Tracking random walk of individual domain walls in cylindrical nanomagnets with resistance noise
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Tracking random walk of individual domain walls in cylindricalnanomagnets with resistance noise
Amrita Singh ∗ , Soumik Mukhopadhyay † , and Arindam Ghosh Department of Physics, Indian Institute of Science, Bangalore 560 012, India
Abstract
The stochasticity of domain wall (DW) motion in magnetic nanowires has been probed by mea-suring slow fluctuations, or noise, in electrical resistance at small magnetic fields. By controlledinjection of DWs into isolated cylindrical nanowires of nickel, we have been able to track the motionof the DWs between the electrical leads by discrete steps in the resistance. Closer inspection ofthe time-dependence of noise reveals a diffusive random walk of the DWs with an universal kineticexponent. Our experiments outline a method with which electrical resistance is able to detect thekinetic state of the DWs inside the nanowires, which can be useful in DW-based memory designs. ∗ electronic mail:[email protected] † electronic mail:[email protected] H ) or by electric current, is intimately connected tothe magnetization reversal mechanism, and hence influenced by the geometry and magneticanisotropy properties, as well as intrinsic disorder within the nanowires that act as pinningcenters. A consequence of this is the non-deterministic, or stochastic, kinetics of the DWswhich is a subject of great fundamental and technological importance [1, 4–9]. Randomnessassociated with the magnetization reversal process is the major source of stochasticity, anda common mechanism involves random walk of DWs through Barkhausen avalanches, wherethe DWs are treated as particles undergoing Brownian motion according to the Langevinequation [10]. Such a mechanism has been established in magnetic thin films, where theBarkhausen statistics reflects in universal scaling exponents in the avalanche size/durationdistribution functions [4, 7]. The viscous DW flow has also been observed in Mn-doped semi-conductor epilayers [11], but a clear signature of H -driven random walk of DWs in magneticnanowires has not been observed experimentally. This issue is of particular interest in thecase cylindrical magnetic nanowires, where the DWs behave as “massless” particles withzero kinetic energy [12], and hence whether the general mechanisms of stochastic kineticscan at all be relevant in this case is unknown. In this work, we provide the first unam-biguous evidence of diffusive random walk of DWs in cylindrical high-aspect ratio magneticnanowires by tracking their motion at small H (just above the depinning field). We showthat not only the stochasticity in DW kinteics within magnetic nanowires can be describedby Brownian diffusion with universal exponents, but the nature of the stochasticity can beemployed to probe the kinetic state of the DWs themselves.Quantifying stochasticity with conventional probes such as the Kerr effect, X-ray/electronor force microscopy, involve analyzing magnetization burst size, variations in DW displace-ment or the depinning fields etc, where the sensitivity to the evolution in the DW motion intime domain can be limited. Time of flight probing, for example in the context of first timearrival [4], or planar Hall effect [11, 13, 14], have been useful in locating DWs or measuretheir average velocity between spatially separated probes. Here we have adopted a differentroute, and measured the low-frequency fluctuations in longitudinal electrical resistance ( R )of magnetic nanowires at small H above the depinning threshold. In disordered metallic2ystems these fluctuations, often known as 1 /f -noise, are extremely sensitive to slow re-laxation of defects (dislocations, cluster of point defects etc). Random movement of thescatterers, even at a scale ∼ Fermi wavelength ( λ F < ∼ R of the nanowires. This can occur eitherthrough direct reflection when DW width ∆ ∼ λ F [16], or by spin-dependent scattering ofelectrons by the disorder inside the DWs [17, 18]. Recently, the fluctuations in R in differentforms of nano-magnetic structures have been associated with the motion of DWs [19, 20],although the details of the time dependence of R due motion of individual DWs remainunexplored.We have used nickel nanowires that were electrochemically grown inside anodic aluminatemplates - a well-characterized system in the context of magnetic storage [21–23]. Wehave used nanowires of average diameter ≈
200 nm, where strong shape anisotropy (aspectratio > N aOH solution. Nanowires were then drop-casted on flat silicon oxidesubstrates, after which electron-beam lithography was used to form Ti/Au contact padson the nanowire for electrical measurements. SEM micrograph of the device used for thepresent experiments appears in Fig. 1a where the length ( L ) of the nanowire between thevoltage probes (indicated by V + and V − ) was ≈ . µ m. Edge roughness, and also thebranching/clusters attached at the end of the nanowire, reduce the DW nucleation barriersubstantially [21]. To measure small changes in R , a dynamically balanced ac Wheatstonebridge arrangement was used (excitation frequency of 226 Hz). [see Ref. [20] and [25]for more details on noise measurements.] All measurements were performed with a verylow excitation current density ( < ∼ A/m ) to avoid heating, electromigration, or thespin-torque effect. The background fluctuations consisted mainly of Nyquist noise, and theresolution to change in R was ∼
