DOMAIN WALL DEPINNING PROCESS IN BISTABLE GLASS-COATED MICROWIRE

A simple analytically solvable model for description of dynamics of the domain wall depinning process from the closure domain structure in bistable microwires was proposed. In this model closure domain structure is modelled by a single domain wall located in quadratic potential well. Critical parameters of rectangular magnetic field pulse needed to release this wall from the potential well were calculated. Theoretical dependence obtained in this way was fitted to experimental data measured on glass-coated Fe77.5B15Si7.5 microwire. Information about the order of closure domain structure dimension and about the mass of the domain wall, which is depinned from the wire end were obtained in this way.


INTRODUCTION
Glass-coated amorphous ferromagnetic microwires are very promising materials for technical application as well as for basic research due to their magnetic properties and simple and cheap fabrication [1,2,3].
A large single Barkhausen jump is typical for magnetization reversal process in Fe-based microwires.It is a result of strong axial anisotropy due to high positive magnetostriction, internal mechanical stresses originating from rapid quenching and presence of glass coating.Stray field causes that closure domain structures is formed at the wire ends.Axial magnetization reversal starts by depinning of domain wall from the wire end and subsequently the reversal continues by propagation of a single domain wall along the wire.Velocity of this domain wall can be very high and so this phenomenon can be very interesting for technical applications.On the other hand the study of a single domain wall dynamics can give important information for understanding the process of magnetization reversal [4][5][6][7][8] and about dynamic characteristics of a single domain wall [9,10].
In the presented paper a simple model, which gives possibility to obtain information about characteristic dimension of closure domain structure as well as about dynamic parameters of a domain wall, is described.Model is confronted with experiment results.

MODEL
A simple analytically solvable model is proposed for description of single domain wall depinning from the wire end.In the framework of this model closure domain structure at the wire end, in which the wall is initially trapped, is modelled by parabolic potential well.The wall is set into motion by applied homogeneous magnetic field pulse.If parameters of the field pulse (length and magnitude) are large enough, the wall is released from the potential well at the wire end.Inertial motion of the wall after switching off the pulse is taken into account.
Free energy G of the wire is given by: where E p is potential well energy (Taylor series of potential energy around its potential minimum at x = 0 up to the third term), E H is contribution of an external field H to the wire energy; K is a positive constant and x is a displacement of the wall from its equilibrium position, S is the area of the wall projection onto a plane perpendicular to the wire, μ 0 is magnetic constant and M s is saturation magnetization.It is also considered that the potential well has characteristic dimension x 0 .In other words the Eq. ( 1) describes G only in the interval (-x 0 , x 0 ).For x ≥ x 0 we assume that E p ≤ E p (x 0 ).Free energy as function of the wall position x is depicted in Fig. 1.At the figure bottom corresponding position of the wire is plotted.
Fig. 1 Free energy as function of normalized horizontal coordinate x/x 0 (x 0 -position of right well border, i.e. half-width of the well), vertical axis unit equals to the depth of the well.Legend: the slope of function E H (see Eq. ( 2)) As can be seen in Fig. 1 application of external field causes that the well depth with respect to its right border decreases.If magnitude of the field pulse is high enough the wall propagating under the action of this field to the right can reach the right well border.However, there is some minimum pulse magnitude under which it is not possible.If the wall does not rich the right border during the pulse there are two possibilities.The first one is the case when at the instant when the pulse is switched off the wall velocity is high enough for the wall to travel to the right well border due to its inertial motion and the wall can be released from the well.If this is not the case the wall remains trapped in the well.
Eq. ( 1) can be used to derive corresponding force acting on the wall: For a moving wall also damping force has to be taken into account where β is damping coefficient (it will be considered as constant in this model) and v is wall velocity.The net force acting on the wall is equal to F p + F H + F β , so using Eqs.(3,4,5) the wall equation of motion can be expressed as where m is inertial mass of the wall.
After some rearrangements this equation can be written in the form where Using substitution equation for damped harmonic oscillator is obtained from Eq. (7).So solution of Eq. ( 7) for the domain wall position x and the wall velocity v as a functions of time are given by the expressions Constants A and φ can be obtained from initial conditions.Two different situations should be considered: is real and solution of Eq. ( 7) can be expressed by harmonic functions, as written in Eqs.(8,9).
. The solution of Eq. (7) in this case can be expressed by hyperbolic functions.This expression can be obtained directly from Eqs. (8,9) substituting iω (i is the imaginary unit, ω is real) for the square root expressions.Formulas are formally the same only harmonic functions sin and cos are replaced by hyperbolic function sinh and cosh.This holds for all relevant expressions in the article, if not mentioned otherwise.
We do not know which kind of solution corresponds to our experiment; both possibilities will be discussed in the next section, in which the results obtained from the theoretical model will be compared with experimental data.
Since the inertial motion of the wall after switching off the field pulse is taken into account the Eq. ( 7) was solved for two time intervals: 1.
for which H = H pc and initial conditions at t = 0 are x = 0 and v = 0.The wall reach a position x p (x p ≤ x 0 ) with velocity v p at the time τ c .The obtained solution is .
, for which H = 0 and initial conditions at the beginning of this interval are x = x p and v = v p .At the end of this time interval the wall just reaches the well border x 0 with zero velocity.Solution obtained for this interval is Using Eqs.(10,11) we can obtain the lowest possible value of the field pulse magnitude H pc min for which the wall can be released from the potential well if the length of this critical pulse τ 0 is long enough.For H pc min in the instant of its switching off τ 0 the wall just reaches position x 1 (t = τ 0 ) = x 0 with velocity v 1 (t = τ 0 ) = 0. (For higher field the velocity is higher than zero at the well border, for lower field with the same length of pulse the wall never reaches the right well border at position x = x 0 ).So, from boundary conditions at time τ 0 : x 1 = x 0 and v 1 = 0 using Eqs (10,11) we obtain in the case: where h pc min = H pc min k / m.For the pulse with the length τ satisfying condition τ < τ 0 there exists some value of the field pulse magnitude H pc > H pc min for which the wall reaches position x 0 with zero velocity.In this case also the second solution (inertial motion after the field pulse) has to be taken into account.

