Theory of domain wall motion in a random medium is extended to the case when
the driving field is below the zero-temperature depinning threshold and the
creep of the domain wall is induced by thermal fluctuations. Subject to an ac
drive, the domain wall starts to move when the driving force exceeds an
effective threshold which is temperature and frequency-dependent. Similarly to
the case of zero-temperature, the hysteresis loop displays three dynamical
phase transitions at increasing ac field amplitude $h_0$. The phase diagram in
the 3-d space of temperature, driving force amplitude and frequency is
investigated.Comment: 4 pages, 2 figure

A. Middleton

B. Drossel

B. K. Chakrabarti

D. S. Fisher

G. Blatter

G. Grinstein

H. Ji

I. F. Lyuksyutov

J. P. Sethna

J. Villain

K. Dahmen

L. B. Ioffe

L. Radzihovsky

L. Roters

M. Kardar

M. V. Feigelman

O. Narayan

T. Halpin-Healy

T. Nattermann

U. Nowak

V. M. Vinokur

V. Pokrovsky

English

arXiv

Crossref

Physical Review Letters

Hysteretic Dynamics of Domain Walls at Finite Temperatures

Nattermann, T.

Pokrovsky, V.

Vinokur, V. M.

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:cond-mat/0106281v1  [cond-mat.dis-nn]  14 Jun 2001Hysteretic dynamics of domain walls at finite temperaturesT. Nattermann1, V. Pokrovsky2,3 and V.M. Vinokur41 Institut fu¨r Theoretische Physik, Universita¨t zu Ko¨ln, D-50937 Ko¨ln, Germany2Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA3Landau Institute for Theoretical Physics, ChernogolovkaMoscow District, 142432, Russia4Argonne National Laboratory, Argonne, IL 60439, USA(June 26, 2018)Theory of domain wall motion in a random medium is extended to the case when the driving fieldis below the zero-temperature depinning threshold and the creep of the domain wall is induced bythermal fluctuations. Subject to an ac drive, the domain wall starts to move when the driving forceexceeds an effective threshold which is temperature and frequency-dependent. Similarly to the caseof zero-temperature, the hysteresis loop displays three dynamical phase transitions at increasing acfield amplitude h0. The phase diagram in the 3-d space of temperature, driving force amplitude andfrequency is investigated.Pinning dominated driven dynamics of elastic mediain random environment is a paradigm for a vast diversityof physical systems. Examples include vortices in type IIsuperconductors, charge density waves (CDW) in solids,stripe phases, Wigner crystals, dislocations in crystals,domain walls in magnets and many others [1]. Havingappeared first in the context of dislocation dynamics [2],the scaling theory of glassy dynamic state of random elas-tic media came to fruition in the context of CDW [3] andvortex lattices in high temperature superconductors [4,3],and enjoyed an impressive success in explaining a wealthof phenomenology of the low temperature vortex state[5]. A closely related subject is the zero temperaturedepinning transition first studied for CDWs [6,7] and do-main walls [8,9]. Despite the significant recent progress,several key questions specific to glassy dynamics are yetpoorly understood. One of such fundamental key issues,although known and extensively studied for more thanhundred years in magnets, is hysteresis of interfaces sub-ject to the applied ac drive and related aging and memoryeffects. A quest for urgent progress in understanding hys-teretic behavior of magnetic domain walls is motivatedalso by emerging technological nano-scale magnetic sys-tems whose ac properties are controlled by the hystereticdynamics of interfaces.A step towards theoretical description of hysteretic be-havior of disordered interfaces has been undertaken in[10], where the cyclic motion of the domain wall at zerotemperature under the ac field was investigated and theresulting magnetization hysteretic loop was described. Afinite temperature may change drastically the interfacedynamics: thermally activated creep motion becomespossible at any small drive.In this Letter we develop a unified description of ther-mally activated and over-threshold