On the instabilities of the Walker propagating domain wall solution
A powerful mathematical method for front instability analysis that was recently developed in the field of nonlinear dynamics is applied to the 1+1 (spatial and time) dimensional Landau-Lifshitz-Gilbert (LLG) equation. From the essential spectrum of the LLG equation, it is shown that the famous Walker rigid body propagating domain wall (DW) is not stable against the spin wave emission. In the low field region only stern spin waves are emitted while both stern and bow waves are generated under high fields. By using the properties of the absolute spectrum of the LLG equation, it is concluded that in a high enough field, but below the Walker breakdown field, the Walker solution could be convective/absolute unstable if the transverse magnetic anisotropy is larger than a critical value, corresponding to a significant modification of the DW profile and DW propagating speed. Since the Walker solution of 1+1 dimensional LLG equation can be realized in experiments, our results could be also used to test the mathematical method in a controlled manner.
pacs:75.60.Jk, 75.30.Ds, 75.60.Ch, 05.45.-a
I I. Introduction
The past century witnessed the quantum leap of the semiconductor industries which gave birth to the computer science and information technology. We are now in an era in which information keeps being generated at a skyrocketing pace such that the net volume of information produced per day might be comparable to that accumulated after years one century ago. As an important participant, magnetic data recording now has assumed the major task of information documentation, through video tapes, hard disks, etc. In order to cope with the exponentially growing information volume, the need to develop data storage devices with higher capacity and faster read/write operation speed is demanded. This intrigue the development of spintronics-the pursuit to employ, in addition to the charge of electrons, their spin properties into applications. As one major branch, magnetic domain wall (DW) propagation along nanowires has attracted considerable attention in recent years due to its potential in achieving, for instance, high-intensity information storage, nonvolatile random access memory and DW logic circuit [1, 2, 3, 4, 5, 6, 7].
It has already been known for almost 40 years that the 1+1 (spatial and time) dimensional Landau-Lifshitz-Gilbert (LLG) equation , which universally governs magnetization dynamics, admits a well-known exact Walker propagating DW solution for a biaxial nanowire. It predicts that, in the presence of an external magnetic field, the DW subjects to a rigid-body translational motion which is valid when the magnetic field is in a proper regime. Despite its attractive simplicity and elegance, and the fact that this Walker solution has played a pivotal role in our current understanding of both current-driven and field-driven DW propagation in magnetic nanowires [9, 10, 11, 12, 13], whether or not this solution is genuine, i.e., describing a realistic physical system, is still an open question. As one necessary touchstone, genuine solution of a physical system must be stable against small perturbations. By now there is no proof of the stability of the Walker solution and the validity of it for a 1D wire is always taken as self-evident. Any deviation in experiments or numerical simulations are assumed to be attributed to the quasi-1D nature or other effects . However, there are signs [12, 13] that this solution may be unstable. For instance, in Reference , it is shown that under a huge hard-axis anisotropy, a DW motion damped by spin-wave emission occurs after the field exceeds a critical value. In addition, only stern waves were observed therein. In Reference , a propagating DW dressed with spin-waves was also captured both in the absence and presence of the Gilbert damping, and unlike , the spin-waves observed emit both stern and bow waves. Moreover, apparent deviation of DW velocity and deformation of DW profile from Walker predicted values were also observed. Unlike the microscopic DW profile which is sensitive to any errors incurred in simulations, the speed of DW manifests collective behavior of spins composing the DW; thus it is capable of reflecting the macroscopic physics that are invulnerable to the self-averaging microscopic perturbations when a large number of spins are involved. Therefore, this velocity deviation, as a more conspicuous fingerprint of the DWï¿½ï¿½s destabilization, shall also be addressed in regards to its origin in order for a deep understanding of DW propagation in nanowires. On the other hand, applications of spintronics devices require accurate description of DW motion [14, 15, 16, 17]. Thus, the stability of the Walker propagating DW solution becomes vital in our understanding of DW propagation along a magnetic wire.
