We study current-induced domain-wall dynamics in a thin ferromagnetic
nanowire. The domain-wall dynamics is described by simple equations with four
parameters. We propose the procedure to determine these parameters by
all-electric measurements of the time-dependent voltage induced by the
domain-wall motion. We provide an analytical expression for the time variation
of this voltage. Furthermore, we show that the measurement of the proposed
effects is within reach with current experimental techniques.Comment: 5 pages, 5 figures, update to published versio

Abanov, Ar.

Liu, Y.

Tretiakov, Oleg A.

English

arXiv

Journals published by the American Physical Society can be found at http://journals.aps.org/Current-induced domain-wall dynamics is studied in a thin ferromagnetic nanowire. The domain-wall dynamics is described by simple equations with four parameters. We propose a procedure to unambiguously determine these parameters by all-electric measurements of the time-dependent voltage induced by the domain-wall motion. We provide an analytical expression for the time variation of this voltage. Furthermore, we show that the measurement of the proposed effects is within reach of current experimental techniques

Tretiakov, O. A.

Abanov, Artem.

OAKTrust Digital Repository (Texas A&M Univ)

PHYSICAL REVIEW B 84, 052403 (2011)Electrical signature of magnetic domain-wall dynamicsY. Liu, O. A. Tretiakov, and Ar. AbanovDepartment of Physics & Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA(Received 18 July 2011; published 16 August 2011)Current-induced domain-wall dynamics is studied in a thin ferromagnetic nanowire. The domain-wall dynamicsis described by simple equations with four parameters. We propose a procedure to unambiguously determine theseparameters by all-electric measurements of the time-dependent voltage induced by the domain-wall motion. Weprovide an analytical expression for the time variation of this voltage. Furthermore, we show that the measurementof the proposed effects is within reach of current experimental techniques.DOI: 10.1103/PhysRevB.84.052403 PACS number(s): 75.78.Fg, 75.60.Ch, 85.75.−dIntroduction. Recently, applications for future memoryand logic devices, as well as important fundamental physicsquestions, have stimulated a number of experimental1–8 andtheoretical9–11 studies of the current-driven domain wall (DW)dynamics in ferromagnetic nanowires. It has been shownthat DWs can be moved by a current either parallel1–6 orperpendicular to the wire.7,10,11 In some of the experimentsshort current pulses were employed to depin a DW frompinning sites.2,3,6 Furthermore, the topological electromotiveforce induced by DW dynamics in a vortex DW has beenstudied both experimentally and theoretically.8,12A conventional experimental method to study the DWdynamics in nanowires is to measure the average DW velocityusing Kerr polarimetry,13 x-ray microscopy,4 or electronmicroscopy.5,14 These types of experiments require a com-plicated setup which is separate from the one needed for theDW manipulation. This situation is neither ideal for studies ofDW dynamics nor for further technological advances.In this Brief Report we propose a way to use the sameexperimental setup for both current DW manipulation andsimultaneous measurements of DW dynamics. Our mainresults are that the time-dependent voltage induced by theDW motion15,16 can be used to fully and comprehensivelydetermine the effective parameters of the DW dynamics.This proposal follows from the fundamental properties of thecurrent-induced DW motion, namely, (i) Applied dc current(above critical value) produces voltage with ac components.