domain wall dynam-ics in impure magnets. We demonstrate that at finitetemperature new scales of length, activation energy andforce appear leading to emergence of a new, temperatureand frequency-dependent threshold field in the case ofac drive. The latter is the first in a series of dynamicalphase transitions. To be specific, we will speak on mag-netic domain walls. Accordingly we will be using eitherof terms ”force” or ”field” equivalently.Finite temperature dc dynamics The essential of thezero-temperature dynamic behavior of an elastic mediumin a random environment is the existence of the finitethreshold depinning force hp, separating immobile ath < hp and sliding at h > hp states of the system. Nearthe threshold the sliding velocity v shows a critical be-havior [6–9] v ∼ (h − hp)β . At finite temperatures andh ≪ hp thermally activated drift motion controlled bythe static rugged energy landscape occurs. The latter isgoverned by the interface free energyH =∫dDx{12Γ(∇Z)2 + V (x, Z)− hZ(x)}(1)where Γ is the interface stiffness, h is the external drivingforce and V (x, Z) is the random impurity potential. D-dimensional vector x is the coordinate along the inter-face, and Z is the coordinate of the transverse interfacedisplacement. In the following we assume that the dis-order average of the random potential vanishes. Thereare two different types of impurities, random bond (RB)and random field (RF) type in terms of magnetic models.The RB potential obeys the Gaussian statistics with:VRB(x, Z)VRB(0, 0) = v2lD+1 δ(x) δ(Z) (2)where v2 = v20c. v0, c and l denote the strength, the con-centration and the correlation length of the impurity po-tential. In the RF case VRF (x, Z) =Z∫0h(x, Z ′) dZ ′ wherethe RF h(x, Z) has properties similar to VRB(x, Z).The static interface in a random environment becomesrough. Its roughness obeys the scaling law [13]:1w2(L) = (Z(x)− Z(0))2≈ l2(LLp)2ζ; L = |x| (3)where the roughness exponent is ζ = 4−D3 for RF andζ ≈ 0.2083(4−D) for 4−D ≪ 1 and ζ = 2/3 for D = 2,for RB impurities, respectively [14,13]. The rough con-figuration develops over length scales L ≥ Lp, where theLarkin length Lp is a distance at which a typical fluc-tuation of the pinning forces, balanced by elastic forces,produces the transverse displacement w ∼ l:Lp ≈ l (Γ/v)2/(4−D). (4)Note that the typical slopes w(L)/L of the wall vanish forL≫ Lp since ζ < 1. The energy barriers which must beovercome to depin a segment of the wall with the linearsize L is [2]:EB,0(L) ≈ Tp(L/Lp)χ; χ = D − 2 + 2ζ. (5)Here Tp ≡ EB,0(Lp) ≈ Γl2LD−2p is a typical pinning en-ergy on a scale Lp. At temperature T > Tp the effectiveforce necessary for depinning drops rapidly with the tem-perature [2]. If an external driving force h is applied, thetotal energy barrier EB(L) becomesEB(L) ≈ EB,0(L)− hLDw(L)≈ EB,0(L)(1− (L/Lh)2−ζ)(6)where Lh ≈ Lp(hp/h)1/(2−ζ). The energy barrier reachesits maximum EB,max at L ≈ (χD+ζ )1/(2−ζ)Lh which givesEB,max ≈ Tp(hp/h)µ and µ = χ/(2− ζ).For completeness we note here that equilibrium RFsystems in d = D + 1 = 2 bulk dimensions have no longrange order at length scales L ≫ ξRF ≈ l exp c(Γ/v)4/3[16]. For weak disorder ξRF is very large and can easilyexceed the system size. As soon as an external field isapplied such that Lh < ξRF , domain wall motion is dom-inated by the forces on scales Lh and the absence of truelong range order can be ignored.As first found by Middleton [17], close to hp the small-est energy barrier vanishes as (h− hp)θ with θ = 2. Wetherefore get the effective energy barrier at h <∼ hp if wereplace in the above expressions Tp by T˜p = Tp(hp−hhp)θ.The time scale to overcome this barrier is of the orderτ(Lh) ≈ τ0 exp(E˜B,max/T), where τ0 is a microscopichopping time, which leads to an average velocity:v(h) ≈ γh exp[−TpT(hp − hhp)θ (hph)µ](7)where γ is the effective friction coefficient [2,3,15].At low temperature T ≪ Tp the dynamic threshold hpseparates the creep regime from the active sliding regime.As can be seen from (7) a characteristic crossover fieldhT plays the role of the depinning