However, unlike stability analysis of solutions of linear and nonlinear ordinary differential equations which can be easily done by using the linearization techniques and Lyapunov-exponent concept [18, 19], it is hard in general for nonlinear partial differential equations like the LLG equation. Although numerical approaches can provide clues and hints, an analytical approach is lack for Walker solution until recently the progress of traveling wave analysis suggests a feasible way. In this paper, we shall present in detail the method and results of our stability analysis of the Walker exact propagating DW solution of a 1+1 dimensional LLG equation . It is shown that a propagating DW is always dressed with spin waves so that the Walker solution is not stable against spin-wave emission. In the low field region, only stern spin waves are emitted while both stern and bow waves emerge under high field. When the transverse magnetic anisotropy is larger than a critical value and the external field is sufficiently high, the solution is convective or absolute unstable, corresponding to severe distortion of the propagating DW profile. This shall lead to noticeable deviation of DW speed from the Walker formula besides that the DW is dressed with spin waves. The paper is organized as follows. The model and theoretical formulation are explained in the next section. Section III is the results and discussions, and the conclusion is in Section IV.
Ii Ii. Model and Theoretic Formulation
To study the stability of Walker exact propagating DW solution under an external field, we consider the dimensionless LLG equation ,
This LLG equation describes the dynamics of the magnetization of a magnetic nanowire schematically shown in Fig. 1. With the easy axis along the wire ( direction) and the width and thickness being smaller than the exchange interaction length, exchange interaction dominates the stray field energy caused by magnetic charges on the edges; the DW structure tends to be homogeneous in the transverse direction , i.e., behaves effectively 1D. We are interested in the behavior of a head-to-head DW under an external field shown in Fig. 1. In Eq. (1), is the unit direction of the local magnetization with saturation magnetization and is the phenomenological Gilbert damping constant. The effective field (in units of ) is where , , and are respectively the easy axis anisotropy coefficient, the hard axis anisotropy coefficient, and the exchange coefficient. is the external magnetic field parallel to . The time unit is , where is the gyromagnetic ratio. Using polar angle and azimuthal angle for as shown in Fig. 1, this LLG equation has a well known Walker propagating DW solution ,
Here is the Walker breakdown field and is the DW width which will be used as the length unit () in the analysis below. is the Walker rigid-body DW speed that is linear in the external field and the DW width, and inversely proportional to the Gilbert damping constant. Solution (2) is exact for .
In the following analysis, the meaning of stability/instability of the DW is confined to Lyapunov definition, i.e., the DW is stable if any other solution of Eq. (1) starting close enough to the Walker solution will remain close to it forever; otherwise it is unstable. We will prove the instability of solution (2) against spin-wave emission by performing a spectrum analysis according to a recent developed theory for a general travelling front, such as a propagating head-to-head DW shown in Fig. 1. To prove the instability of solution (2), we follow a recently developed theory (Sandstede and Scheel  and Fiedler and Scheel ) for stability of a general traveling front, that is, a solution connecting two homogeneous states, such as a propagating head-to-head DW shown in Fig. 1. A modus operandi is to perturb Eq. (1) by a small deviation from the solution (2) via which an evolution equation governing this deviation can be derived. Note that if we directly perturb Eq. (1) by , and , with (, and are components of Eq. (2) in the Cartesian coordinates), the three components of were not independent due to the preservation of . A convenient way to circumvent this problem is, instead of analyzing in the Cartesian space, to work with the polar-coordinate form of Eq. (1) in which the two variables and satisfy :
where single and double prime denote the first and the second derivatives with respect to . By assuming and the solution of Eq. (3) with , , and by keeping only the terms of the first order in and , the linearized equations of and in the moving DW frame of velocity (with the coordinate transformation and , where ) are, in a two-component form of (superscript means transpose),
where , , and are matrices that depend on through . has the following matrix elements: , , sesh, . Here is the Gudermannian function and . , can be expressed explicitly in terms of as:
Eq. (4) is a linearized equation, and its general solutions are linear combinations of basic solutions of the form,
where is a proper complex number that supports nontrivial solutions (not constant zero) for equation
where . Then all such define the spectrum of . It is straightforward to verify that, due to translational invariance of solution (2), always belongs to the spectrum, with the corresponding eigenfunction If none of in the spectrum has positive real part, the spectrum is said to be stable; otherwise it is unstable. For a stable spectrum, any moderate deviations from the Walker solution must either decay exponentially with time  or undergo periodic motion by retaining its amplitude . When the spectrum encroaches the left half plane, exponentially growing modes () exist. We shall use so-called essential and absolute spectra of to decide the stabilities/instabilities of domains and DW profile.
