# Precessing Ball Solitons in Kinetics of  a Spin-Flop Phase Transition

Authors: V. V. Nietz
Journal: Physics Research International
 (2011)

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## Abstract

The fundamentals of precessing ball solitons (PBS), arising as a result of the energy fluctuations during spin-flop phase transition induced by a magnetic field in antiferromagnets with uniaxial anisotropy, are presented. The PBS conditions exist within a wide range of amplitudes and energies, including negative energies relative to an initial condition. For each value of the magnetic field, there exists a precession frequency for which a curve of PBS energy passes through a zero value (in bifurcation point), and hence, in the vicinity of this point the PBS originate with the highest probability. The characteristics of PBS, including the time dependences of configuration, energy, and precession frequency, are considered. As a result of dissipation, the PBS transform into the macroscopic domains of a new phase.


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## Body

## 1. Introduction

Magnetic solitons with spherical symmetry, which can arise in crystals with magnetic ordering during the phase transitions induced by a magnetic field, were considered in some papers [1–5]. The cases when it is possible to confine oneself to uniaxial symmetry of the crystal are of particular interest. For such a crystal, in addition to the amplitude (with corresponding configuration) and pulse, the solitons have the third parameter: the frequency of precession. In these cases, each value of an external field relates to a continuous spectrum of solitons by frequency and corresponding energy. Near boundaries of metastability, for each value of a field, the spectrum of precessing ball solitons (PBS) has the bifurcation point where the probability of PBS spontaneous origin is increasing abnormally. The frequency dependencies of energy and PBS configuration, as well as the process of PBS spontaneous origin at spin-flop transition in antiferromagnets have been discussed in [5]. However, the analysis of time evolution connected with dissipation of energy in PBS is not correct in this article.In the present article, it is shown that dissipation of energy is accompanied not only by the change of configuration of solitons but also by the change of the precession frequency. Taking into account this time dependency, we can carry out a more comprehensive analysis of the quantities of PBS and to consider the whole process of transformation of PBS into macroscopic domains of a new phase.In the first chapter, we represent a more correct analysis of equations and expressions for PBS than in [4, 5]. In the second one, the character of the time change of PBS that connected with dissipation is shown. In the third chapter, the influence of the movement on the form of PBS is discussed. Finally, in the fourth chapter, we estimate the influence of the demagnetizing fields.