10 ppm. Fig. 1b shows the magnetoresistance curves atthree different angles ( θ ) between H and the electric current density (nanowire long axis).The nanowire exhibits anisotropic magnetoresistance (AMR) where the switching field H sw ,identified by the dip in the AMR, increases continuously from ≈
250 Oe at at θ = 0 to the3aximum of ≈
850 Oe at θ = 90 ◦ . This indicates the magnetization reversal to occur viacurling mode as expected in nickel nanowires with diameter > ∼
45 nm [22].To find signature of the DWs, expected to be of vortex type in our case [2], we haverecorded R as a function of time at fixed values of H applied parallel to the nanowire axis. H was increased monotonically in small steps, starting from H = 0, and at every steptime dependence of R was measured over ≈
60 min. In Fig 2, time series recorded at fourdifferent H are shown. At very low H ( ≪ R are featureless witha power spectral density (PSD) of noise, S R /R ∝ f − α , where α ≈ H > ∼ R appear as a function of time as shown in the twolower panels of Fig. 2 for H = 1 . H somewhat obscuresthe visibility of the multi-level states, which disappear completely for H > ∼ H sw (time seriesnot shown). The same sequence was repeated over many magnetization cycles, indicatingthe phenomena to be due to application of H , and not due to relaxation of internal disorderdriven by temperature or electric current.Before analyzing the time-dependence of the fluctuations, we address the origin of thediscrete jumps in R at H > ∼ increase in R from itsbase value R ( ≈ . R or 2∆ R , where ∆ R ≈ H is kept fixed, AMR or Lorentz contributions to R do not change, and hence a naturalexplanation involves the DWs, which nucleate at the defect sites and travel intermittentlyacross the voltage probes. Increasing R by ∆ R and 2∆ R then corresponds to fitting adomain partially (one DW) or fully (two DWs) between the voltage probes, respectively.Indeed, the positive correction ∆ R can be quantitatively understood from the Levy-Zhangmodel of spin-mixing due to disorder scattering inside the DWs [17], which estimates thefractional change in R from the inclusion of a single DW between the voltage probes as(∆ /L ) × [1 + ξ ( ρ ↑ − ρ ↓ ) / ρ ↑ ρ ↓ ] ≈ . ρ ↑ and ρ ↓ correspond to resistivities of theup and down spin channels respectively with ρ ↑ /ρ ↓ ≈ ξ = π ¯ h k F / m e J ∆ ≈ k F = 1 . × m − , nickel exchange energy J = 4 . × − J, andDW width ∆ = 24 nm [23]. Experimentally, we find ∆
R/R . K ≈ . − .
5% which agreeswith the expected DW contribution within a factor of two (we used the low temperatureresidual resistance R . K ≈ . H shows direct jumps of 2∆ R which could be due tonucleation of domains within the region between the voltage probes. In the time domain,4he jumps did not show any regular pattern or sequence, which prompted us to focus on thefrequency domain through power spectral analysis.The PSD of the fluctuations in R over the entire ( ∼ hour long) time series was found tovary as S R /R = A R /f α (Fig. 3a), where both noise amplitude ( A R ) and α depend strongly(and non-monotonically) on H (Figs. 3b and 3c). Three regimes can be clearly identified,and understood in term of the DWs: (1) At H < δR /R = R ( S R /R ) df in Fig. 3c) is low and α is ≈ − . H -independent background noise arisesfrom slow relaxation of disorder (such as dislocations, vacancy clusters etc.), which has aPSD ∼ /f α , where α ≈
1, due to the broad distribution of associated time scales. (2)For intermediate H (1 Oe < H < ∼ H sw ), we identify a sharp increase in both δR /R and α ( ∼ . − . ∼ /f α with α ∼
2, and second, the fluctuations in R generated by any given DW during its flight betweenthe probes. The latter causes R to fluctuate in a given state, and embodies the stochasticityof DW propagation which will be treated separately. (3) Finally, for H > H sw the numberof domains diminish, and both δR /R and α return to their zero-field background values.Can the stochasticity in DW propagation be extracted from the kinetics of resistancenoise? To answer this we return to Fig. 2, and focus on the fluctuations only in the R = R +2∆ R state, which would correspond to one propagating domain ( i.e. two DWs) betweenthe voltage probes. For a preliminary time-of-flight analysis, we note that the time ( τ H ) thatthe system stays in this state corresponds to the time the domain takes to travel from onevoltage probe to the other. In Fig. 4 two histograms of τ H obtained at H = 1 . τ H decreases with increasing H , due toincrease in the DW velocity (inset). Two important points are to be noted here: (1) Thewidth of the velocity distribution deceases with increasing H , which can be attributed tothe H -induced reduction in the effective propagation barrier that suppresses (lower) partof the barrier energy distribution. (2) Secondly, the typical velocity is about five orders ofmagnitude lower than thin film-based magnetic nanostrips, which can be understood fromthe suppression of DW mobility ( ∼ ∆ . ) at greatly reduced DW width in nanowires ofcylindrical cross section [16].The PSD of noise in the high resistance state (spanning over τ H ) shows a strikingly uni-5ersal behavior. At all τ H segments (see typical time traces in Fig. 5a), the PSDs vary as S R /R ∼ /f α , where α = 1 . ± .