Iii Iii. Results and Discussions
iii.1 A. Essential Instability
In order to compute the spectrum of , it is convenient to rewrite Eq. (6) in the first order differential form by using ,
is the identity matrix. All that supports nontrivial solutions to Eq. (7) form its spectrum. Eq. (6) and Eq. (7) have the same spectrum because they are equivalent. We shall focus hereafter on the spectrum of Eq. (7). To do so, we need to obtain the conditions under which Eq. (7) has nontrivial solutions. Let us first divide axis into four regions: , , and with . Notice that depends on only through that varies with only within the DW, Eq. (7) is essentially,
in region and
in region . The two asymptotic matrices are,
can be directly obtained from Eq. (8) by replacing with for and with for . In region () and for each given , has 4 eigenvalue and eigenvector pairs, , and Eq. (9) ((10)) has solution of form (). can then be denoted by for () being the number of with positive (negative) real parts. Obviously, we have except on the so-called Fredholm borders explained below in detail. can then be ordered descending by their real parts as . Each solution () in region () can be continued into region () as (). Suppose we are interested in a nontrivial bounded solution of Eq. (7), i.e., , then must be the linear superposition of those  in () whose corresponding eigenvalues () have positive (negative) real parts. Note that the number of () with Re (Re) is (), whether or not such exists is equivalent to whether or not we can find nontrivial solution satisfying
This is the condition of the continuation of at . The spectrum of Eq. (7) is the set of all such that Eq. (12) has at least one nonzero solution of for and . Obviously, there are variables and 4 equations. The existence of such a solution is then . The explict solutions of require the knowledge of that is normally not known analytically because of the complicate -dependence of . Numerical method such as the shooting algorithm used in the Schrodinger equation may be use here by numerically integrating Eq. (7) starting from (where all linear independent solutions of Eqs. (9) and (10) are known) and ending at . Correct set of shall make the shooting of from end with the same value at . The shooting algorithm, proved to be efficient for the Schrodinger equation whose spectrum is on the real line, may become excessively arduous for the LLG Eq. where the spectrum extends to the whole complex plane. As we shall see, this formidable task can be partly dodged as far as only the essential instability is considered which is pertinent to spin wave emissions.
Similar to the energy spectrum of a quantum system, the spectrum of Eq. (7) can be discrete and continuum. The continuum is also called the essential spectrum. The essential spectrum is not sensitive to the so called relatively compact perturbations to Eq. (7). Here a relatively compact perturbation can be understood, in some senses, as a local perturbation to a Schrodinger equation
This continuous spectrum will not be changed by a of finite potential range, such as a potential well or barrier, although wave functions are altered and point spectrum may be introduced. According to [26, 25, 24, 23], a similar local perturbation to Eq. (7) (or in general to any linearized equation of a system around a front solution) preserve the essential spectrum so that we can replace (Fig. 1 (c)) by , where is the Heaviside step function. The new equation with the same essential spectrum as that of Eq. (7) is
Eq. (15) have nonzero solution if the number of variables and , , is greater than 4. Thus, Eq. (13) has nontrivial solution bounded at for all whose . If one allow other types of solutions at , then the general condition is . Indeed according to the theory of Refs. [23, 22, 25, 24, 26], the essential spectrum of (also of Eq. (7) and/or Eq. (13)) is the union of all closed sets of (boundaries included) whose indices and satisfy . The boundaries of each region, known as the Fredholm borders, must be those lines crossing which either or changes its value by 1. Then along each Fredholm border, either or must possess pure imaginary eigenvalue (not a hyperbolic matrix); thus these lines can be determined by with [23, 22, 24]. Each of the two equations has two branches of allowed denoted as . Note that Eqs. (9) and (10) admit pure plane wave solution when is on (). Therefore an encroachment of these borders to the right half plane implies spin wave emission. We refer to the type of instability characterized by the presence of essential spectrum on the right half plane as the essential instability.