(ii) Applied ac current induces phase shifted ac voltage. Themagnitude of the proposed effects is calculated to be withincurrent experimental resolution.Similar techniques have already shown promise in magneticfield driven DW systems.17 This method should make itmore feasible to utilize DW dynamics for device applications.Furthermore, the proposed systematic approach can be usedto compare the extracted phenomenological parameters of theDW dynamics for a system described by arbitrary underlyingHamiltonian to those of microscopic theories.Model. The dynamics of the magnetization S in a quasi-one-dimensional wire is described by Landau-Lifshitz-Gilbert(LLG) equation with current j ,18,19˙S = −S × He − j∂zS + βjS × ∂zS + αS × ˙S, (1)where He = −δH/δS is the effective magnetic field givenby the Hamiltonian H of the system, S = M/|M| is a unitmagnetization vector, α is the Gilbert damping constant, βis the nonadiabatic spin torque constant, ∂z ≡ ∂/∂z where zˆis along the wire, and the time is measured in units of thegyromagnetic ratio γ0 = g|e|/(2mc). DWs in a ferromagneticwire can be modeled by a spin HamiltonianH which containsexchange, spin-orbit,20 and dipolar interactions. In a thin wire,the latter can be approximated by two anisotropies: a stronganisotropy along the wire (λ) and a weak anisotropy transverseto it (K). In realistic systems α,β  1 and K  λ.In a thin wire, a lowest-energy magnetization configuration(at j = 0) is uniformly ordered along the z or −z direction.A static DW is the next low-energy configuration with theboundary conditions Sz(±∞) = ±1 or Sz(±∞) = ∓1. DWscan be injected in the wire using different techniques. A sketchof a wire with a DW of width, determined by the Hamiltonianparameters, is depicted in Fig. 1.For small enough applied currents, it can be shown that theDW in a thin wire is a rigid spin texture14 and its dynamics canbe described in terms of only two collective coordinates.21,22These coordinates correspond to the two softest modes of theDW motion: the DW position along the wire z0 and the rotationangle φ of the magnetization in the DW around the wire axis(see Fig. 1). It has been shown22,23 that the equations of motionfor the DW in a thin ferromagnetic wire are model independentand can very generally be written in the formz˙0 = Aj + B[j − jc sin(2φ)], (2)˙φ = C[j − jc sin(2φ)]. (3)Here all current nonlinearities are neglected, since the largecurrents leading to observable nonlinear effects would burnthe nanowire. For a dc current below the critical value jc, thatis, j < jc, Eq. (3) implies that the DW tilts from the transverseanisotropy plane by the angle that satisfies sin(2φ) = j/jcaround the wire axis and then moves along the wire with aconstant velocity Aj . For j > jc, the DW constantly rotateswhile moving.The coefficients A, B, C, and the critical current jc arethe parameters that fully describe the DW dynamics. Theycan be calculated microscopically for certain toy models,22but in general they vary for different wires and depend onthe temperature and nanofabrication details. Therefore, in thisBrief Report we propose a way to determine these coefficientsby model-independent measurements of an induced ac voltagedirectly from an experiment suitable for all-electric DWmanipulation. As we show below, this ac voltage can beinduced by applied dc currents and by certain time-dependent052403-11098-0121/2011/84(5)/052403(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW B 84, 052403 (2011)FIG. 1. (Color online) A moving head-to-head domain wall ofwidth . The DW is centered at z0 and is tilted by an angle φ.current pulses with parameters similar to those achieved inrecent experiments.24,25Microscopically the dynamics parameters can be obtainedin the following way. The energy of a static DW, E0(z0,φ) =∫ H[S0(z,z0,φ)]dz, where S0 is a solution of a static LLGwith K = 0, in general depends on both z0 and φ. However,assuming that the wire is translationally invariant (pinningcan be neglected), E0 would not depend on the DW positionz0 and therefore ∂z0E0 = 0. The only contribution to E0 thatdepends on the angle φ comes from the small anisotropy inthe transverse plane, E0(φ) = −κ cos(2φ).22,26 This allows usto find the coefficients in Eqs. (2) and (3) in terms of theparameters of the LLG (1).22,27 Up to first order in α and βthey areA =˜βα˜, B = α˜ −˜βα˜(1 + α˜azφ), (4)C = (α˜ − ˜β)azz, jc = α˜α˜ − ˜β κ, (5)where α˜ = αD, ˜β = βD, D =√azzaφφ − a2zφ , azz =12∫dz(∂zS0)2, aφφ = 12∫dz(∂φS0)2, and azφ = 12∫dz∂zS0 ·∂φS0. Equations (4) and (5) are consistent28 with the expres-sions for A, B, C, and jc found in Ref. 22.We now outline the