PHYSICAL REVIEW B VOLUME 55, NUMBER 18 1 MAY 1997-II Dynamics of domain walls in weak ferromagnets N. Papanicolaou Department of Physics, University of Crete, and Research Center of Crete, Heraklion, Greece Received 3 September 1996; revised manuscript received 3 December 1996 The dynamics of domain walls in a model weak ferromagnet is shown to be governed by a suitable extension of the relativistic nonlinear model to account for the Dzyaloshinskii-Moriya anisotropy and an applied magnetic field. Our analytical results are confirmed by a numerical calculation in a discrete spin model and significantly amend earlier treatments. Thus we provide a detailed description of static domain walls and subsequently study their dynamics. A virial theorem is derived that underlies the existence of a terminal state and allows a simple calculation of the mobility at low fields for both Bloch and NeŽel walls. We further establish the existence of a critical field above which a driven domain wall is always NeŽel, whereas a bifur- cation takes place below the critical value where the two types of walls behave rather differently. The terminal states as well as the mobility curves are obtained for practically any strength of the applied field. Implications for the phenomenology of domain walls in orthoferrites and in rhombohedron weak ferromagnets are discussed briefly. S0163-1829 97 10017-0 I. INTRODUCTION to interpolate between the expected linear behavior at low fields v h, where is the wall mobility and a limiting Weak ferromagnets WFM's are basically antiferromag- velocity (v c) in the opposite limit (h ). Formula 1.1 nets AFM's in which a small permanent magnetization shows no sign of a critical field but is in reasonable agree- arises thanks to an antisymmetric exchange interaction dis- ment with experimental data.4 covered and studied by Dzyaloshinskii1 and Moriya.2 An The actual picture in, say, YFeO early account of the main properties of weak ferromagnets 3 is more involved in that Eq. 1.1 provides only a rough envelop of the experimental may be found in the review article of Moriya3 while a recent curve. The latter is interrupted by at least two plateaus in the book4 focuses on the dynamics of topological magnetic soli- tons such as domain walls. vicinity of v 4 and 7 km/sec which are identified with the At first sight, it is natural to assume that the situation is velocities of longitudinal and transverse sound and suggest a similar to that of the extensively studied ferromagnetic FM resonant coupling between magnetic degrees of freedom and domain walls where the essential dynamical features are cap- lattice vibrations. Such magnetoelastic anomalies are absent tured by an analytical solution derived by Walker.5 When a in ordinary ferromagnets because the maximum Walker ve- FM wall is subjected to an external field h, it reaches a locity is usually smaller than the speed of sound. Since the terminal state with constant velocity v v(h) for field pure WFM wall dynamics already confronts us with a non- strengths below a certain critical value h trivial problem, magnetoelastic couplings will be neglected w but undergoes a complicated evolution for h h in the present work but could be included on a future occa- w . A related fact is that the maximum velocity achieved by the wall in the region h sion. h The semiempirical relation 1.1 may be thought of as a w is typically small, of the order of a few hundred meters per sec. Although important boundary effects in ferromag- relativistic extension of the linear mobility relation v h. netic films render the Walker solution inapplicable in its de- Indeed all previous attempts at a theoretical derivation of Eq. tails, the overall picture is essentially correct and has been 1.1 were based on a phenomenological continuum model the main source of intuition for many refinements that that is a generalization of the relativistic nonlinear model.4 followed.6 On the other hand, our recent study of AFM domain walls7 However, experimental studies of WFM walls4 have re- suggests that some tricky issues arise in the derivation of a vealed a significantly different dynamical behavior; no trace continuum approximation. The first objective of the present of a critical Walker field has been found and the observed paper is then to repeat the analysis of Ref. 7 in the presence wall velocities are typically much greater than those encoun- of the Dzyaloshinskii-Moriya anisotropy. We shall find that tered in ferromagnets. Instead of a Walker maximum one relation 1.1 survives in a subtle and interesting way but it observes a limiting velocity c that coincides with the phase certainly falls short of explaining the whole story. For ex- velocity of the magnons associated with the underlying anti- ample, it is tacitly assumed in the work reviewed in Ref. 4 ferromagnetic exchange interaction. As a consequence, the that the terminal velocities of driven Bloch and NeŽel walls limiting velocity is rather high, reaching the value c are given by two independent copies of relation 1.1 distin- 20 km/sec in the most typical example of an orthoferrite guished only by the respective mobilities. Instead, we find that exhibits weak ferromagnetism (YFeO3). One may then that only NeŽel walls are described by the simple relativistic invoke the simple formula formula 1.1 whereas the mobility curve of Bloch walls is significantly different. Nevertheless, a critical field h h c exists v above which a driven Bloch wall is dynamically converted 1.1 1 h/c 2 into a NeŽel wall. Above hc both types of walls are described 0163-1829/97/55 18 /12290 19 /$10.00 55 12 290 © 1997 The American Physical Society 55 DYNAMICS OF DOMAIN WALLS IN WEAK FERROMAGNETS 12 291 by a single mobility curve of type 1.1 applied with a mo- bility appropriate for NeŽel walls; see Fig. 9 at the end of the paper. In view of the great diversity as well as crystallographic complexity of realistic weak ferromagnets, it is useful to de- velop a simple microscopic model that entails only the es- sential features of the dynamics. In a sense, our aim here is to obtain the analog of the idealized Walker solution. Hence, in Sec. II, we introduce the simplest discrete spin model that embodies the main interactions present in a typical weak ferromagnet. The model is then used to calculate the profile of static domain walls by a relaxation algorithm applied di- rectly on the lattice. The calculated domain walls are repro- duced very precisely by analytical solutions derived within the continuum approximation worked out in Sec. III. The same section lays down the foundation for a complete study of the dynamics of driven domain walls, which is carried out in Sec. IV. The main conclusions are summarized in Sec. V and some calculational details are relegated to the Appendix. II. THE DISCRETE SPIN MODEL In order to obtain a manageable theoretical framework we consider a strictly one-dimensional 1D discrete spin model FIG. 1. Illustration on a short chain of labeling conventions and in which magnetic ions are placed on a chain whose sites are the dimerization process first row , of the two degenerate ground states second and third rows , and of the two types antikink and labeled by i 1,2, . . . , . The spin Hamiltonian consists of kink of prototype domain walls fourth and fifth rows . three terms, W W weak ferromagnets are characterized by a sixth-order single- E WDM WA , 2.1 ion anisotropy in the basal plane which would technically corresponding to the exchange, Dzyaloshinskii-Moriya complicate the theoretical development. We thus prefer to DM , and single-ion anisotropy contributions. The exchange complete our model by considering instead the most general interaction is taken to be antiferromagnetic, i.e., rhombic anisotropy W 1 E J Si*Si 1 , 2.2 1 2 3 i WA 2 g1 Si 2 g2 Si 2 g3 Si 2 , 2.4 i with J 0, and we consider an antisymmetric DM interaction 1 2 3 of the form where Si , Si , and Si are the Cartesian components of spin along the principal axes and g1 , g2 , and g3 are anisotropy constants. WDM 1 i 1D* Si Si 1 , 2.3 Actually a rhombic anisotropy is suitable for the study of i orthoferrites such as YFeO3. But a strictly 1D model is not where D is a vector of constant direction and magnitude directly relevant in this case because the interacting magnetic D. A microscopic explanation of the sign alternation present Fe ions form a 3D lattice.10 Recall, however, that domain in the sum of Eq. 2.3 may be inferred from the discussion walls are 1D structures embedded in a crystal in such a way of Moriya3 and is crucial for the occurrence of weak ferro- that substantial variations of spin occur along a single direc- magnetism. Lack of sign alternation would instead lead to a tion. Therefore, when a continuum approximation is appli- spiral spin state. cable, domain walls are effectively described by a differen- A strictly 1D model describes fairly well rhombohedron tial equation that is formulated in terms of a single spatial weak ferromagnets such as MnCO3 where the magnetic Mn variable in addition to time. In such a context all memory of ions interact significantly only along the crystallographic c the original lattice is reflected in appropriate renormaliza- axis.8 Between any two successive Mn ions on the c axis tions of the microscopic parameters by simple functions of there exists a CO3 complex such that the relative orientation the coordination number. Hence we shall assume that the 1D of the triangle formed by the three oxygen atoms alternates discrete spin model applies to the domain-wall dynamics of at any two successive bonds, in direct correspondence with orthoferrites, with due caution on the determination of the the sign alternation in Eq. 2.3 . This example suggests the relevant microscopic parameters. Some indirect conclusions more abstract notation9 employed in the first row of Fig. 1 to will also be drawn for rhombohedron weak ferromagnets. illustrate a short chain where open circles stand for the mag- Within the limits of the 1D model the chain direction need netic ions and up down triangles located on the bonds in- not coincide with any of the principal axes used for the dicate positive negative signs in the sum 2.3 . specification of the anisotropy constants in Eq. 2.4 . In other It would be natural to pursue the discussion of the