In order to understand numerical results in Ref. , parameters of yttrium iron garnet (YIG)  are assumed in our analysis with , , , and . is used and is a varying parameter. Fig. 2 plots the essential spectrum for (in units of that is about 10 times larger than ). The qualitative results are very similar to the early results : In the absence of an external field, the two branches of the spectrum of are the same, , shown in Fig. 2(a). Since the spectrum encroaches the right half plane, unstable plane waves shall exist and spin wave emission are expected. Similar conclusion was also obtained in early study , but for . Solid lines are for negative group velocity [determined by Im], thus these are stern modes. The dashed lines indicate positive group velocity, corresponding to bow modes. The green dots are zero group velocity points. According to Fig. 2(a), all unstable modes have negative group velocities so that DW can only emit stern waves in the low fields. As the external field increases, and will separate, and the area of essential spectrum in -plane becomes bigger and bigger (shadowed regimes in Fig. 2(b). The green dots also moves toward Im-axis and cross it at [Fig. 2(b)]. Upon further increase of , the unstable modes have both positive and negative group velocities although the most of them have the negative ones. One shall have propagating DW to emit both stern and bow waves. The stern waves should be stronger than the bow waves as schematically shown in the right figure of Fig. 2(b). This is exactly what were observed in numerical simulations for stern wave emission in low field  and stern-and-bow wave emission in high field . In a realistic wire with damping, emitted spin waves will be dissipated after a short distance, and are hard to be observed in experiments.
iii.2 B. Transient/convective/absolute Instability
The essential spectrum decides the instability of domains. DW propagation will generate spin waves in domains when the essential spectrum encroaches the right half of the plane. The fact that the essential spectrum is not affected by the variation of DW profile means that the essential spectrum cannot determine the instability of DW profile that is important for many quantities such as the DW velocity. Interestingly, DW instability is determined by the so-called absolute spectrum explained below. It can be classified into three categories. Absolute instability [AI, Fig. 2(c)iii] occurs when at each fixed point on the -axis, the disturbance grows exponentially with time. It is associated with the emergence of nontraveling unstable modes in the absolute frame (the moving frame that we adopted); thus coins its name. This point-wise growth feature of AI is in sharp contrast with the other two types of instability, which albeit grows in the total norm, decays locally at each fixed point on the -axis. They happens when all unstable modes are transported to infinities at fast enough velocities. It is called a transient instability (TI, Fig. 2(c)i) if the disturbance generated locally in the DW region transports to infinity in one direction (either towards or ), while it is called convective (CI, Fig. 2(c)ii) if it can transport in both directions. Intuitively, transient instability shall have the least influence on the DW property since once generated, it will leave the DW region quickly and will not interact with the DW hereafter. Convective instability is stronger than the transient one since although transported outside the DW region, it could influence the DW through second order effect in which new bidirectional unstable modes excited by the convecting wave packets can collide and interact with the DW again. Absolute instability is the most severe one in the sense that once a nontraveling disturbance is generated, it can stay within and keep interacting with the DW, leading to dramatic modification on the DW profile. For this reason, physical quantities depending on DW profile, such as the DW velocity, are expected to be strongly affected.