method to find A, B, C, and jc directlyfrom all-electric measurements. It is based on measuring theac voltage V induced by a moving DW. To find V one has toknow the time evolution of the total energy (per unit area ofthe wire’s cross section) in the system,˙E =∫dzδHδS ·˙S(z). (6)In general, DW energy has two contributions: the powersupplied by an electric current and a negative contributiondue to dissipation in the wire. Using the general solution ofthe LLG, Eq. (1), one can obtain the derivative of the energyas22,27˙E = 2[βazzz˙0 + (1 − βazφ) ˙φ]j − α∫dz ˙S20. (7)The last term on the right-hand side of Eq. (7) describes thedissipation and is therefore always nonpositive. Meanwhile,the first term is proportional to the current density j and givesthe power Vj supplied by the current. With the help of Eqs. (4)and (5) and adopting the approximation D  1 of Ref. 22 weobtain the expression for the induced DW voltage,29V = A2CBj + C(1 + A)[j − jc sin(2φ)]. (8)Note that Eq. (8) gives the contribution to the voltagedue to DW motion. This contribution is in addition to theusual Ohmic one. The voltage V in Eq. (8) is measured inunits of PgμB/(eγ0) and the current density is measured inunits 2eM/(PgμB ), where P is the current polarization. Weemphasize that unlike in the previously studied cases,8,12 thisvoltage is not caused by the motion of topological defects(vortices) transverse to the wire.Measurement of coefficients A, B, C, and jc. In order to findcoefficients A, B, and C, we propose three independent mea-surements of the voltage induced by a moving DW. Althoughthere are various factors affecting the nanowire resistance,the contributions from most of them are independent of DWmotion and therefore give only a constant component of theresistance. To characterize the DW dynamics, one has toconcentrate only on the resistance variations in time. Ourestimates show that the amplitude of voltage oscillationsdue to DW motion is of the order of 10−7 V and thereforeexperimentally measurable.Equation (8) implies that the voltage of the DW can give allthe necessary information about DW dynamics. Namely, onecan obtain C by measuring the voltage changing with time andparameters A and B by measuring the amplitude of the voltageoscillations.Slopes measurement. In Refs. 23 and 30 it was proposed toobtain A, B, and jc by measuring the drift velocity of the DW,〈z˙0〉. It is important to note that Eq. (8) has the same form asEq. (2). Thus, instead of measuring the drift velocity, whichrequires a more complicated experimental setup, we proposeto perform all-electric measurements. Namely, to measure theaverage voltage of DW (〈V 〉) as a function of dc current.From Eq. (8) one can see that 〈V 〉 = A2CBj for j < jc, whereas〈V 〉 = A2CBj + (1 + A)C√j 2 − j 2c for j > jc (see Fig. 2). Thecritical current is determined by the end of the region linear inj for small currents. The measurement of slope k1 at j < jcand slope k2 at j 	 jc gives the two independent quantities:k1 = A2CB, k2 − k1 = (1 + A)C. (9)Instead of measuring voltage average for dc current, one canapply a linearly increasing time-dependent current j (t) = qtbelow the critical value jc. At sufficiently small q the voltagewill also be linear in time, V (t) ≈ A2CBqt . By measuring thisvoltage one can findV (t)j (t) =A2CB. (10)Once C is determined, Eqs. (9) give A and B. The drawbackof this measurement is that it might be hard to disentangle k1and k2 from the Ohmic contribution. However k2 − k1 is freefrom the Ohmic resistance of the wire.In order to find C, the most intuitive approach is to input adc current slightly above jc. Then the voltage induced by the052403-2BRIEF REPORTS PHYSICAL REVIEW B 84, 052403 (2011)FIG. 2. (Color online) Dependence of average voltage 〈V 〉 on dccurrent j for C > 0 and C < 0, respectively, see Eq. (8). The slopeat j < jc gives A2CB, whereas at j 	 jc it gives A2CB + (1 + A)C.moving DW will oscillate with the period of the double angleφ (see the insets of Fig. 3). The half-width of the peak (dip) forC > 0 (C < 0) is given by arccos(jc/j )/(|C|√j 2 − j 2c ). Themeasurement of the voltage oscillations period T0 (which weestimate