## 2. Equations for Precessing Ball Solitons (PBS)

To analyze magnetic solitons in an antiferromagnet with uniaxial anisotropy, we used the following expression for the thermodynamic potential:(1)W=-A2l2+B2m2+C4(ml)2+K12(|m⊥|2+|l⊥|2)-K24(|m⊥|2+|l⊥|2)2-mzH+αxy2×[(∂m∂X)2+(∂m∂Y)2+(∂l∂X)2+(∂l∂Y)2]+αz2[(∂m∂Z)2+(∂l∂Z)2].
Here m and l are nondimensional ferromagnetism and antiferromagnetism vectors; l⊥=lx+ily,m⊥=mx+imy; the absolute value of the vector l at H=0 equals 1; K1>0,K2>0; magnetic field H is directed along the anisotropy axis Z.The equations of motion for thel and m vectors, taking into account the energy dissipation, are
(2)∂l∂t=2μBℏ(m×δWδl+l×δWδm)+Γ(m×∂l∂t+l×∂m∂t),∂m∂t=2μBℏ(m×δWδm+l×δWδl)+Γ(m×∂m∂t+l×∂l∂t).The solutions of (2) can be presented in the following form:
(3)l⊥(r,τ)=q(r,τ)ei(ω(τ)τ-k(τ)x),m⊥(r,τ)=p(r,τ)ei(ω(τ)τ-k(τ)x).
For the simplicity, it is supposed that the excitations advance along the x-axis. The time dependencies of the ω and k values are necessary for analyzing the time evolution of PBS.From (2), we have the following system of equations:
(4)k1-0.5(ω+Δ+h)q=-(1-k2k1(q2+p2)+k2)(mzq+lzp)+ak1(mzq-lzp)+1k1(mzq-lzp)+mzΔq-qΔmz+lzΔp-pΔlz-QA1,(5)k1-0.5(ω+Δ+h)p=-(1-k2k1(q2+p2)+k2)(lzq+mzp)+lzΔq-qΔlz+mzΔp-pΔmz-QA2,(6)∂p∂τ=k1[2k(lz∂q∂x+mz∂p∂x)-Q(ω+Δ)(lzq+mzp)2k(lz∂q∂x+mz∂p∂x)],(7)∂q∂τ=k1[2k(mz∂q∂x+lz∂p∂x)-Q(ω+Δ)(mzq+lzp)2k(mz∂q∂x+lz∂p∂x)],(8)∂lz∂τ=2k1[k(p∂q∂x+q∂p∂x)+Q(ω+Δ)qp],(9)∂mz∂τ=-k1[kddx(q2+p2)-Q(ω+Δ)(q2+p2)kddx(q2+p2)],
where Δ=((dω/dτ)τ-(dk/dτ)x) and
(10)A1=(q∂mz∂τ-mz∂q∂τ+p∂lz∂τ-lz∂p∂τ),(11)A2=(q∂lz∂τ-lz∂q∂τ+p∂mz∂τ-mz∂p∂τ).
In (3)–(11), h=HB-0.5K1-0.5, k1=K1/B, k2=K2/B, a=A/B,c=C/B and the dimensionless coordinates and time are used, x=K10.5αxy-0.5X, y=K10.5αxy-0.5Y,z=K10.5αz-0.5Z; τ=2μB(K1B)0.5ℏ-1t; Q=Γ/k1.First of all, take notice that inserting (6)–(9) into (4) and (5), we can obtain a system of two modified equations describing the soliton configuration but not containing the evident time dependency. However, the time dependency of the parameters p,q,lz,andmz characterizing the soliton form and described by (6)–(9), is remaining. It means that during the soliton evolution connected with energy dissipation not only the configuration of PBS but also the frequency of precession ω and k value are changing too. For each time moment, the definite configuration of PBS, frequency ω, and wave value k correspond. The correspondence between the changing values of ω, k, and configuration is explicitly defined by such modified equations (4)–(5).To analyze the approximate soliton solutions of (4)–(9) system, it is convenient to transform this system into one equation. First of all, we will receive expression for magnetization that depends on q and p. The relaxation time of ferromagnetic moment is less for some orders than the relaxation time for the l vector. Therefore, for mz component we use its quasiequilibrium value, which can be obtained from δW/δmz=0 (see [5])
(12)mz≅mz0=1B+Clz2[Hz-C2lz(l⊥m⊥*+l⊥*m⊥)]=δk1h-(1-δ)lzpqlz2+δ(1-lz2)≅-pqlz+δlz2(k1h+pqlz),
where δ=B/(C+B)=χII/χ⊥ is the ratio of two magnetic susceptibilities.Taking into consideration thatk1≪1, k2(q2+p2)≪1 in (5) and for long-wave oscillations k1Δq≪q,k1Δlz≪lz,k1Δmz≪mz,k1Δp≪p, from (4) and (13) we obtain the following dependence
(13)p≅-k1(ω+Δ+h)qlzq2+lz2[1-a(2q2-1)]+δ(1+a)k1qlz(h-(ω+Δ+h)q2)-k1k1k2(ω+Δ+h)qlz(2q2-1)-QlzA1.Using the (13) correlation, from (5) we obtain the following equation for PBS:(14)Δq-qlzΔlz=[1-(1+a)(ω+Δ+h)2+k2+δ(1+a)(ω+Δ+h)hlz2]×q-[(ω+Δ+h)2lz2k2k1-2a×(ω+Δ+h)2+2k1(ω+Δ+h)2k2+δ(1+a)(ω+Δ+h)2lz2]q3+QA2lz.Let’s confine ourselves to low temperature approximation, that is, suppose that(lm)=0, in such case a=δ=0, q2+p2+lz2+mz2=1,(15)A1=2k1k[3qp∂q∂x+(2q2+p2-1)∂p∂x-12pddx(q2+p2)]+Q(ω+Δ)p,(16)A2=2k1k[3pq∂p∂x+(2p2+q2-1)∂q∂x-12qddx(q2+p2)]+Q(ω+Δ)q,
and the equation for PBS is as follows:(17)Δq-qlzΔlz=(1-(ω+Δ+h)2+k2)q-(k2k1+2k1(ω+Δ+h)2k2)q3+QA2lz.This equation should be supplemented by the following expression obtained from (7), (12), (13) at a=δ=0 and describing the evolution of PBS:(18)∂q∂τ≅k1(ω+Δ+h)×[2k(3q2-1)∂q∂x-Q(ω+Δ)q(2q2-1)∂q∂x].Only in the case of immovable PBS, that is, atk≡0, (17) has the solutions with a spherical symmetry. For such case, the equation has the following form:(19)d2qdr2+2rdqdr+q1-q2(dqdr)2=q[(1-q2)(1-(ω+h)2-k2k1q2)+Q2ω(1+12q2)]
(taking into account that lz2≅1-q2). In this equation, the frequency ω, as well as q values, depends on time. Thus we make the replacement (ω+(dω/dτ)τ)→ω(τ), which is possible considering a rather slow change of the precession frequency in comparison with the change of precession phase. Note that in (19), the addition connected with the dissipation is negligibly small, Q2|ω|≪1 (in our examples Q=0.02,|ω|<0.03). Therefore, in further calculations of PBS configuration we will neglect this addition.Transforming (19) with respect to lz parameter, we can obtain the following equation that can be used to analyze the PBS originated during the reverse phase transition, at decrease of the field:(20)d2lzdr2+2rdlzdr+lz1-lz2(dlzdr)2=((ω+h)2-1+k2k1)lz-((ω+h)2-1+2k2k1)lz3+k2k1lz5.Using (12) and (13) at δ=0,a=0 and k1≪1, from (1) we obtain the following expression for the energy of PBS:(21)Es=8πM0αxyαzk1B×∫0∞{[1+(ω+h)22-(ω+h)h]q2-k24k1q4+12(1-q2)(dqdr)2+k22q2}r2dr
(M0 is the magnetization of each sublattice). The last term in this integral corresponds to the kinetic energy of PBS.