The three types of transportation behavior, either unidirectional, bidirectional or non-travelling, are determined by the so-called absolute spectrum and the branching points [23, 22, 25, 24, 26, 30, 28, 29]. To introduce the absolute spectrum and the branching points, we recall that, for each in the complex plane, there are four () for , ordered by their real parts as . Then is said to belong to the absolute spectrum () if and only if or . The branching points are special points in the absolute spectrum, denoted as , satisfying . To have a better feeling about the absolute spectrum and differences in unidirectional/bidirectional transportation and nontraveling modes of a wave , we introduce the concept of pointwise decay and growth. A wave is said to be pointwise decay iff for any fixed . The opposite ( instead of 0) is said to be pointwise growth. Let us first consider a wavelet that may exemplify a transient disturbance transporting to the right along the z axis:
This is an unstable mode if . At each fixed point , () if (). In another word, an unstable disturbance moving fast enough can lead to pointwise decay (vanish in a long time at each fixed point) although its norm increases exponentially with time. Interestedly, (16) can be brought to be stable when if an exponential weight is used
Note that the integral is finite whenever . Therefore any makes such that the new norm (17) decay exponentially with time. However, if , either (17) diverges with time or is infinity for any . In another word, the mode becomes stable under a proper exponential weight for , and a transient disturbance traveling towards fast enough can be stabilized by a positive since then the multiplier balance the growing modes at .
In general, the exponentially weighted norm denoted by for a real number is defined as
The transient and convective instabilities behave very differently under the norm. For a given , its eigenmode is transient unstable if it has an exponentially growing factor that travels towards (or ). Under an exponentially weighted norm with a proper choice of () for mode traveling to (), the growth at () can be absorbed by the multiply . Therefore the essential spectrum calculated under the exponential norm With () can be transferred to the left half of the plane for the unidirectional modes traveling towards (). Mathematically, this corresponds to a proper choice of the origin of the plane in some sense. Thus, with the proper definition of the norm by choosing a large enough , all unstable unidirectional eigenmodes of eigenvalues (essential spectrum in the right half of the plane) are removable because all such can be transferred to the left half of the plane. This treatment fail to the modes traveling to both directions of (bidirectional eigenmodes). They are not removable since an exponential weight can only suppress the growth in one direction and blow up in the other direction. The ability/inability of using an exponential weight (18) to stabilize/destabilize transient/convective modes leads to the following properties: TI occurs if all unstable essential spectrum can be move to the left half plane under a proper exponentially weighted norm while it is CI or AI if part of the unstable essential spectrum cannot be stabilized by the norm. A naturally raised question then is which part of the essential spectrum cannot be removed by this weight.
The answer is quite simple: The absolute spectrum cannot be moved around in the plane by introducing an exponential weight. It must locate to the left of the rightmost Fredholm border. If it encroaches the right half of the plane, then the essential spectrum cannot be stabilize no matter how one chose the exponential weight . To see why this is so, it is noticed that we need to introduce a weight  in order to move the Fredholm border determined by Eq. (9) [(10)]. Thus, by using the exponential weight of , it is equivalent to shift the eigenvalues of [23, 22, 25, 24, 26, 30, 28, 29] by
and accordingly the indices of the are transformed as
Now suppose with belongs to the Essential spectrum but not on the Fredholm border, which means . For the LLG equation and independent of the norm we use, in the right hand side of the rightmost Fredholm border has the indices of , then all possible combinations of in the regions right after passing through the rightmost Fredholm border can only be one of the four cases: , , , . Consider for instance , then obviously and . If we also have , i.e. , we could always find the aforementioned proper weight as, for instance:
which means that, for essential spectrum calculated under this norm, is well to the right of the essential spectrum. We can thus remove all unstable (i.e., ) in this way if there is no on the right half of the plane. However, if belongs to such that , it is easy to verify that no such pair of exist. This absolute spectrum is exactly the set of which could not be stabilized by the aforementioned proper weights . Therefore we conclude that the absence of in the right half of the plane indicate Transient instability in which all eigenmodes are unidirectional while the presence of the absolute spectrum means emergence of bidirectional eigenmodes.