above words, spin rotations act as an internal group without specific example through to its conclusion. However, rhombohedron reference to a coordinate system in real space. In order to 12 292 N. PAPANICOLAOU 55 keep with the interpretation alluded to in the preceding para- graph we will adopt the real-space conventions commonly used in discussions of orthoferrites4,10 but the direction of the chain will be left arbitrary. We introduce the unit vectors e1 (1,0,0), e2 (0,1,0), and e3 (0,0,1) along the three principal axes and take the constant vector D to point along the second axis: D De2 . Spins will be treated as classical vectors with constant magnitude s, S2i s2, which is a simple multiple of the Planck constant; e.g., s 52 for the Fe ions in YFeO3. Because of this constraint one of the anisotropy con- stants may be set equal to zero, e.g., g1 0, without loss of generality. The remaining constants g2 and g3 are often taken to be equal and positive, a choice that leads to a uniaxial anisotropy with the easy axis in the first direction. The restriction of equal magnitudes will not be made in the present work but we will assume for the moment that both g2 and g3 are positive so that the first axis is still the easy axis. Thus we are ready to address the first important question FIG. 2. The two degenerate ground states in the absence of an concerning the nature of the ground state. If the DM anisot- external field, which are related to each other by the parity trans- ropy were absent (D 0) the minimum energy configuration formation a,b a, b . would be the usual NeŽel state with spins polarized along the first axis. For D 0, the ground state is also achieved with preted as the magnetization and is seen to assume a nonva- spins alternating between two distinct values. In the notation nishing value, hence leading to weak ferromagnetism. of Fig. 1 all spins to the left of an up triangle take the value The two types of ground states described above are shown A and those to the right of such a triangle the value B. In schematically in the second and third rows of Fig. 1, in terms of the corresponding unit vectors a A/s and b B/s preparation for the definition of the prototype domain walls the energy per site measured in units of s2J, i.e., w given in the fourth and fifth rows of the same figure. By W/s2J , is given by convention, these configurations will be referred to as anti- kink and kink and correspond to the two distinct ways of 1 g g w a*b 2 2 2 3 2 2 connecting the two degenerate ground states between the two J D* a b 4J a2 b2 4J a3 b3 . ends of the chain. More general domain-wall configurations 2.5 may be constructed on long chains by retaining the The sign alternation present in Eq. 2.3 is crucial for the asymptotic characteristics of the prototype walls while validity of Eq. 2.5 and for the implied repetition of the pair choosing the intermediate spin values more or less at ran- a,b along the chain. Now the energy 2.5 is minimized by dom. the two distinct canted spin configurations shown in Fig. 2, However our aim here is to obtain true domain walls that which lie in the 13 plane and are related to each other by are local minima of the energy functional and are thus stable the parity transformation (a,b) ( a, b). In both cases spin configurations, even though their energy is greater than the energy is given by the energy of the ground state. It is then important to con- sider the dynamics associated with the Hamiltonian 2.1 ­ D g 2.4 . The equation of motion for the spin vector Si treated as w cos2 3 classical may be put in the standard Landau-Lifshitz form J sin2 4J 1 cos2 , 2.6 and the canting angle is found by minimizing Eq. 2.6 to Si 2 s2, 2.10 obtain t Si Fi , Si D where the effective field Fi is given by the general relation tan2 J g3/4. 2.7 W F , 2.11 Figure 2 also depicts the two vectors i Si 1 1 or, more explicitly, by m 2 a b , n 2 a b , 2.8 Fi J Si 1 Si 1 1 iD Si 1 Si 1 which will play a special role in the following. These vectors 1 2 3 may be expressed in terms of the canting angle as g1Si e1 g2Si e2 g3Si e3 . 2.12 m 0,0,sin , n cos ,0,0 , 2.9 Our first concern is to search for static solutions which satisfy Eq. 2.10 with the time derivative absent. It is a where the choice corresponds to the two degenerate straightforward matter to verify explicitly that the ground- ground states shown in Fig. 2. The vector m may be inter- state configurations may be viewed as the simplest static 55 DYNAMICS OF DOMAIN WALLS IN WEAK FERROMAGNETS 12 293 solutions. Nontrivial solutions are difficult to obtain analyti- exercise. Anticipating the discussion of the continuum ap- cally but their existence is guaranteed by the following argu- proximation in Sec. III we search for variables that may pos- ments. Note that the prototype domain walls do not solve Eq. sess a smooth continuum limit at least in some regions of the 2.10 thanks to the obstruction created at the interface. This parameter space. Thus the 2N sites of the chain are situation is slightly different from the case of pure AFM grouped into dimers labeled by a sublattice index walls studied in Ref. 7. Therefore, if a prototype wall is 1,2, . . . ,N as illustrated in the first row of Fig. 1. This somehow created, it will evolve according to Eq. 2.10 in a mode of dimerization is not unique in that every dimer con- complicated precessional mode. Nevertheless, if some dissi- tains a bond that carries an up triangle. Discussion of the pation is at work, it will eventually relax in a spin configu- dual dimerization in which all dimers contain down triangles ration that solves the static equation and is a local minimum is deferred for the moment. Now it is convenient to relabel of the energy functional. This minimum inherits the topo- the two spins contained in the th dimer according to logical structure of the prototype wall and is thus distinct from the absolute ground-state minimum. S2 1 A , S2 B . 2.14 The preceding remarks also suggest a simple numerical The advantage of the new spin variables A method for the calculation of static domain walls through a and B is that each one of them is expected to be smooth as the index relaxation algorithm described in Ref. 7. At this point one moves from one discrete value to the next. An even more must specify the parameters employed in the numerical cal- convenient set is provided by the two linear combinations4 culation. The spin magnitude s and the exchange constant J can be scaled out of static solutions and the only relevant 1 1 parameters are the dimensionless ratios formed by scaling m the DM and single-ion anisotropy constants with the ex- 2s A B , n 2s A B 2.15 change constant. These ratios are chosen to belong to a pa- which satisfy the constraints rameter regime that is appropriate for orthoferrites but no special effort is made at this stage to select constants that m 2 2 *n 0, m n 1. 2.16 correspond precisely to a specific substance. Hence we adopt the values D/J 10 2, g The idea is simply to present the numerical data for the 1 0, g2 /J 10 4 g3 /J in all nu- merical calculations and consider other possibilities on the variables m and n as histograms calculated at basis of analytical solutions derived in subsequent sections. 1,2, . . . ,N. As it turns out, these histograms approach The small canting angle calculated from Eq. 2.7 , namely continuous curves at small values of the parameter intro- 2 0.57°, is typical of orthoferrites. To complete the dis- duced in Eq. 2.13 . We may then drop the index in the cussion of parameters we introduce the equivalent set g vectors m and n and plot their three components as functions 1 0 and of the position variable g D 2 0 , 1,2, . . . ,N, 2.17 g3 2 J , 2 g , d joining discrete points in the graph by the graphics routine. 3 J , 2.13 Here 0 is an arbitrary constant that sets the origin of the whose theoretical significance will become apparent as the coordinate system. Nevertheless, it is convenient to set the discussion progresses. In our standard numerical example, origin at the center of the domain wall which coincides in the 10 2, 1, and d 1. present calculation with the center of the open chain and The numerical calculation was performed on an open hence 0 (N 1)/2. The resulting graphs are shown in Fig. chain with an even total number of sites 2N where N is 3 and make it apparent that a smooth continuum limit has also even; these are technical assumptions of no great sig- indeed been reached for the small value 10 2 used in the nificance and will be commented upon at later stages. We numerical calculation. consider only the antikink configuration illustrated for a It is clear that both m and n exhibit a more or less stan- small ( 8) chain in the fourth row of Fig. 1, the discus- dard domain-wall structure. A closer look at the numerical sion of the kink being completely analogous. The initial data reveals that the vectors m and n quickly approach con- prototype wall was prepared by assigning the pair of spin stant values far from the wall center which are in excellent values (a,b) throughout the first half of the chain and the agreement with the ground-state values 2.9 calculated with pair ( a, b) on the second half. This configuration was a canting angle derived from Eq. 2.7 . Also note that the then used as initial condition in the relaxation algorithm of components of m and n along the second axis vanish. In Ref. 7 applied for an effective field Fi now given by Eq. other words, spins are confined in the 13 plane in order to 2.12 . The resulting relaxed state is a static domain wall optimize the energy cost imparted by the DM anisotropy. In whose interface spreads out to a half width given roughly the context of orthoferrites such configurations are called by 1/ 100 sites, where 10 2 is the parameter intro- ac or Bloch domain walls.4 duced in Eq. 2.13 . To avoid interference from the bound- The numerical calculation just presented accomplishes the aries the total number of sites must satisfy the inequality main goal of this section but its content cannot be fully ap- 1/ . In our calculation we used a long chain with preciated before we tie some loose ends. First we return to 5000 sites in which domain walls fit quite comfortably. the use of an open chain with an even total number of sites Because of the implicit antiferromagnetic