force, wherehThp=(T˜p(hT )T)1/µ=(TpT(1−hThp)θ)1/µ(8)At h ≈ hT , the drift velocity increases rapidly and atlarger fields it displays almost linear behavior v ≈ γh.Note that hT is a monotonously decreasing function oftemperature with a maximum hT = hp at T = 0.In a close vicinity of the threshold field hp, the effec-tive energy barrier becomes small and even small thermalfluctuations may be sufficient to overcome it. At finitetemperatures and h <∼ hp the wall moves via thermal ac-tivation process with velocity given by eq. (7). Strictlyspeaking, it means that at finite temperatures the criticalpoint shifts from h = hp to h = 0. Yet there remains amemory of the critical behavior around h ≈ hp displayingitself in a crossover behavior at finite but low tempera-tures. The crossover is seen as a rounding of the h − vcharacteristics v(h ≈ hp) ∼ Tβ/θ. We now can write aninterpolation formula for the velocity which is valid in awide range of variables:v(h, T ) = γhF (x, y); x = h/hp; y = Tp/T, (9)F (x, y) =Θ(1− x)1 + (yx−µ)β/θexp[−yx−µ (1− x)θ](10)+Θ(x− 1)[11 + (yx−µ)β/θ+(1−1x)β]Here Θ(x) is the step function equal to zero at x < 0and equal to 1 at positive x. The interpolation for-mula (10) satisfies following requirements: (i) v(h, T ) =γh at any fixed T and h ≫ hT ; (ii) v(h, T ) =γh exp[−TpT(hp−hhp)θ]for hp − h ≪ hp and T ≪ Tp;(iii) v(h, T ) ≈ γh exp[−TpT(hph)µ]for T ≪ Tp, h≪ hpand EB,max/T ≫ 1 ; (iv) v(h, T ) ≈ γhp(h/hp − 1)βfor (h/hp − 1) ≪ 1 and T ≪ Tp(h/hp − 1)θ; (v)v(hp, T ) ≈ γhp(T/Tp)β/θ for T ≪ Tp.So far we assumed that the propagating interface isself-affine. This is confirmed by numerical simulations inD > 1 interface dimensions for systems with weak disor-der [18]. In D = 1 dimensions the situation is less trans-parent: in simulations which use a bounded distributionof random fields the interface appears to be self-affine[19] or faceted [20] depending on whether lattice effectsare avoided or admitted, respectively. We ignore here thepossibility of faceted growth which occurs only in systemswith narrow magnetic domain walls. For an unboundeddistributions of random fields however a percolative self-similar domain wall propagation was observed [21]. Inthe following we will always assume, that the randomfields distribution is bounded such that the domain walls2remain well defined. This is also confirmed by our earliersimulation outside of the critical region [10].Alternating fields. If the external drive is oscillatingwith frequency ω, h = h0 sinωt, the barriers for whichω τ(L) > 1 cannot be overcome during one cycle of the acfield. From the condition ω τ = 1 we find a new frequencyand temperature dependentmagnetic field hω which obeyshωhp=(TpTΛ(1 −hωhp)θ)1/µ(11)where Λ = ln 1/(ωτ0). hω plays the role of the dynamicthreshold. At low fields h0 < hω there is no macroscopicmotion of the wall, its segments oscillate between themetastable states with close energies giving rise to dis-sipation [2]. Drift of the wall starts at h0 > hω. Weassume ωτ0 ≪ 1, so that hω < hT . Various regimes ofdomain wall motion are summarized in Fig.1.                                                                                                                                                                                                                                                                                                                                                                     Thω hT(2)hh(t)hpTp creepv = 0h(t) (1)critical regimelinear  regimeFIG. 