more electrons are reflected, in this case the domain wall is driven by the non-adiabatic momentum transfer effect Fel > Tel/∆.
Starting also with the free electron Stoner model, equation (5.2) Xiao and Zangwill [20] cal-culated quantum mechanically the non-adiabatic effects from continuously varying magne-tization in one dimension. They find that the non-adiabatic torque only occurs for domain walls with widths w that are smaller than the characteristic length
L= EF Eex
1 kF = kF
kB2 . (5.6)
They calculated that the non-adiabatic torque is oscillatory and nonlocal in space, be-cause of spin precession of the single-electron spin moments around the local effective field Bef f(x) from the local magnetization. The strength of the non-adiabaticityb(x) decreases monotonically for wider domain walls as,
max|b(x)| ∝ 1
we−γw/L, (5.7)
where γ is a constant that describes the the probability of a majority electron retaining its spin when passing through the domain wall, it depends on the sharpness of the domain wall. In figure 5.2a the relative strength of the non-adiabatic spin torque is plotted as a function of the domain wall width w [20].
In a similar way Bohlens et al. [22] derived the oscillating non-adiabaticity in the case of ballistic spin and diffusive charge transport in narrow domain walls. They find that not only the strength, but also the sign of the non-adiabaticity ξ depends on the domain wall widthλ, as shown in figure 5.2b. What is more, they predict that by tuning of the domain wall width, the ratio ξ/α can be changed from ξ > α to ξ < α. This calculation is in contrast to the calculations of Xiao, but both models yield an enhanced non-adiabatic spin transfer torque in one dimensional narrow domain walls.
5.1. Origin of the non-adiabaticity of the spin transfer torque 67 In summary, the non-adiabatic transport in narrow domain walls is enhanced in two ways, the electrons are more strongly “reflected”e when passing narrow domain walls with large domain wall resistance and due to enhanced spin accumulation and electron spin precession of the conduction electrons around the local magnetic moments within the domain wall.
Non-adiabatic transport due to spin diffusion
Recently Manchon et al. [23] rigorously followed the derivations of Zhang and Li [19]
and calculate the additional spin torque exerted on the magnetic texture from the spin diffusive term (∝ ∇2δm). They find the spin torque exerted on the magnetic texture T=Tren+Tst+Td, whereTrenis the renormalization torque (∝∂tM) which renormalizes the gilbert damping parameter γ and the phenomenological damping parameter α, Td is the additional damping term due to the spin motive force (SMF) [119] and
Tst = (u·∇)M
| {z }
Adiabatic spin torque
− β MS
M×(u·∇)M
| {z }
Non-adiabatic spin-torque
+λ2ex MS
∇2[M×(u·∇)M]
| {z }
Spin diffusion term
. (5.8)
This term scales with the spin drift velocity u, including the adiabatic and non-adiabatic spin torque terms and additionally the spin diffusion term, λ2ex = D0τex/(1 +β2) is the transverse spin diffusion length in metallic spin valves, whereD0 is the diffusion coefficient [23]. They derive the effective non-adiabaticity for a transverse or a Bloch domain wall
βef f =β+ λ2ex
2∆2, (5.9)
and for the vortex structure
βef f =Cβ+28λ2ex
3ρ2V , (5.10)
where C = 1 + 0.5 lnRρVV [63] and RV, ρV are the radii of the vortex structure and the vortex core, respectively. The spin torque due to spin diffusion enhances the effective non-adiabaticity both, in the case of a Bloch wall and of a free vortex core, inverse proportionally to the square of the domain wall width∝∆−2or the vortex core radius∝ρ−V2. Furthermore the effect is enhanced by a factor of ≈ 4 in the case of a vortex structure compared to the abrupt Bloch wall. This is due to the two dimensional nature of the vortex structure, where magnetization gradients along the horizontal ∂xM and the vertical ∂xM axis are present, which is not the case in a one dimensional Bloch wall.
Finally, let us discuss the relevance of the theoretical models to recent experimental find-ings. From thermally activated hopping of magnetic domain walls Eltschka et al. [116]
measured a non-adiabaticity βT W ≈ α for transverse domain walls in permalloy. On the other hand, for vortex domain walls they find a significantly larger non-adiabaticity βV W ≈8α, in agreement with [9]. These results are in agreement with the measurements
of the displacement of a vortex core under constant current [44], where a non-adiabaticity
of the displacement of a vortex core under constant current [44], where a non-adiabaticity