II. (ω 0 < b)
In this case, equations corresponding to Eqs. (10,11) are Now it is possible for a given value of h pc to calculate the corresponding value of τ c (i.e. to obtain function h pc = f (τ c )) for which the wall just reaches the well border at x 0 .
Conditions at time t 0 after the end of pulse are: x 2 = x 0 and v 2 = 0. From these conditions and using Eqs.(12,13) we can obtain expressions for t 0 and h pc = H pc k / m as function of τ c .
In particular cases we obtain:

EXPERIMENTAL RESULTS AND DISCUSSION
Glass-coated microwire Fe 77.5 B 15 Si 7.5 prepared by Taylor-Ulitovski method with parameters: length 12 cm, diameter of metallic nucleus about 15 μm, thickness of glass coating about 9 μm was studied in the experiment Experimental setup is shown in Fig 2 .Solenoid (So), 15 cm long can generate homogeneous magnetic field along the wire.The wire end is placed into a narrow coil (PC), 2 cm long with diameter of 1mm, which can generate local magnetic field pulse that can set the wall into motion.At some distance from this wire end a pick up coil (PuC), 1 cm long with diameter of 1mm, is located and gives possibility to obtain information about magnetic state of the wire.Detailed description of the experiment is given in [11].The dependence of critical pulse length vs. critical pulse magnitude obtained in experiment is shown in Fig. 3.As can be expected critical pulse magnitude increases with decreasing of critical length of the pulse.Experimental results can be analysed using the simple theoretical model presented in the previous section.From these fitting parameters it is possible to obtain information about domain wall mass and about typical dimension of potential well.
To obtain the mass of domain wall we express the wall mobility as a function of the wall mass.In the simplest model of the wall propagating under action of constant homogeneous magnetic field with constant velocity the equation of motion Eq. ( 7) has a form 2bv = h.The velocity as function of external field v = λH, where λ is mobility, can be measured.Using these equations the expression for wall mobility as function of domain wall mass is where 0 / m m S = .Using the expressions for parameters defined in Eqs.(7,19,27), the following equations for the wall mass and the well half-width can be obtained

CONCLUSIONS
A simple analytically solvable model for description of dynamics of the domain wall depinning process from the closure domain structure in bistable microwires was proposed.In the model the closure domain structure is modelled by quadratic potential well.A comparison with results of experiment, in which critical parameters of the field pulse for which the wall is just released from the closure domain structure, gave possibility to obtain important information about the order of closure domain structure dimension and about the mass of the domain wall which is depinned from the wire end.

Fig. 2
Fig. 2 Schematic view of wire location in the experimental set up.So -solenoid, PC -pulse coil, PuC -pick up coil

Fig. 3 Fig. 4
Fig. 3 Dependence of critical pulse parameters magnitude H pc vs. length τ c for wall depinning This model gives possibility to obtain parameters b and ω from experimental data using fitting procedure.As can be seen in Fig. 3 the value of minimum magnitude of the field pulse is H pc min ≈ 70 A/m.For fitting we used dependence H pc min / H pc = h pc min / h pc .Fitting procedure for the case I (ω 0 > b) was not successful (not convergent).For the case II (ω 0 < b) fitting was successful and it was possible to obtain fitting parameters.The fitting function in this case was obtained using Eqs.(12,23) experimentally obtained values of parameters H pcmin ≈ 70 A/m (see Fig.3), λ ≈ 3 m 2 A -1 s -1 , µ 0 M s ≈ 1.56 T[6], Eqs.(25,26) and Eqs.(28,29), the characteristic values of m 0 and x 0 were calculated. in qualitative agreement with known experimental results[9,10,2].