discontinuity of 2N where N is also even. Once a domain wall has been the spin values as one moves from site to site, presenting the realized on the open chain, removing one or more spins from explicit results in a concise fashion is in itself an interesting either side and reiterating the relaxation algorithm would af- 12 294 N. PAPANICOLAOU 55 dimerization. The best way to illustrate the question is to present the same numerical data as those employed in Fig. 3 in conjunction with the dual dimerization where all dimers contain down triangles. Specifically, we simply omit the two end points of the chain and consider the dimers 23 , 45 , . . . , ( 2, 1) labeled consecutively by an integer 1,2, . . . ,N 1. We then construct the fields 1 1 m 2s S2 S2 1 , n 2s S2 S2 1 , 2.18 which are the direct analogs of Eq. 2.15 in the dual dimer- ization scheme. We again consider the histograms for m and n by plotting the data as functions of the variable 2 0 , 1,2, . . . ,N 1, 2.19 with discrete points joined smoothly through the graphics routine. Here we may set the origin of the coordinate system at the center of the wall by choosing the arbitrary constant as 0 N/2. The resulting curves are shown in Fig. 4 and should be compared to those of Fig. 3. The observed significant differences between the two fig- ures are at first disturbing. However a closer examination reveals that these differences are quite natural and, indeed, necessary for the consistency of the entire calculation. For example, the field n has flipped sign and now appears as a kink configuration in contrast to the antikink of Fig. 3. On the other hand, the asymptotic values of m and n must now be given by 1 1 m 2 b a , n 2 b a , 2.20 instead of the values 2.8 in the original dimerization. Therefore, the necessity of a sign flip in the field n becomes self-evident at least in the asymptotic region. The same ar- gument suggests that the asymptotic values of the field m FIG. 3. The profile of a static domain wall calculated numeri- must remain the same, as actually observed in Figs. 3 and 4, cally within the discrete spin model. The numerical results are pre- or that the kink antikink character of m is preserved. The sented using the standard dimerization scheme discussed in the text last statement could have been anticipated on physical and are very accurately reproduced by the continuum solution grounds, for a nonvanishing m in the ground state signals the 3.30 and 3.31 for a static Bloch wall applied for 10 2, d occurrence of weak ferromagnetism and cannot depend on 1, and 1 . the mathematical process of dimerization. Nevertheless, no such simple explanation of the observed fect the wall only mildly provided that 1/ . In particular, curious differences around the wall center can be given until the wall need not be located at the center of the chain, as a complete analytical solution is obtained within the con- long as it stays sufficiently appart from the end points, nor tinuum approximation in Sec. III. At this point we merely does its center have to coincide with the middle of a bond. state that the answer to any physically relevant question is On the contrary, when we study the dynamics, we shall ex- independent of the specific mode of dimerization, provided tensively deal with domain walls that glide through the lat- that the mathematical framework is not overinterpreted. For tice. instance, if m is interpreted literally as magnetization, one The possible occurrence of interesting surface states may wonder whether Fig. 3 or Fig. 4 will describe the results around the end points of the chain, which may or may not be of an actual measurement. In fact, either figure can be used related to domain walls, is a separate issue that is not studied as long as experimental resolution is such that spin values in the present paper. This issue has recently attracted consid- can be measured at every site. Otherwise, one should expect erable attention within the pure AFM model applied to a to observe a fuzzy magnetization curve around the wall cen- Fe/Cr superlattice.11,12 One can only expect that adding to ter, which becomes progressively sharper as one moves away the model the DM anisotropy would lead to a more involved from the wall where the field m attains definite values that picture. are independent of the mode of dimerization. Yet we must address an apparent ``ambiguity'' that is not A number of physically relevant questions will be asked related to the size of the chain but rather to the process of and answered unambiguously in the continuation of the pa- 55 DYNAMICS OF DOMAIN WALLS IN WEAK FERROMAGNETS 12 295 figuration on the reduced chain is now a domain wall with total moment s. But a wall with total moment s also exists on the reduced chain and is again obtained by reversing the signs of all spins. Therefore for any finite chain with an even number of sites domain walls develop a net moment s while the moment of the ground state vanishes. The last re- mark is pertinent to the possibility of removing only one spin from either side of the chain. The resulting spin configura- tion carries a vanishing total moment, but the ground state of the reduced chain with an odd number of sites 1 2N 1 is now doubly degenerate and carries a moment either s or s. Hence the total moment of the domain wall again differs from that of either ground state by an amount s or s. The above examples strengthen the earlier conclusion that the tiny wall moment s is not localized and is to some extent elusive. Therefore such a moment is hardly relevant for macroscopic properties, such as those discussed in the present paper, but could be important in, say, a semiclassical quantization of pure AFM domain walls in a quantum anti- ferromagnetic chain.13 The process of dimerization, or any other substitute, is an inevitable fact of life in the derivation of a continuum approximation for antiferromagnets. It is then important that the continuum model derived in the following section cope with apparent paradoxes, as is discussed further after Eq. 3.33 . III. THE NONLINEAR MODEL The numerical calculation of static domain walls in the discrete spin model makes it clear that a suitable continuum approximation should be possible to obtain in some region of the parameter space. The appropriate region is actually sug- gested by the specific choice made in Sec. II. Indeed a rela- tively simple continuum model emerges for parameters such that g1 0 and FIG. 4. The same numerical results as those of Fig. 3 now D g g presented using the dual dimerization scheme and very accurately , 2 , 3 1. 3.1 reproduced by the continuum solution 3.30 and 3.31 applied for J J J 10 2, d 1, and 1 . These inequalities are generally satisfied in realistic weak per. This section is concluded by recalling an example of ferromagnets and will be invoked in the following without such a question that was posed within the pure AFM model exception. We must also consider the effects of an externally in Ref. 7. At small , AFM domain walls acquire a nonvan- applied field as well as dissipation. The latter is taken to be ishing total magnetic moment equal to s with respect to the of the standard Landau-Gilbert form and Eq. 2.10 is further ground state. However, it is impossible to ascertain where extended to include the effect of a uniform magnetic field H, the moment of a pure AFM wall is actually located, in view of the fact that the local values of m are sensitive to the mode of dimerization; in this respect, the result of Ref. 7 was Si Si S overstated. Nevertheless the total moment is unambiguously t Si t i Fi g0 0H , 3.2 defined on any finite chain irrespectively of the mode of dimerization, the latter being only a technique for obtaining a where is the dissipation constant, g0 2 is the gyromag- continuum approximation. Specifically, let us return to a do- netic ratio, and 0 e/2mec is the Bohr magneton divided main wall of the type shown schematically in Fig. 1 of Ref. by the Planck constant. In our conventions the combinations 7 on a long finite chain with an even number of sites of parameters s and g0 0H/sJ are dimensionless and may 2N. Such a wall carries a total moment s at sufficiently assume any values within the discrete spin model. However, weak anisotropy, whereas a wall with total moment s can for the validity of a continuum description, inequalities 3.1 be obtained on the same chain by reversing the signs of all must be supplemented by spins. Now suppose that one removes both the leftmost and the rightmost spin of the original configuration, a move that g0 0H amounts to reducing the total moment by 2s. The spin con- s , sJ 1, 3.3 12 296 N. PAPANICOLAOU 55 which are sufficiently nonstringent for all practical purposes. 1 1 More convenient rationalized quantities are defined by m 2s A B , n 2s A B , 3.9 2s g 0 0H and introduce the rescaled time variable , h 2 sJ , 3.4 and extend the set of parameters introduced in Eq. 2.13 . 2 sJt. 3.10 Inequalities 3.1 and 3.3 then read We further recall the set of parameters g1 0, , , and d of , , d, , h 1 3.5 Eq. 2.13 , which we extend slightly by defining a vector d whose magnitude is equal to d and its direction coincides and are conditions for the validity of the continuum model with the DM axis (d de derived in this section. 2), and the rationalized dissipation constant and field h of Eq. 3.4 . The continuum model is derived by a method already em- In the strict continuum limit m and n satisfy the reduced ployed in the simpler context of Ref. 7. We adopt the dimer- constraints ization scheme of Fig. 1 and again defer discussion of the dual dimerization. The Landau-Lifshitz equation 3.2 is then m*n 0, n2 1, 3.11 rewritten as a system of two coupled equations for the sub- lattice spins A and B introduced in Eq. 2.14 : m is expressed entirely in terms of n as A A t A t A F g0 0H , m 3.6 2 n n n n d n n h , 3.12 B Ba and the field n satisfies the differential equation t B t B G g0 0H , n f n 0, 3.13 where the effective fields F and G are given by where we have separated the dissipative term and the effec- F J B 1 B D B 1 B tive field f reads g 1 2 3 1A e1 g2A e2 g3A e3 , 3.7 f nš n 2 h n h d n*h h n*d d G J A A 1 D A A 1 2n2e2 n3e3 . 