1. The phase diagram for the domain wall hystere-sis at finite temperature and frequency. Solid and verticallines separate no sliding, thermal creep and mechanical driftregimes. See also explanations in the text.Having derived the domain wall velocity as a functionof the driving field in the different h− T -regions we con-sider now the magnetic hysteresis following from the mo-tion of a single wall under the influence of an oscillatingfield. Since substantial length scales are larger than Lp,where slopes are small, the domain wall will be consideredas a straight line (plane) characterized by one coordinateZ [10]. Its dynamics is determined by equation of motionZ˙ = v(h(t)) (12)Z varies between limiting values 0 and L. Here L is thelinear size of the sample in the case of a single domainwall or, in the multi-domain case, equal to the averagedistance between expanding nuclei . For harmonically os-cillating field h = h0 sinωt, equation (12) can be rewrit-ten in terms of h only:dZdh=v(h)ω√h20 − h2; Z(h = 0) = 0 (13)Equations (12) and (13) are valid for h > hω. The fieldregion h < hω where the motion has zero drift velocitywill not be considered here. The value hω plays the samerole as the threshold field hp plays at zero temperature.At h > hω the hysteresis is dominated by the activationprocesses. In this respect it is similar to the nucleationdominated hysteresis described in [10] with two essentialdifferences. First, the activation relates to the formationof a nucleus on the interface, not in the bulk. Second,the activation energy depends on the magnetic field as apower function ∼ h−µ due to the distribution of barriersdepending on their length scale.hc)hha).hω.0hrhchh0ωhhh hω hr c= = h0...hω h h r h0.cb)d)M MMMFIG. 2. Schematic pictures of hysteresis loops(HL). (a) In-complete HL for h0 < ht1. (b) Symmetric HL for h0 = ht1.(c) The HL for ht1 < h0 < ht2. (d) The HL for h0 > ht2.The values hp, hc, hr and h0 are all marked in all figuresWe start with large temperatures T > Tp at which thezero-temperature threshold field hp plays no role (fieldsweep (2) in Fig.1). In this case the motion of the do-main wall is determined by equation ( 13) for h > hω.As in the case of zero temperature [10] three dynamicphase transitions take place when the amplitude h0 in-creases gradually at a fixed value of frequency. Thesetransitions change the shape (symmetry) of the hystere-sis loop. At the first of them proceeding at h0 = hωthe hysteresis loop first appears, at smaller amplitudesh0 < hω the magnetization remains unchanged. Thehysteresis loop appearing at h0 > hω is characterized byincomplete magnetization reversal and reflection symme-try h→ −h, M →M as shown in Fig. 2a. This symme-try as well as incomplete magnetization reversal persistsuntil the next dynamic phase transition at h0 = ht1.At h > ht1 the magnetization reversal becomes com-plete and hysteresis loop symmetry changes to inversionh → −h, M → −M (see Figs. 2b, 2c). The value ht1 isdetermined by a requirement that the domain wall pro-ceeds from one sample boundary to another for half aperiod. At the next dynamic phase transition the sym-metry of the hysteresis loop remains unchanged, but thepart of the cycle becomes reversible. Visually the hys-teresis loop acquires characteristic ”whiskers” as shownin Fig. 2d. The point of this transition ht2 is determinedby a requirement that the domain wall proceeds from onesample boundary to another for quarter period.3Starting from the transition amplitude h0 = ht1 eachhysteresis loop goes through three important points. Oneof them is hω, at which the motion of domain wall starts.Two others the so-called coercive field hc and reversalfield hr. At coercive field the magnetization turns intozero, at reversal field the magnetization becomes com-pletely reversed (see Fig. 2b-2d). Note that for h0 = ht1hr = hω and hc = h0; for h0 = ht2 hr = h0. All thesefields can be found in our case. Equations for ht1, ht2are: ∫ htnhωv(h)dh√h2tn − h2=nωL2, n = 1, 2 (14)For the case T > Tp the equations (14) read:g(xn) =nωL2γhT, g(x) =x∫xωye−y−µ(x2 − y2)−1/2dx (15)where xn = htn/hT , n = 1, 2; xω = hω/hT = 1/Λ1/µ.Thus, x1, x2 are functions of a dimensionless parameteru = ωL/γhT , where L is the size of the system or an av-erage size of domains. Its asymptotic at small u results inxn ≈ [ln(2/nu)]−1/µ. The fields ht1; ht2 are close in thiscase: (ht2 − ht1)/ht1 ≈ ln 2/(µ lnu). The requirementht1 > hω is satisfied if γhT τ0 < L. The coercive fieldhc and the reversal field hr are determined by equations:M(hc) = 0; |M(hr) |=Ms.A simple analysis at small u results in hc ≈ ht1; hr ≈ht2. The area A of the