Finally, the presence of unstable non-traveling modes is associated with the branching points’s presence on the right half plane. It is straightforward to verify that the eigenmodes associated with these branching points have zero group velocity, as follows. Denote the secular polynomial of as: Then for satisfying , it must hold that:
where . Then the group velocity of modes associated with is
Thus, branching points are non-travelling eigenmode [30, 33]. For , the absolute spectrum in the right half of the plane is generated by . Fig. 3(a) shows two branches . They are well separated by the real axis for and no absolute spectrum could be found in the right half plane. As the field increases, the two branches get closer with each other and at an onset field , depending on , two branches tangent at the real axis and then separate again in horizontal direction as shown in Fig. 3(b) for . At this moment, unstable absolute spectrum begins to emerge on the real axis (enclosed by the dashed circle). Fig. 3(c) is the enlarged vision showing the absolute spectrum (the segment between two branching points (green solid dots)). The dependence of or on between these two points is shown in Fig. 3(d).
According to Refs. [30, 28, 29], wavepackets would be emitted if the essential spectrum encroaches the right half -plane. There are three types of instability [23, 22, 24, 25, 30, 28, 29]. The instability is called transient (TI) if the essential spectrum encroaches the right half plane and absolute spectrum are either in the left half plane or does not exist. The propagating DW emits stern waves shown in Fig. 2(c)i. The instability is called convective if both essential and absolute spectrum encroaches the right half -plane. In this case, the emitted waves can propagate in both direction as shown by Fig. 2(c)ii. For an convective instability, if any branching point is also in the right half -plane, the instability is called absolute. An absolute instability can then emit non-traveling (zero group velocity) waves as illustrated in Fig. 2(c)iii. For LLG equation, since the absolute spectrum is the segment connecting two branching points and [Fig. 3 (c), (d)], the absolute instability (AI) and convective instability (CI) co-exist. It is known that transient instability is very weak that can be removed under proper mathematical treatment [23, 32]. Thus, we should not expect to have great physical consequences. On the other hand, the absolute instability move with the DW, and cause the change of DW profile [31, 30, 32]. It is known  that field-induced DW propagating speed is proportional to the energy damping rate that is sensitive to DW profile. Therefore absolute instability, which deform propagating DW profile, shall substantially alter DW speed. This may explain why the field-induced DW speed start to deviate from the Walker result only when the field is large enough to emit both stern and bow waves in simulations .
Fig. 3(e) is the calculated phase diagram in and plane. A transition from transient instability (denoted as TI in the figure) to absolute/convective instability (AI/CI) occur at a critical field as lng as at which . It means no absolute/convective instability exist for , and one shall not see noticeable change in famous Walker propagation speed mentioned early. This may explain why many previous numerical simulations on permalloy, which have small transverse magnetic anisotropy, are consistent with Walker formula. A snapshot of the convecting wavepackets could be identified in Fig. 2 in Reference  where wavepackets can be seen in the vicinity of the traveling DW and travel to both directions.
It should be noticed that the effects of point spectrum have not been analyzed. In principle, it can also affect the stability of the Walker solution, and should be a very interesting subject too. Unfortunately, there are not many theorems on the point spectrum yet. Thus, one can only rely on a numerical method to find a point spectrum of operator and to find out whether it can also induce any instability on a propagating DW.
Iv Iv. Conclusions
In conclusion, we present a powerful recipe for analyzing the stability of a front of partial differential equation. For the Walker propagating DW solution of the LLF equation in 1+1 dimension, it is found that DW will always emit stern waves in a low field, and both stern and bow waves in a higher field. Thus the exact Walker solution of LLG equation is not stable. The true propagating DW is always dressed with spin waves. In a real experiment. the emitted spin waves shall be damped away during their propagation, and make them hard to be detected in realistic wires. For a realistic wire with its transverse magnetic anisotropy larger than a critical value and when the applied external field is larger than certain value, a propagating DW may undergo simultaneous convective and absolute instabilities. As a consequence, the propagating DW will not only emit both spin waves and spin wavepackets, but also change significantly its profile. Thus, the corresponding Walker DW propagating speed will deviate from its predicted value, agreeing very well with recent simulations.
This work is supported by Hong Kong RGC Grants (604109 and 605413).
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