3.14 g 1 2 3 1B e1 g2B e2 g3B e3 . The dot stands for differentiation with respect to the time A sign alternation is no longer present in the DM contribu- variable of Eq. 3.10 and the prime with respect to the tions but its effect has been correctly accounted for in Eqs. spatial variable of Eq. 2.17 . It is understood that the 3.7 in relation to the specific mode of dimerization. strong inequalities 3.5 are enforced and terms of order 2 The main assumption supported by the numerical data is and higher have been neglected. that the sublattice spins A Therefore the ``magnetization'' m may be viewed as an and B approach smooth con- tinuum limits A A( ) and B B( ) where is the discrete auxiliary field and the dynamics is governed mainly by Eq. variable 2.17 that becomes continuous in the limit 0. 3.13 at the heart of which lies the relativistic nonlinear The dimensionless variable provides a measure of position model. The latter corresponds to the first two terms of the along the original chain. The actual distance on the chain is effective field f which originate in the pure antiferromagnetic given by a / where a is the physical distance between two interaction. The fifth and sixth terms amount to a redefinition magnetic ions. However the lattice constant a will not be of the anisotropy constants due to the applied field and the used in any stage of the theoretical development except when DM interaction. The third term in f is special in that it breaks quantities such as distance, velocity, etc., will have to be Lorentz invariance at nonvanishing field, whereas the term translated in physical units. Thus we make the replacements (h d) introduces a direct coupling between the applied field A and the DM anisotropy. A and B B in Eqs. 3.6 and 3.7 together with In order to facilitate a direct comparison to the early work 1 it is also useful to derive the effective field from an action A 1 A 2 A 2 2 2A , principle, 3.8 1 A B f 1 B 2 B 2 2 2B , n , 3.15 where the prime denotes differentiation with respect to . where A is the action Subsequent steps of the argument differ from those of Ref. 7 only in the length of the required algebra and will be relegated to the Appendix to avoid obscurring the simplicity A L d d 3.16 of the final result. Let us consider the continuum analogs of the fields 2.15 , i.e., and L the corresponding Lagrangian density: 55 DYNAMICS OF DOMAIN WALLS IN WEAK FERROMAGNETS 12 297 1 L 2 n 2 n 2 h* n n h d *n m 2 n d n e2 3.20 1 and 2 2 2 n*h 2 n*d 2 2n2 n3 . 3.17 n f 0, f n d2 2 n2e2 n3e3 . 3.21 This result agrees for the most part with Eq. 2.30 of Ref. 4 restricted to an antisymmetric DM interaction with the im- It proves useful to rewrite the effective field f in the abstract portant exception of the crossed term (h d)*n which is form absent in the above reference; so is a parity-breaking gradi- ent term in Eq. 2.28 of the same reference. It should be F f mentioned here that the possible existence of a parity- n , 3.22 breaking contribution in the field m had been anticipated on symmetry grounds14 but such a possibility was apparently where F is an energy functional given by overlooked in the literature for a long time.7,13 This contri- bution is sensitive to the symmetry of the lattice and may be 1 F n 2 d2 2 n2 n2 d . 3.23 special to the model considered here. But it should also be 2 2 3 clear that the crossed term (h d) in the effective field f is not related to the parity-breaking contribution and its conse- Resolving the constraint n2 1 by the standard spherical pa- quences are more drastic for the dynamics of domain walls. rametrization, The implications of these differences will be discussed in the following as the need arises. n1 sin cos , n2 sin sin , n3 cos , Before proceeding with detailed applications of the de- 3.24 rived continuum model we must comment on the rational- yields ized physical units employed throughout this paper. The spin magnitude s carries dimension of action, sJ of frequency, 1 and s2J of energy. The constants s and J as well as the F 2 sin2 2 d2 2 lattice constant a do not appear explicitly in the dynamical 2 equations which are formulated in terms of the dimension- sin2 sin2 cos2 d , 3.25 less ratios , , d, , and h. In particular, the spatial coordi- nate of Eq. 2.17 and the time variable of Eq. 3.10 are and static solutions are stationary points of F with respect to both dimensionless. A related fact is that the ``velocity of and . Hence we are led to the system of ordinary differ- light'' associated with Eq. 3.14 is equal to unity. Recalling ential equations that the actual distance on the chain is given by a / and taking into account the definition of time in Eq. 3.10 we 1 2 d2 2 sin2 cos sin 0, conclude that velocity is measured in units of 3.26 sin2 d2 2 sin2 cos sin . c 2asJ, 3.18 Bloch domain walls are confined in the 13 plane and which coincides with the phase velocity of magnons in the thus satisfy the simpler system long-wavelength limit of the underlying pure 1D antiferro- magnet and also provides an expression for the limiting ve- 0, cos sin 0, 3.27 locity c discussed in the Introduction. On this occasion, we wish to return to our earlier remarks concerning the use of whose solution reads the 1D model in relation to orthoferrites. The rationalized continuum equations of the field n for the description of domain walls within the 3D crystal will have the same form sin tanh , cos cosh , 3.28 as those derived above but the interpretation of constants will be slightly different. For instance, the limiting velocity is where the ``kink number'' and the ``polarity'' are given given more generally by by 1, 1, 3.29 c 2asJ z2, 3.19 taken in any combination. Therefore the vector n (n1 ,n2 ,n3) is given explicitly by where z is the lattice coordination number. As a first application of the continuum model we consider the derivation of static domain walls at vanishing field. One n1 tanh , n2 0, n3 cosh , 3.30 may then neglect field-dependent terms as well as time de- rivatives in Eqs. 3.12 ­ 3.14 and further insert the special and the corresponding expressions for m (m1 ,m2 ,m3) are form of the vector d de2 to obtain calculated from Eq. 3.20 using as input Eq. 3.30 : 12 298 N. PAPANICOLAOU 55 speaking necessary and is probably unattainable on lattices m 1 2 cosh cosh d , whose symmetry allows the appearance of parity-breaking gradient terms.14 This digression is concluded noting that the m dimerization process becomes more intricate in higher- 2 0, dimensional lattices, as discussed in a forthcoming article on the dynamics of topological solitons in 2D m3 antiferromagnets.15 2 tanh cosh d . 3.31 We may then return to the standard dimerization scheme Applied for 10 2, d 1, and 1 the above of Fig. 1 to which we will consistently adhere in the rest of continuum approximation is found to be graphically indistin- the paper. An immediate dynamical consequence of the rela- guishable from the numerical solution of Fig. 3. A good tivistic invariance of Eq. 3.13 at vanishing field and dissi- estimate of the relative accuracy is already given by the pation is that domain walls moving with a constant velocity asymptotic values of the fields 3.30 and 3.31 evaluated at v 1( c) can be derived by elementary means. For a freely : moving Bloch wall the field n is obtained simply by a Lor- entz transformation of the static solution 3.30 : m 0,0, d/2 , n 1,0,0 . 3.32 n1 tanhu, n2 0, n3 coshu , 3.34 For the specific numerical example, Eq. 3.26 yields m( ) (0,0,0.005) and n( ) (1,0,0) which are where consistent with the accurate values given in Eq. 2.9 . More generally, for parameters satisfying the inequalities 3.1 , the v canting angle of Eq. 2.7 may be approximated by u . 3.35 D/2J d/2 and Eq. 3.32 is consistent with Eq. 2.9 1 v2 applied for sin and cos 1. An analogous statement of The magnetization m is then computed from Eq. 3.12 relative accuracy holds true for all values of . where the first three terms gradient, dynamical, and DM are We are now in a position to clarify the dimerization am- now all important and the fourth term is absent at vanishing biguity discussed in Sec. II. Had we derived the continuum field: approximation using the dual dimerization centered around down triangles would simply amount to the replacement d d in Eqs. 3.12 ­ 3.14 . Therefore the numerical re- m 1 1 sults of Fig. 4 should also be predicted by Eqs. 3.30 and 2coshu 1 v2 coshu d , 3.31 applied for d 1 and for some suitable choice of the kink number and polarity which are still free to take the v values 1 and 1 in any combination. It is not m2 2 1 v2 coshu , 3.36 difficult to see that Fig. 4 is reproduced very precisely by applying the continuum solution 3.30 and 3.31 for d 1 and 1 . More generally, the mapping of solutions m between the two modes of dimerization is given by the 3 2 tanhu 1 1 v2 coshu d . simple rule d d, , . Therefore the field n changes by an overall sign while the corresponding changes In addition to an apparent Lorentz contraction we note that in m may be inferred also from the general relation 3.12 the field m develops a nonvanishing component in the sec- subjected to the transformation ond direction due entirely to the wall motion. Incidentally we mention that the continuum approximation breaks down in n n, d d. 3.33 the ultrarelativistic limit (v 1) where the wall width re- duces to a few lattice spacings. The above rule summarizes the manner in which the con- This section is completed with a corresponding discussion tinuum model copes with the dimerization ambiguity. In par- of NeŽel domain walls. We consider first static solutions for ticular, the asymptotic values and the kink antikink charac- which the vector n is confined in the 12 plane and system ter of the field m are invariant under transformation 3.33 as 3.26 reduces to is evident from Eq. 3.32 . Furthermore the continuum model handles quite efficiently questions such as the elusive total moment of a pure AFM wall. Indeed, although the con- 2 , d2 2 cos sin . 3.37 tinuum approximation is by construction oblivious to the cut- ting and pasting of a finite chain, it copes with the various Solutions of the second equation are given by moves described in the concluding paragraphs of Sec. II by mapping solutions of the underlying nonlinear model onto other solutions of the same model obtained through the sym- cos tanh , sin cosh , 3.38 metry transformation n n. Having understood this point, no real ambiguity is present in comparing careful numerical where the kink number and the polarity take the same calculations with continuum solutions. Moreover a definition values as those of Eq. 3.29 and is the rescaled spatial of a magnetization m with definite local values is not strictly coordinate 55 DYNAMICS OF DOMAIN WALLS IN WEAK FERROMAGNETS 12 299 d2 2 . 