hysteresis loop at u << 1 andh0 > ht1 does not depend strongly on the amplitude h0and becomes A ≈ 4hrMs. Fig.2 shows typical hysteresisloops and illustrates the geometrical meaning of the fieldhω, hc, hr. The dependence of magnetization on magneticfield is given by equation: M(h) =Ms(2Z(h)L − 1).Finally, in the range of moderately low temperatureT < Tp the more complete expression (10) and (11) haveto be used in integrating equation (14).It interesting to note that the similar (dynamic) tran-sition from incomplete to complete hysteresis was ob-served even in a standard simplistic mean-field model forpure magnets with the reaction described by the Bril-loin function [12]. This suggests that this kind of dy-namic transition we discuss, may be a generic propertyof nonlinear systems. Another note is in order: hysteresisin large multidomain magnet samples is a very complexphenomenon and cannot be always reduced to motion ofa single domain wall (see for example numerical simula-tions of random Ising model in [11], where hysteretic andmemory efects unlikely reducible to motion of a sible DWwere revealed).In conclusion, we have investigated critical creep mo-tion at low, T ≪ Tp, and at high, T > Tp, temperatures,Tp being the depinning temperature, and constructed thedynamic phase diagram. At low temperatures creep ath ≈ hp retains features of the critical behavior and ex-hibits the rounding of the h− v characteristic, accordingto [17]. At finite frequencies ω, a new characteristic fieldhω < hp comes into play, and the transition from thesliding regime to pinning dominated activation motion isshifted to hω.Acknowledgements. We are grateful to A. Middletonfor a valuable discussion. One of us (V.P.) acknowl-edges the support of this work by the NSF under thegrant DMR-97-05182, DMR-00-72115 and by the DOEunder the grant DE-FG03-96ER45598 as well as fromHumboldt-foundation. The work at ANL was supportedby the by the U.S. DOE, Office of Science under contractNo. W-31-109-ENG-38.[1] for recent reviews see e.g. M. Kardar, Phys. Rep. 301,85 (1998), D.S. Fisher , Phys. Rep. 301, 113 (1998),[2] L. B. Ioffe and V.M. Vinokur, J. Phys. C 20, 6149 (1987).[3] T. Nattermann, Phys. Rev. Lett. 64, 2454 (1990).[4] M.V. Feigelman, D.B. Geshkenbein, A.I. Larkin, andV.M. Vinokur, Phys. Rev. Lett. 63, 2303 (1989).[5] G. Blatter, M.V. Feigelman, V.B. Geshkenbein, A.I.Larkin, and V.M. Vinokur, Rev. Mod. Phys. 66, 1125(1998);[6] D.S. Fisher, Phys. Rev. Lett. 50, 1486 (1983).[7] O. Narayan and D. S. Fisher, Phys. Rev. B 46, 11520(1992).[8] T. Nattermann, S. Stepanow, L.H. Tang, H. Leschhorn,J. Physique 2,1483 (1992).[9] O. Narayan and D.S. Fisher, Phys. Rev. B 48, 7030(1993).[10] I.F. Lyuksyutov, T. Nattermann, and V.L. Pokrovsky,Phys. Rev. B 59, 2262 (1999).[11] J. P. Sethna, K. Dahmen, S. Kartha, J.A. Krumhansl,B.W. Roberts and J. D. Shore, Phys. Rev. Lett. 70, 3347(1993); K. Dahmen and J. P. Sethna, Phys. Rev. Lett.71, 3222 (1993) .[12] B. K. Chakrabarti and M. Acharyya, Rev. Mod. Phys.71, 847 (1999).[13] T. Halpin-Healy and Y-C. Zhang, Phys. Rep. 254, 215(1995).[14] D. S. Fisher, Phys. Rev. Lett. 56, 1964 (1986).[15] For a RG-derivation of 7 see L. Radzihovsky, APS MarchMeeting 1998 and P. Chauve, T. Giamarchi and P. LeDoussal, Europhys. Lett. 44, 110 (1998).[16] J. Villain, J. Phys. Lett. (Paris) 43, L551 (1982); G.Grinstein and S.-K. Ma, Phys. Rev. Lett. 49, 685 (1982).[17] A. Middleton, Phys. Rev. Lett. 68, 670 (1992).[18] L.Roters, A. Hucht, S. Lubeck, U. Nowak, and K.D.Usadel, Phys. Rev. E 60, 5202 (1999), B. Koiller andM.O. Robbins, arXiv:cond-mat/0004183 and referencestherein.[19] U. Nowak and K. D. Usadel, Europhys. Lett. 44, 634(1998).4[20] H. Ji and M.O. Robbins, Phys. Rev. B44, 2538 (1991).[21] H. Ji and M.O. Robbins, Phys. Rev. B46, 14519 (1992),B. Drossel and K. Dahmen, Eur. Phys. J. B3, 485 (1998).5

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Hysteretic dynamics of domain walls at finite temperatures

Hysteretic dynamics of domain walls at finite temperatures

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