05 at both H = 1 . α was found to be ≈ . − . α = 1 . R originates from the movements of the DWs themselves.We suggest a mechanism with the help of the schematic shown in the inset of Fig. 5b,and the Levy-Zhang model of electron scattering within the DWs by disorder that mixesthe spin-up and spin-down channels [17]. As the DW moves the scatterers move to theopposite direction with respect to the DW. Hence the wave function of the electrons withinthe DW, which depends on the mistracking of the electron spin to local magnetization,“see” a time-varying layout of the scatterers. This will cause a time-dependent mixing ofthe spin-channels, i.e. ρ ↑ /ρ ↓ will fluctuate with time, leading to fluctuations in the measured R . The diffusive kinetics of the scatterers indicated by the PSDs in Fig. 5c, then impliesthat the DW itself moves by diffusive random walk from one the pinning center to the other,providing the first evidence of such a behavior in magnetic nano-systems. A distributionfunction of resistance jumps in these states is difficult to compare with the theoretical modelsthat associate universal exponents to distribution of DW displacements [8, 10], but we doobserve a power law behavior in such constructions with an exponent of ≈ .
7, which ispresumably non-universal (inset of Fig. 5c). Nevertheless, observation of α ≈ . α that is closer to unity.In conclusion, we have shown that low-frequency fluctuations in electrical resistance ofmagnetic nanowires can be a sensitive probe to domain kinetics under an applied magneticfield. Both noise magnitude and spectral exponent can detect the number fluctuation andpropagation stochasticity of the domain walls. We find the first evidence of random walkin the propagation of individual domains along the nanowire at small magnetic fields, that6isplay an universal kinetic exponent.We acknowledge the Department of Science and Technology, Government of India, forfunding the work. [1] S. P. Parkin, M. Hayashi, and L. Thomas, Science
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180 (2001).[24] J. H. Scofield, Rev. Sci. Instrum. , 985 (1987).[25] U. Chandni, S. Kar-Narayan, A. Ghosh, H. S. Vijaya, and S. Mohan, Acta Materialia ,6113 (2009).[26] U. Chandni, A. Ghosh, H. S. Vijaya, and S. Mohan, Phys. rev. Lett. , 7802 (1986)[28] A. Ghosh et al. , J. Phys. D: Appl. Phys. L75 (1997). IG. 1: (color online): (a) Scanning electron micrograph of the device used in the present experi-ments. The voltage and current probes are indicated as V + /V − and I + /I − , respectively. (b)Anisotropic magnetoresistance (AMR) for three different angles between the current and externalmagnetic field ( H ) . IG. 2: (color online): Time variation of resistance at four different magnetic field applied parallelto the long axis of the nanowire. The high resistance state for H = 1 . τ H . The dashed horizontal lines identify the discrete resistancestates observed in the time traces. IG. 3: (color online): (a) Noise power spectral density (PSD) at different values of H . Non-monotonic variation of (b) the spectral exponent and (c) the normalized variance in noise. In (b)different symbols signify different magnetization cycles. The switching field H sw obtained from theAMR measurements is also indicated in (b) and (c). IG. 4: (color online): The distribution of τ H for two magnetic fields ( H = 1 . v = L/τ H , where L is the distance betweenthe voltage probes, at the same values of H . IG. 5: (color online): (a) Resistance-time behavior within three high resistance states (see Fig. 2also). Power spectral density (PSD) of resistance fluctuations in this states is shown for (b) H = 1 . H = 3 Oe. Inset of (b): Schematic of electron scattering events within adomain wall which becomes time dependent as the wall moves between the voltage probes. Insetof (c): Distribution of resistance jumps (in the high resistance state) for H = 3 Oe.= 3 Oe.