3.39 whereas the field m is calculated from Eq. 3.12 using as input Eq. 3.43 and h 0: The vector n reads tanhw m d2 2 d2 2 n 1 2 1 v2 cosh2w , m2 2 1 v2 coshw , 1 tanh , n2 cosh , n3 0, 3.40 3.45 and the vector m is calculated from Eq. 3.20 to yield v m d2 2 3 2 1 v2 coshw d tanhw . m 1 2 d2 2 cosh2 , IV. DRIVEN DOMAIN WALLS tanh m 2 The main point of this work is the study of the dynamical 2 d2 2 cosh , response of a domain wall, either Bloch or NeŽel, to an exter- nally applied magnetic field in the presence of dissipation. m During the initial steps of the development it is conceptually 3 2 d tanh , 3.41 simpler to work strictly within the discrete spin model. The where we note that all three components are now different continuum description will be invoked at a later stage and from zero. will prove more powerful in establishing the complete pic- The numerical calculation of Sec. II did not produce evi- ture. For definiteness let us assume that the initial configura- dence for NeŽel domain walls because it was based on a re- tion is the static Bloch wall calculated in Sec. II for the laxation algorithm that inevitably leads to a local minimum discrete system Fig. 3 which is subjected to a uniform field of the energy functional. As a result a NeŽel wall would decay that is turned on at t 0 and points in the third direction, into a topologically equivalent Bloch wall with lower energy. h 0,0,h he A more precise statement is that 13 walls have lower en- 3 . 4.1 ergy than 12 walls for parameters such that The mathematical problem consists of solving Eq. 3.2 with initial condition supplied by the static wall. One must then g g study the ensuing evolution and possibly ascertain the for- d2 2 1 or D 2 2 3 J J J . 3.42 mation of a terminal state where the spin configuration moves rigidly with constant velocity v. In particular, one This inequality is certainly satisfied for the uniaxial anisot- must determine the nature of such a state and the terminal ropy (g2 g3) used in our numerical calculation and is also velocity v as functions of the applied field. typical of orthoferrites. However, at least one example is To appreciate the results of an explicit numerical solution quoted in the literature, the dysprosium orthoferrite we first examine the behavior of the spin configuration far DyFeO3, where the inequality is reversed below 150 K and from the wall center or, equivalently, determine the fate of the role of Bloch and NeŽel walls is interchanged.4 Hence, in the two degenerate ground states after the field is turned on. the bulk of the paper, we shall assume that Eq. 3.42 is It is clear that the applied field lifts the degeneracy and cre- satisfied and defer discussion of the consequences of the op- ates an imbalance between the two sides of the wall. Actu- posite inequality for the end of the argument. Since NeŽel ally a field by itself would merely set the ground state in walls are unstable one may question whether or not they are eternal precession. The role of dissipation is also important relevant for the phenomenology of weak ferromagnets. Ac- in that precession eventually dies out and two new static tually the inequality 3.42 is marginally satisfied in orthof- ground-state configurations emerge that are both local errites and NeŽel walls are substantially stable; they can be minima of the energy functional but now have different en- experimentally produced and studied apparently without ergies thanks to the magnetic field. The precise nature of great difficulty. Furthermore they will prove to be crucial in these minima is again determined by optimizing the simpli- our theoretical analysis of driven domain walls. The situation fied energy function 2.5 extended to include a Zeeman is different in rhombohedron weak ferromagnets where the term: two types of walls are separated by a wide energy gap and NeŽel walls are rather unstable. 2 2 2 We conclude this line of reasoning by quoting an analyti- w a*b d* a b 4 a3 b3 h a3 b3 . cal solution for a freely moving NeŽel wall at vanishing field 4.2 and dissipation. The field n is obtained by a Lorentz trans- formation of the static solution 3.40 , Here we have anticipated that the optimal configurations are confined in the 13 plane hence a2 0 b2 and have also expressed parameters in their rationalized form. n1 tanhw, n2 coshw , n3 0, 3.43 The original ground state depicted in the first row of Fig. 2 evolves into a state with a field-dependent canting angle with satisfying the algebraic equation 2 w d2 2u d2 2 1 sin2 dcos2 hcos 0, 4.3 1 v2 v , 3.44 4 12 300 N. PAPANICOLAOU 55 2 d h , 2 d h , 2 h d . 4.6 This approximation becomes progressively questionable for very strong fields in the region h 1/ . Therefore, when the field is turned on, the two ground- state configurations domains on the two sides of the do- main wall are expected to adjust to those of Fig. 5 at some characteristic time interval 0 . During the transient period, 0 , precession effects are strong and the wall behaves in a complicated manner. However, for 0 , the two sides have adjusted to the new static domains one of which has higher energy density. We thus expect the whole spin con- figuration to reach a terminal state where the domain with the lower energy expands perpetually at the expense of the other; whence the motion of the domain wall with a constant terminal velocity v v(h). The qualitative picture described above can be confirmed by a straightforward numerical calculation in the discrete spin model. We solve the initial-value problem for Eq. 3.2 for our standard choice of parameters ( 10 2, 1 d) together with a typical dissipation constant s 10 2 or 2. Simulations were performed for a number of values of the rationalized field h, using as an initial condition the static Bloch wall calculated numerically in Sec. II at vanishing field. FIG. 5. The fate of the two degenerate ground states of Fig. 2 in Explicit results for h 1/2 are given in Figs. 6 and 7. The the presence of an external field h. Degeneracy is lifted because the wall motion was monitored by tracking the point where the second ground state displays a different canting angle ( ) and first component of n vanishes, using linear interpolation to higher energy. The third row of the figure illustrates the second locate its actual position between two lattice sites. The cor- ground state in connection with a mild crossover that takes place at responding velocity is plotted in Fig. 6 a and displays a h d. transient period after which it quickly approaches a terminal which reduces to Eq. 2.7 at vanishing field (h 0). Simi- value v 0.214 that should be accurate to all three significant larly the ground state shown in the second row of Fig. 2 figures. Figure 6 b illustrates the time evolution of the becomes a state with a different canting angle given by ground-state configurations away from the wall center con- centrating on the third component of m whose initial ( 2 0) asymptotic ( ) values are those of Eq. 2.9 . Af- 1 4 sin2 d cos2 h cos 0. 4.4 ter the same transient period m3 approaches the terminal value m3 sin 0.007 499 39 far to the left of the wall and The two states are depicted symbolically in the first and sec- the value m3 sin 0.002 501 08 far to the right, ond rows of Fig. 5 and are no longer related by a parity which are in excellent agreement with the accurate roots of reflection because . A simple numerical solution of the the algebraic equations 4.3 and 4.4 and in good agree- algebraic equations and a corresponding calculation of the ment with the approximate roots 4.6 . Finally Fig. 7 dis- energy 4.2 establishes that both states are local minima but plays the terminal state of the wall with numerical data for the energy of the second solution is higher. The third row in the spin presented using the technique discussed in Sec. II in Fig. 5 illustrates the manner in which the second state connection with Fig. 3. evolves for a sufficiently strong field (h d). Note that for A simple comparison of Figs. 3 and 7 reveals that the h d the solution of Eq. 4.4 is 0. A new element in the terminal state has acquired a significant NeŽel component region h d is that the vector m points in the same direction (n2 0) and is thus appreciably different from the original the direction of the magnetic field for both types of ground Bloch wall. In particular, the terminal state of the driven states but its magnitude is different in the two cases. The Bloch wall has no resemblance to the freely moving Bloch canting angle is calculated from wall of Eqs. 3.34 ­ 3.36 . To push this picture further we repeated the calculation for a stronger field (h 3/2) which 2 was expected to lead to a stronger Bloch-NeŽel hybridization. 1 4 sin2 d cos2 h cos 0, 4.5 The results of Fig. 8 came as a surprise in that the original Bloch wall (n2 0) had turned completely into a NeŽel wall which is related to Eq. 4.4 by the substitution . (n3 0). Incidentally we note that we are now in the field Within the domain of validity of the continuum approxima- regime h d, because h 3/2 and d 1, where the vector m tion the various canting angles introduced above can be ap- points in the field direction on both sides, as was anticipated proximated by by the discussion of Fig. 5 and is evident in Fig. 8 b . How- 55 DYNAMICS OF DOMAIN WALLS IN WEAK FERROMAGNETS 12 301 FIG. 6. Dynamical response of the static Bloch wall of Fig. 3 to FIG. 7. The terminal state of the Bloch wall of Fig. 3 driven by an applied field h 1/2, calculated numerically within the discrete a field h 1/2, calculated numerically within the discrete spin spin model. a The wall velocity approaches a terminal value v model. Note a significant Bloch-NeŽel hybridization (n2 0). 