to be ∼10−7–10−6 s) determines C at a given j :|C| = 1T0∫ π0dφj − jc sin(2φ) =πT0√j 2 − j 2c. (11)For j − jc  jc, the period diverges but the half-width∼1/(Cjc) stays finite. To obtain the period T0, one can performthe Fourier transform of V (t) to find the frequency f0 = 1/T0(see Fig. 3).To determine coefficient A in the same experiment, onecan measure V = Vmax − Vmin = 2(1 + A)|C|jc (see insetsof Fig. 3). ThenA = V2|C|jc − 1. (12)Note that V = 2(k2 − k1)jc and therefore this experimentcan also provide a crosscheck with the aforementionedmeasurement of the slopes.FIG. 3. (Color online) Fourier transform of the voltage V as afunction of frequency f at the dc current 1.1jc. The insets show V asa function of time t for C > 0 given by α = 0.02 and β = 0.01; andfor C < 0 given by α = 0.01 and β = 0.02. The voltage period isT0 = 1/f0. In the inset for C < 0, the voltage varies between Vmax =0.041jc/ and Vmin = −0.019jc/.FIG. 4. (Color online) Input current j (dashed line) and measuredvoltage V (solid line) as functions of time t . (a) and (b) show the phasedelay θ between the current maximum and voltage extremum forC > 0 and C < 0, respectively. (c) and (d) depict V (t) at θ = 0 forthe same C > 0 and C < 0, respectively.Phase shift experiment. Another method to measure thecoefficient C is by applying an ac current j = j0 sin ωt withj0 > jc, which has only a short time interval where j > jc, sothat there is only one period of voltage within the period ofj (t). One can measure the phase delay θ between the currentmaximum and voltage extremum31 (see Fig. 4). Next, one fixesthe amplitude j0 and tunes the frequency ω until θ = 0. Inthis case, for j0 − jc  jc, we can use half of the time intervalfor which the current pulse is above jc to approximate theperiod of φ by dc current j0 as12ω(π − 2 arcsin jcj0)≈ π|C|√j 20 − j 2c. (13)FIG. 5. (Color online) Voltage (solid line) evolution after thecurrent (dashed line) is turned off at time ti for C > 0 given byα = 0.02 andβ = 0.01. Inset: the same dependencies forC < 0 givenby α = 0.01 and β = 0.02. The measurement of Vf is performed atti + t . The region encircled by the dotted line cannot be describedwithin our approach but it is too small to effect our results.052403-3BRIEF REPORTS PHYSICAL REVIEW B 84, 052403 (2011)For j0 − jc  jc, Eq. (13) can be further simplified to give|C| ≈ πω2(j0 − jc) . (14)In other words, when ω ≈ C(j0 − jc) which correspondsroughly to ω ∼ 107 Hz, the current pulse covers only oneperiod of voltage. Our simulations show that the expression(14) works sufficiently well for j0  1.3jc. The sign of C isdetermined by the extremum of the measured voltage: C > 0if V has the minimum and C < 0 if V has the maximum.Our simulations show (Fig. 4) that in addition to the largepeak (dip) of voltage there is a smaller one with the oppositecurvature. This is because when j (t) reaches jc, the angle φhas not yet rotated to the angle corresponding to sin(2φ0) = 1due to the cumulative phase delay between current and voltage.Abrupt current pulse experiment. It is also possible tomeasure the coefficient C for currents below the critical valuejc. The constant |C|jc determines the internal time scale ofthe DW motion. After one switches the subcritical current offat time ti , the voltage asymptotically decays as exp(−2|C|jct)(see Fig. 5). To measure the decay of V (t) with time, one inputsa dc current below jc, then measures voltage Vi immediatelyafter turning off the current at ti , and then later measuresvoltage Vf at time ti + t . We note that right after turning offthe current there is a short time period when the DW dynamicscannot be described by Eqs. (2) and (3). It corresponds tothe dynamics of fast degrees of freedom. This process has acharacteristic time ∼10−11 s which is typically much smallerthan the voltage decay time ∼10−8 s. Thus we can safelyassume that the rotation angle φ does not change much duringthis time interval, and we find|C|  12tjcln2Vi/Vf1 +√1 − j 2/j 2c , (15)which is valid for Vi/Vf 	 1. For example, estimatingVi/Vf = 10 we find |C| ≈ 1.17/(tjc). The sign of C canbe easily determined by the form of voltage decay (see Fig. 5).To summarize, we propose several all-electric measure-ments of the parameters fully describing domain-wall dy-namics in thin ferromagnetic nanowires. These measurementsare based on the voltage induced by a moving DW inresponse to certain current pulses. Our proposal allows forexperiments which are suitable not only for all-electric DWmanipulation but also for the simultaneous measurement of theDW dynamics. These findings give an experimental methodto determine the DW dynamics parameters, which can thenbe compared to microscopic theories. The procedure wedescribed works for a given temperature regime. It may also beused to investigate the temperature dependence of the effectiveparameters. Future work will include accounting for pinningeffects, which break translational invariance in the wires.32We thank I. V. Roshchin, J. Sinova, and E. K. Vehstedt forvaluable discussions. This work was supported by the NSFGrant No. 0757992 and Welch Foundation (A-1678).1A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo,Phys. Rev. Lett. 92, 077205 (2004).2L. Thomas et al., Nature (London) 443, 197 (2006).3L. Thomas et al., Science 315, 1553 (2007).4G. Meier, M. Bolte, R. Eiselt, B. Kruger, D. H. Kim, and P. Fischer,Phys. Rev. Lett. 98, 187202 (2007).5J. Rhensius, L. Heyne, D. Backes, S. Krzyk, L. J. Heyderman,L. Joly, F. Nolting, and M. Klaui, Phys. Rev. Lett. 104, 067201(2010).6D. Ilgaz et al., Phys. Rev. Lett. 105, 076601 (2010).7C. T. Boone, J. A. Katine, M. Carey, J. R. Childress, X. Cheng, andI. N. Krivorotov, Phys. Rev. Lett. 104, 097203 (2010).8S. A. Yang, G. S. D. Beach, C. Knutson, D. Xiao, Q. Niu, M. Tsoi,and J. L. Erskine, Phys. Rev. Lett. 102, 067201 (2009).9See, e.g., G. S. D. Beach, M. Tsoi, and J. L. Erskine, J. Magn.Magn. Mater. 320, 1272 (2008) and references therein.10P. Yan and X. R. Wang, Appl. Phys. Lett. 96, 162506(2010).11P. Yan, Z. Z. Sun, J. Schliemann, and X. R. Wang, Europhys. Lett.92, 27004 (2010).12S. A. Yang, G. S. D. Beach, C. Knutson, D. Xiao, Z. Zhang,M. Tsoi, Q. Niu, A. H. MacDonald, and J. L. Erskine, Phys. Rev. B82, 054410 (2010).13G. S. D. Beach et al., Nat. Mater. 4, 741 (2005).14M. Kla¨ui, P. O. Jubert, R. Allenspach, A. Bischof, J. A. C. Bland,G. Faini, U. Rudiger, C. A. F. Vaz, L. Vila, and C. Vouille, Phys.Rev. Lett. 95, 026601 (2005).15Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77, 134407(2008).16R. A. Duine, Phys. Rev. B 79, 014407 (2009).17A. Singh, S. Mukhopadhyay, and A. Ghosh, Phys. Rev. Lett. 105,067206 (2010).18Z. Li and S. Zhang, Phys. Rev. Lett. 92, 207203 (2004).19A. Thiaville et al., Europhys. Lett. 69, 990 (2005).20It results in crystalline anisotropy, Dzyaloshinskii-Moriya interac-tion, etc.21G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004).22O. A. Tretiakov and Ar. Abanov, Phys. Rev. Lett. 105, 157201(2010).23O. A. Tretiakov, Y. Liu, and Ar. Abanov, Phys. Rev. Lett. 105,217203 (2010).24M. Kla¨ui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini,E. Cambril, L. J. Heyderman, F. Nolting, and U. Rudiger, Phys.Rev. Lett. 94, 106601 (2005).25L. Heyne et al., Appl. Phys. Lett. 96, 032504 (2010).26For the Hamiltonian of Ref. 22 this constant is κ =πK2/ sinh(π) which reduces to κ = K for   1.27O. A. Tretiakov and Ar. Abanov (unpublished).28For the Hamiltonian used in Ref. 22, D = 1, azφ = , azz =(1 + 22)/, and  = D/J , where D and J are, respectively,the Dzyaloshinskii-Moriya and exchange interaction constants.29Since in majority of materials both α  1 and azφ =   1, wecan be safely neglect α compared to 1 in Eq. (8).30O. A. Tretiakov, Y. Liu, and Ar. Abanov, J. Appl. Phys. 109, 07D505(2011).31Our simulations show that the initial phase of angle φ does not affectthe phase delay, since the time it takes for the current to increasefrom 0 to jc is long enough to adjust the initial angle to the onecorresponding to (2φ) ≈ j/jc.32Y. Liu, O. A. Tretiakov, and Ar. Abanov (unpublished).052403-4

Electrical signature of magnetic domain-wall dynamics

Y. Liu

O. A. Tretiakov

Ar. Abanov

Crossref

Physical Review B

Texas A&M University

https://oaktrust.library.tamu.edu/bitstream/1969.1/127356/1/PhysRevB.84.052403.pdf