0.214 after a transient period 0 5. b Response of the ground state monitored by the values of m3 far to the left of the wall upper change the situation because the effective anisotropies along curve and far to its right lower curve . The corresponding terminal the second and third axes now appear with coefficients d2 values are given in the text. 2 and 1 h2. At low fields, where the inequality d2 ever the mild crossover at h d is not the important issue in 2 1 h2 is still satisfied, 13 walls continue to be stable Fig. 8. Rather this calculation suggests the existence of a but may develop a small 12 component due to the applied field. However, at a sufficiently strong field where the in- genuine critical field hc above which a driven Bloch wall is equality is reversed, 13 walls become relatively unstable to always converted into a NeŽel wall. The critical field hc need 12 walls and a complete dynamical conversion takes place not coincide with the crossover value h d. in the terminal state. The critical field is then estimated from A detailed investigation of this important issue based only d2 2 1 h2 or on numerical simulations would be tedious. Hence we recall at this point the continuum model which will prove to be a h very powerful tool. For instance, the continuum model may c d2 2 1, 4.7 be used to provide a simple explanation for the existence of in units specified by Eq. 3.4 , and is clearly not related to a critical field which will also yield a rough estimate of its the crossover value h d discussed in connection with Fig. actual value. The important issue is again inequality 3.42 5. Now, applied for d 1 , the above estimate yields hc that governs the relative stability of Bloch and NeŽel walls. 1 which explains the Bloch-NeŽel hybridization observed in Suppose that Eq. 3.42 is indeed satisfied and thus NeŽel or Fig. 7 for h 1/2 as well as the complete dynamical conver- 12 walls are relatively unstable to Bloch or 13 walls. A sion of a driven Bloch wall at h 3/2 shown in Fig. 8. The simple inspection of Eq. 3.14 applied for d de2 and h picture is completed in three steps described in the following he3 suggests that the presence of an applied field may three subsections. 12 302 N. PAPANICOLAOU 55 and the problem is accordingly reduced to the solution of ordinary differential equations. In this subsection we shall not attempt to find explicit solutions of the above equations. Instead we will derive a general virial relation that can be used for a direct calculation of the mobility. The method is an elementary adaptation of related work in the theory of magnetic bubbles.16 An equivalent form of Eq. 4.9 is obtained by taking the cross product of both sides with n and using the constraint n2 1: vn f f*n n. 4.11 Next we contract both sides with the vector n and use the identity (n*n ) 0 which is also a consequence of the con- straint: vn 2 f*n . 4.12 The right-hand side of Eq. 4.12 may be written as f*n , 4.13 which is indeed an identity if f is taken from Eq. 4.10 and is given by 1 2 1 v2 n 2 2 h d *n n*h 2 n*d 2 2n2 2 2 n3 . 4.14 Equation 4.12 is then written as vn 2 , 4.15 whose advantage is that the right-hand side is a total deriva- tive.The virial theorem is obtained simply by integrating both sides of Eq. 4.15 over all space, FIG. 8. The terminal state of the Bloch wall of Fig. 3 driven by v n 2d , 4.16 a field h 3/2, calculated numerically within the discrete spin model. This state is very accurately reproduced by the continuum where ( ) are the boundary values of . Inspection of solution for a driven NeŽel wall given by Eqs. 4.43 and 4.44 and Eq. 4.14 taking into account that d de2 and h he3 , to- indicates a complete dynamical conversion of the initial Bloch wall. gether with the fact that only the first component of n sur- vives at large distances, yields A. Virial theorem and mobility A driven domain wall that has reached a terminal state v n 2d hd n1 n1 . 4.17 with constant velocity v is described by a field n of the form One may also recall the kink number introduced in Sec. II, n n v . 4.8 1 Time derivatives may then be replaced by n vn where 2 n1 n1 , 4.18 the prime denotes differentiation with respect to either or the entire argument v . This distinction will not be made to write the virial relation in the final form explicit in the following but one must remember that the argument of all fields is v . Equation 3.13 becomes v n 2d 2 dh. 4.19 v n n n f, 4.9 It is important to note that the only contribution on the right- where the effective field f takes the reduced form hand side of this relation originates in the crossed (h d) term discussed earlier, which now proves to be crucial for f 1 v2 n 2v h n h d the very existence of driven domain walls in a terminal state; for, otherwise, the right-hand side of Eq. 4.19 would vanish n*h h n*d d 2n2e2 n3e3 4.10 leading to an obvious contradiction. Therefore, if the La- 55 DYNAMICS OF DOMAIN WALLS IN WEAK FERROMAGNETS 12 303 grangian of Eq. 2.30 in Ref. 4 is taken at face value, driven quencies is to consider the Lagrangian 3.17 at vanishing domain walls in a terminal state would not exist for an anti- field (h 0) expressed in terms of the spherical variables symmetric DM interaction. Relation 4.19 also contradicts 3.24 : the existence of static (v 0) walls in the presence of an applied field (h 0) as well as the existence of a rigidly 1 moving wall (v 0) at vanishing dissipation ( 0) and a L 2 2 sin2 2 2 sin2 2 nonvanishing field (h 0). Virial relation 4.19 will be used in two ways. First, as a d2 2 sin2 sin2 cos2 . 4.24 check of consistency of both numerical and analytical re- Small fluctuations around the ground state /2 and sults. For example, the numerical calculation presented in 0 or are calculated by inserting /2 and Figs. 6 and 7 must be consistent with Eq. 4.19 . Indeed or in Eq. 4.24 and keeping terms that are at most using the input parameters 2, d 1, 1, and h quadratic in and . The resulting quadratic Lagrangian 1/2, the calculated terminal velocity v 0.214 of Fig. 6 a and the field n of Fig. 7 a to compute the integral in Eq. 1 1 4.19 numerically, the virial relation is satisfied to at least L three significant figures. 2 2 2 2 2 2 2 d2 2 2 4.25 A second more tangible application of the virial theorem is an exact calculation of the wall mobility. Equation 4.19 describes two uncoupled free fields with dispersions is consistent with a linear mobility relation at low fields where 2 2 1 k 1 k2, 2 k 2 k2, 4.26 1 where 1 and 2 are the magnon activation frequencies v h, 2 d n 2d , 4.20 1 1, 2 d2 2, 4.27 supplemented by the stipulation that the integral be evaluated using as input the profile of the initial static domain wall, as expressed in the rationalized units employed throughout this is appropriate in the limit of the vanishing field where the paper. velocity also vanishes. The sign of the mobility is not For the moment we consider the dimensionless ratio definite because Eq. 4.20 yields information on both the 2 / 1 and compare it to the right-hand side of Eq. 4.23 to direction and the magnitude of the wall velocity. One should obtain the parameter-free theoretical prediction add that the above result is insensitive to the dimerization ambiguity discussed earlier because Eq. 4.19 is invariant 1 2 . 4.28 under transformation 3.33 . 2 1 We must now distinguish two cases depending on On the other hand, Ref. 10 gives the values whether the initial wall is Bloch or NeŽel. For a Bloch wall 1 11 cm 1 and we may use the static solution 3.30 in Eq. 4.20 to obtain 2 17 cm 1 or 2 / 1 1.55 for the frequencies ac- tually observed in YFeO3, while Ref. 4 adopts values in the d range 1 (11­ 13) cm 1 and 2 (15­ 20) cm 1 or 1.15 v 1h, 1 , 4.21 2 / 1 1.82 . These values are not terribly inconsistent with the measured mobility ratio 1 / 2 1.06, in view of whereas the NeŽel mobility is calculated from Eqs 3.40 and the simplicity of the classical spin model and the fact that 4.20 : anharmonic corrections to the calculated frequencies 4.27 have been neglected. d This subsection is concluded by translating some of the v 2h, 2 quantities calculated above in ordinary units. The definition . 4.22 d2 2 of the dimensionless time variable in Eq. 3.10 implies We also consider the dimensionless ratio that the magnon activation frequencies of Eq. 4.27 are mea- sured in units of 2 sJ. Hence 1 d2 2 4.23 2 g2 2 1 2sJ g3J, 2 2sJ DJ J . 4.29 and relate it to the inequality 3.42 discussed in connection with the potential instability of NeŽel walls. When this in- These results could also be derived by calculating the mag- equality is satisfied NeŽel walls are relatively unstable to non spectrum of the 1D discrete spin model of Sec. II using Bloch walls and their mobility is smaller ( standard spin-wave techniques. The corresponding frequen- 1 2). The measured17,18 mobility ratio in YFeO cies in the 3D model were calculated in Refs. 10 and 19 3 is 1 / 2 1.06. In DyFeO within the harmonic approximation and read in current nota- 3 below 150 K, the inequality 3.42 is reversed, 13 walls become unstable, and their mobility is predicted to be tion smaller than the mobility of 12 walls ( 1 2). One can also relate the mobility ratio to the experimen- z D 2 g2 tally observed magnon activation frequencies in the absence 1 2sJ zg3 2J , 2 2sJ z2 2 J J , of the external field. The simplest way to calculate the fre- 4.30 12 304 N. PAPANICOLAOU 55 where z 6 is the coordination number of the 3D lattice, the chain rule to derive the two independent equations while setting z 2 reproduces the 1D results of Eq. 4.29 . These are further examples of correspondence between the F F microscopic parameters of the 1D and 3D models; see Eq. v , v sin2 , 4.34 3.19 . We also take this opportunity to mention that the positivity condition on g whose explicit forms read 2 may be relaxed to some extent, as long as the arguments in the square roots of Eqs. 4.29 and v 1 v2 hd cos cos 4.30 remain positive. Finally we translate the mobilities 4.21 and 4.22 in 1 v2 2 2vh d2 2 sin2 ordinary units recalling that velocity is measured in units of the limiting velocity c of Eq. 3.18 . Then 1 h2 cos sin 4.35 c g D and 0 0 g3 1/2 1 2s 2sJ J J , v sin2 1 v2 sin2 vh sin2 c g D g hd sin sin d2 2 0 0 D 2 2 1/2 2 2s 2sJ J J J . 4.31 sin2 cos sin . 4.36 We shall not attempt to generalize the above formulas to an This is a rather complicated system of nonlinear differential arbitrary coordination number except to state that the equations and its analytical solution appears to be hopeless. parameter-free prediction 4.28 remains true in any dimen- Nevertheless the system simplifies enormously when we sion. restrict attention to strictly NeŽel walls, i.e., Perhaps the earliest theoretical calculation of the v(h) curves is that of Gyorgy and Hagedorn20 who arrived at two formulas of type 1.1 , one for each kind of wall. This early 2 , 4.37 attempt suffers from two drawbacks. First, the calculated mobilities appear to be proportional to the exchange rather for which the first equation is trivially satisfied and the sec- than the DM constant, which is obviously false on physical ond reduces to grounds because the driving issue is the DM interaction. Sec- ond, as we shall see shortly, a description in terms of two 1 v2 d2 2 cos sin completely independent v(h) curves is also false. The situa- tion was improved in more recent publications21,22 but sev- v hd sin . 4.38 eral issues remained unclear. We now attempt to solve simultaneously the two equations B. Analytical solution for NeŽel walls 1 v2 d2 2 cos sin , One must now consider the case of a field of arbitrary strength and possibly make contact with the semiempirical v hd sin , 4.39 relation 1.1 . Although the numerical simulation is still an in the sense that every solution of Eqs. 4.39 will be a so- option, one might also hope to derive analytical solutions lution of Eq. 4.38 . The first equation gives within the continuum model. Such a hope is partially ful- filled in the present subsection. Thus we return to Eq. 4.11 where the effective field f of cos tanhw, sin coshw , 4.40 Eq. 4.10 is again derived from a variational argument of the form 3.22 by generalizing the energy functional F of where and are the kink number and polarity discussed in Eq. 3.23 to Sec. III and w is the argument of Eq. 3.44 . The important observation is that the angle of Eq. 4.40 also satisfies the 1 second equation in Eq. 4.39 provided that the parameters F 2 1 v2 n 2 2vh* n n 2 h d *n are related by n*h 2 n*d 2 2n2 2 2 n3 d . 4.32 v Expressed in terms of the spherical variables the above func- 1 v2 2h, 4.41 tional reads where 2 is precisely the mobility of a NeŽel wall calculated 1 previously in Eq. 4.22 . The relativisticlike relation 4.41 F 2 1 v2 2 sin2 2 2vh sin2 may then be written as 2hd sin cos d2 2 sin2 sin2 v 2h 4.42 1 h2 cos2 d . 4.33 1 2h 2 One may now use the general form of the effective field f and thus reproduces Eq. 1.1 applied with a mobility from Eq. 3.22 in Eq. 4.11 and a repeated application of 2 appropriate for a NeŽel wall. The lack of a limiting 55 DYNAMICS OF DOMAIN WALLS IN WEAK FERROMAGNETS 12 305 velocity c in Eq. 4.42 is, of course, due to the current use of converted completely into a NeŽel wall. A definitive confir- rationalized physical units where the limiting velocity is mation is achieved by noting that the numerically calculated equal to one. detailed profile of a driven Bloch wall given in Fig. 8 for h To complete the solution we calculate the field n, from 3/2 is reproduced very precisely by the analytical solution Eqs. 4.37 and 4.40 , for a driven NeŽel wall given in Eqs. 4.43 and 4.44 . Yet the results for h 1/2 presented in Figs. 6 and 7 in- n dicate that there exists a field regime (h hc) where the 1 tanhw, n2 coshw , n3 0, 4.43 terminal state of a driven Bloch wall is hybridized and the and the field m from the general relation 3.12 : velocity is not predicted by either Eq. 4.42 or 4.45 ; except for very low fields where v 1h, in accord with our results in subsection A. In other words, the mobility curves for the m d2 2 1 two types of domain walls coincide for h h 2 1 v2 cosh2w , c and are both given by the relativistic formula 4.42 applied with a NeŽel mobility tanhw 2 . Below hc a bifurcation takes place whereby the m d2 2 two curves split and eventually reach the low-field linear 2 2 1 v2 coshw , regime at different slopes; v 1h and v 2h for Bloch and NeŽel walls, respectively. Although we have been unable v m d2 2 to obtain an analytical solution of Eqs. 4.35 and 4.36 for 3 2 1 v2 coshw d tanhw h . 4.44 driven Bloch walls in the region h hc , the bifurcation de- scribed above was unambiguously established by repeating A notable feature of this result is that a driven NeŽel wall the numerical calculation for a number of field values in the differs from the freely moving NeŽel wall of Eq. 3.45 only range 0 h 3/2. The results are summarized in Fig. 9 by an additive field dependent constant in m3 and the fact which is more or less self-explanatory. that the velocity is now a definite function of the applied Our inability to obtain a complete analytical solution is field given by Eq. 4.42 . As a check of consistency one can due to the nontrivial Bloch-NeŽel hybridization that takes verify explicitly that the above solution satisfies the virial place in the region h h relation 4.19 . c see Fig. 7 . A related technical reason is that the contribution of the nonrelativistic term The seeds for the analytical solution presented in this sub- 2(h n ) is now crucial, while it had droped out of Eq. 4.38 section may be found in the paper of Zvezdin21 but its pre- describing driven NeŽel walls. The same fact explains the cise form and physical content remained unclear. In particu- relativistic nature of the NeŽel mobility formula 4.42 and is lar, no distinction was made between Bloch and NeŽel walls. the reason why the corresponding expression for Bloch walls Nevertheless the above work also contains the seeds for the given by Eq. 4.45 is false. At any rate, the picture obtained crossed term (h d) discussed earlier in the present paper. by the combination of analytical and numerical results de- But the formalism of Ref. 21 is unwieldy and was clearly not rived so far is essentially complete and we now turn to the adopted in more recent publications.4 discussion of its implications. First we return to the rough estimate of the critical field C. Dynamical conversion of Bloch walls given by Eq. 4.7 which yields hc 1 for our standard We now resume the study of driven Bloch walls initiated choice of parameters, while the numerical results of Fig. 9 by the numerical simulation presented in the beginning of the indicate a value hc 0.8. Such a discrepancy is not surpris- main section. It would be natural to expect that the corre- ing because the argument leading to Eq. 4.7 ignores more sponding v(h) curve is analogous to Eq. 4.42 , namely, subtle effects from the remaining terms in Eq. 3.14 . As a consequence, Eq. 4.7 is, at best, a rough overestimate of the true critical field. Nevertheless Eq. 4.7 provides a useful v 1h false , 4.45 1 guide especially when it is expressed in terms of the mobility 1h 2 ratio 4.23 to yield where 1 is now the Bloch mobility of Eq. 4.21 . In fact, this appears to be the implicit assumption made in all earlier hc 1 / 2 2 1. 4.46 treatments.4 However, the numerical results presented in In a typical orthoferrite such as YFeO3 inequality 3.42 is Figs. 6, 7, and 8 already disprove such an assumption. satisfied as evidenced by the measured mobility ratio For example, the accurate terminal velocity v 0.214 cal- 1 / 2 1.06. Therefore, two distinct mobility curves v culated numerically for h 1/2 see Fig. 6 a clearly dis- 1h and v 2h emanate from the low-field region that agrees with the value v 0.224 obtained from Eq. 4.45 must join up at a critical value of the driving field and there- applied with 1 1/2; here the Bloch mobility 1 was cal- after follow a single curve given by the relativistic NeŽel for- culated from Eq. 4.21 for our standard choice of parameters mula 4.42 . In view of the small mobility ratio the bifurca- d 1 , 2, and 1. A more impressive statement tion region is expected to be narrow and may easily have can be made at h 3/2 where the numerically calculated ter- been missed in the analysis of existing experimental data, minal velocity (v 0.468) again disagrees with Eq. 4.45 especially because other complications are present such as but is, instead, very accurately predicted by Eq. 4.42 ap- magnetoelastic anomalies.4 However our detailed theoretical plied with a NeŽel mobility 2 1/2& given by Eq. 4.22 . results for both the mobility curves and the profiles of driven This is a concrete confirmation of our earlier statement that a domain walls may help to reassess the experimental situa- critical field hc exists above which a driven Bloch wall is tion. 12 306 N. PAPANICOLAOU 55 approaches the relativistic limiting velocity c and the wall width reduces to a few lattice spacings. Under such extreme conditions one must again resort to the discrete spin model of Sec. II. However a new element arises when Fig. 5 is pushed to extreme field values. The second ground state, now described by the third row of Fig. 5, ceases to be a local minimum and becomes a saddle point of the energy 4.2 at a new critical field hc f(d)/ where f(d) is some function of d that can be determined numerically. In our standard numerical example hc 190. Therefore, when a field with strength above hc is turned on, the imbalance between the two sides of the wall becomes catastrophic and the motion looks more like an avalanche rather than a steady terminal state. In this respect, the field hc may be interpreted as the analog of the critical Walker field in ordinary ferromagnets and seems to be the main preoccupation of Ref. 21. None- theless such a field regime does not appear to be of great practical value because the corresponding wall velocities have practically reached the limiting velocity c. Finally, we mention that the general subject of dynamical conversion of domain walls was discussed previously in dif- ferent physical contexts. For instance, mutual conversion of xy and yz kinks was studied in Ref. 23 for an easy-plane FIG. 9. Mobility curves for driven Bloch and NeŽel walls. The antiferromagnetic chain immersed in an in-plane magnetic NeŽel curve solid line is given by the analytical expression 4.42 applied with a mobility field. Although the above work addresses the question of 2 calculated from Eq. 4.22 ; 2 1/2& for our standard choice of parameters 1, d 1 , and freely moving instead of driven kinks in the absence of 2. The Bloch curve dashed line was obtained by a numerical dissipation, a field-dependent critical velocity was found af- simulation in the discrete spin model and takes off with a slope ter which dynamical conversion takes place. The closest ex- 1 at low fields (v 1h) where 1 1/2 is the Bloch mobility ample to our current work is discussed in Ref. 24 which calculated from Eq. 4.21 . The two curves join up at a critical field considers the effect of a small ``symmetric'' correction to the hc 0.8 and thereafter both follow the analytical NeŽel result of Eq. antisymmetric DM interaction. Freely moving domain walls 4.42 . The inset demonstrates the bifurcation regime in greater de- in the absence of both dissipation and an applied field were tail, with numerical data for the Bloch curve represented by open then shown to undergo dynamical conversion at some critical circles and a dotted straight line indicating the initial slope 1 velocity. On the other hand, we have established that driven 1/2. Bloch walls in the presence of dissipation are dynamically converted for h hc even in the absence of a ``symmetric'' DM interaction; including the latter in our model will only When inequality 3.42 is saturated the critical field van- lead to a compounded effect. Therefore the results of Ref. 24 ishes and the bifurcation region shrinks to zero. Putting it need to be reanalyzed in the light of our current conclusions. differently, the terminal state is always a 12 wall and the mobility curve is that of Eq. 4.42 irrespective of the nature V. CONCLUDING REMARKS of the initial state. This simplified picture remains a fortiori correct when inequality 3.42 is reversed and may describe We believe to have presented a complete study of the the dynamics of domain walls in DyFeO3 below 150 K. domain-wall dynamics within the limits of the simplest non- In contrast, the observed great disparity between Bloch trivial model of a weak ferromagnet which may serve as a and NeŽel walls in rhombohedron weak ferromagnets should prototype for more realistic calculations. From a purely the- be expected to enhance the bifurcation regime and make it a oretical point of view the new elements that are likely to dominant feature in the analog of Fig. 9. As a consequence, survive the specific model are a a clear analysis of the the mobility curve for Bloch walls will depart significantly dimerization ambiguity inherent in all physical systems in- from the relativistic result of Eq. 1.1 for most field values volving antiferromagnetic interactions, b a related deriva- of practical interest. Of course, a detailed calculation in this tion of a parity-breaking gradient term in the magnetization case will require a modification of the Hamiltonian to in- m, and c the identification of a crossed (h d) term in the clude a sixth-order single-ion anisotropy in the basal plane nonlinear model that governs the dynamics of the field and possibly alternative forms of the dissipative term.4 n. These elements are important for a correct understanding We also comment on the domain of validity of the con- of both structural and dynamical properties of WFM domain tinuum model in the presence of an applied field. The con- walls and were for the most part absent in earlier treatments.4 dition h 1 of Eq. 3.5 is well satisfied in Fig. 9 where the At a more practical level we have presented a complete maximum displayed value is h 10 for which h 0.1. For calculation of driven domain walls whose main features are greater field values the continuum approximation begins to a a virial theorem that underlies the existence of a terminal deteriorate and eventually breaks down when h 1. An- state and allows a simple calculation of the low-field mobili- other way for stating the same fact is that the wall velocity ties, b a critical field hc above which Bloch walls are dy- 55 DYNAMICS OF DOMAIN WALLS IN WEAK FERROMAGNETS 12 307 namically converted into NeŽel walls, and c a related bifur- curve for driven Bloch walls will share with Eq. 1.1 some cation process that leads to a new and interesting picture of broad characteristics, such as a low-field linear regime and a the mobility curves. These features should be present also in high-field limiting velocity c, but will differ from Eq. 1.1 realistic weak ferromagnets and could be established experi- in its important details for most field values of practical in- mentally. terest. A closer look at orthoferrites should entail a more detailed Finally, there is some room for theoretical improvements justification of our main results within the proper 3D crystal even within the strict limits of the model considered here. environment. Such a study would remove some uncertainty For instance, our inability to obtain an analytical solution for concerning the correspondence between microscopic param- driven Bloch walls in the subcritical (h hc) regime forced eters and those appearing in the continuum model. Inciden- us to complete the picture in Fig. 9 by a direct numerical tally the original determination of parameters carried out by simulation. It may prove possible to study the neighborhood Treves25 was based on the assumption of a uniaxial single- of the bifurcation point (h hc) analytically and replace the ion anisotropy (g2 g3) and a calculation of susceptibilities rough estimate of the critical field given in Eq. 4.7 by a within the leading classical approximation. It turns out that more accurate value. Inclusion in our basic model of the the classical susceptibilities are not especially sensitive to the magnetoelastic couplings mentioned in the Introduction is precise value of g2 , as long as inequalities 3.1 are en- also a subject for further investigation. forced, which must then be determined by independent mea- surements such as the magnon activation frequencies10 or the ACKNOWLEDGMENTS mobility ratio.17,18 Because of the crudeness of the theoreti- cal models used to describe a rather complex physical situa- I am very grateful to Victor Bar'yakhtar for a number of tion, and a corresponding uncertainty in actual experiments, discussions concerning the work reviewed in Ref. 4 as well it is probably fair to say that a precise knowledge of the as the current effort. The present work was supported in part microscopic parameters is not available at this point. by a grant from the EEC CHRX-CT93-0332 , by a bilateral Although the emphasis in the main text was placed on Greek-Slovak research program, and by the Sossino Founda- orthoferrites, the 1D discrete spin model developed in this tion. paper may prove to be more faithful to the description of rhombohedron weak ferromagnets such as MnCO APPENDIX: THE CONTINUUM LIMIT 3 or FeBO3 iron borate . For such a purpose one must complete In this appendix we provide some of the algebraic details the model by a proper sixth-order single-ion anisotropy in necessary for the derivation of the extended nonlinear the basal plane and then repeat the calculations of the present model of Sec. III. As a first step we insert the Taylor expan- paper. As mentioned already, we anticipate that the mobility sions 3.8 in Eqs. 3.6 to obtain A A t A t A 2J B B 2B 2D B B 2B g1A1e1 g2A2e2 g3A3e3 g0 0H , B B t B t B 2J A A 2A 2D A A 2A g1B1e1 g2B2e2 g3B3e3 g0 0H . A1 This system of equations is not yet fully consistent because it appears to mix different powers of the small parameter . To obtain a consistent system we proceed as in Ref. 7. An equivalent form of Eq. A1 expressed in terms of the fields m and n of Eq. 3.9 and rationalized parameters is given by 1 m 2 m m n n m n m m n n n d m m d n n d n m d m 1 2 n d m m d n 2 2 m2 m e2 n2 n e2 1 2 m3 m e3 n3 n e3 m h , 1 n 2 2 m n n m 2 m n m m n n m d m n d n 2 n m m n 2 n d m m d n 3 m d m n d n 1 1 2 2 2 m2 n e2 n2 m e2 2 2 m3 n e3 n3 m e3 n h . A2 12 308 N. PAPANICOLAOU 55 Here the dot stands for differentiation with respect to the terms of order 2. Taking the cross product of both sides of rationalized time variable of Eq. 3.10 and the prime with Eq. A3 with n and using the reduced constraints yields respect to the space variable of Eq. 2.17 . Simple inspec- tion of the above equations suggests that consistency is ob- m tained if m is of order . The second equation in Eq. A2 2 n n n n d n n h , A4 then becomes, to leading order, which coincides with the expression for the auxiliary field n 2 given in Eq. 3.12 . Finally, A4 is inserted in the first equa- m n n n n d n n h A3 tion A2 to yield, after lengthy but rewarding algebra, the extended nonlinear model 3.13 and 3.14 which is also and the constraints reduce to those of Eq. 3.11 to within correct to within terms of order 2. 1 I. E. Dzyaloshinskii, Sov. Phys. JETP 5, 1259 1957 . 13 B. A. Ivanov and A. K. Kolezhuk, Phys. Rev. Lett. 74, 1859 2 T. Moriya, Phys. Rev. 120, 91 1960 . 1995 . 3 T. Moriya, in Magnetism, edited by G. T. Rado and H. Suhl 14 A. F. Andreev and V. I. Marchenko, Sov. Phys. Usp. 23, 21 Academic, New York, 1963 , p. 85. 1980 . 4 V. G. Bar'yakhtar, M. V. Chetkin, B. A. Ivanov, and S. N. Ga- 15 S. Komineas and N. Papanicolaou unpublished . detskii, Dynamics of Topological Magnetic Solitons-Experiment 16 N. Papanicolaou and T. N. Tomaras, Nucl. Phys. B 360, 425 and Theory Springer-Verlag, Berlin, 1994 . 1991 ; S. Komineas and N. Papanicolaou, Physica D 99, 81 5 N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 1974 . 1996 . 6 A. P. Malozemoff and J. C. Slonczewski, Magnetic Domain 17 R. W. Shumate, J. Appl. Phys. 42, 5770 1971 . Walls in Bubble Materials Wiley, New York, 1981 . 18 C. H. Tsang, R. L. White, and R. M. White, J. Appl. Phys. 49, 7 N. Papanicolaou, Phys. Rev. B 51, 15 062 1995 . 6052 1978 . 8 V. S. L'vov and I. A. Prozorova, in Spin Waves and Magnetic 19 C. H. Tsang and R. L. White, J. Appl. Phys. 49, 6063 1978 . Excitations, edited by A. S. Borovic-Romanov and S. K. Sinha 20 E. M. Gyorgy and F. B. Hagedorn, J. Appl. Phys. 39, 88 1968 . North-Holland, Amsterdam, 1988 , p. 233. 21 A. K. Zvezdin, JETP Lett. 29, 513 1979 . 9 This notation was suggested to the author by V. G. Bar'yakhtar in 22 V. G. Bar'yakhtar, B. A. Ivanov, and A. L. Sukstanskii, Sov. relation to MnCO3. Phys. JETP 51, 757 1980 . 10 R. M. White, R. J. Nemanich, and C. Herring, Phys. Rev. B 25, 23 G. M. Wysin, A. R. Bishop, and J. Oitmaa, J. Phys. C 19, 221 1822 1982 . 1986 . 11 R. W. Wang, D. L. Mills, E. E. Fullerton, J. E. Mattson, and S. D. 24 E. V. Gomonai, B. A. Ivanov, V. A. L'vov, and G. K. Oksyuk, Bader, Phys. Rev. Lett. 72, 920 1994 . Sov. Phys. JETP 70, 174 1990 . 12 C. Micheletti, R. B. Griffiths, and J. M. Yeomans unpublished . 25 D. Treves, Phys. Rev. 125, 1843 1962 .