## 3. Spontaneous Origin of PBS and Their  Evolution into Domains of a New Phase

As in [5], we divide the processes relating to PBS into two stages. At the first stage, the PBS originate spontaneously because of the thermal fluctuations, or by any different way, in the τ=0 moment, and further evolution of PBS is carried out at the second stage. Equations (19) or (20) are used for description of the form of arose PBS. At this stage, we do not take into account the dependence of PBS parameters on the time and dissipation of energy.PBS configurations, which are particular solutions of (19), for different ω values at h=0.99, and dependences of PBS energy and amplitudes on the precession frequency for different values of the field have been shown in [5]. In Figure 1, we present the frequency dependencies for the energy, amplitude (qm≡q(r=0)), and effective radius for PBS at h=0.99 only. In this case, the utmost frequency of PBS ω=0.01 corresponds to the frequency of antiferromagnetic resonance, ωres1=(1-h). In Figure 2, we can see the same dependencies for the reverse spin-flop transition (at the decreasing magnetic field) at h=0.91. In this case, the frequency of antiferromagnetic resonance can be expressed as follows: ωres2=-h+1-k2/k1≅-0.01557. Here and in all subsequent examples the following parameters typical for Cr2O3 have been used: M0=0.33×10-9eVOe-1Å´-3, B=4.9×106Oe,  αxy=αz=3×106OeÅ´2,  K1=700Oe,K2=140Oe,  Q=0.02 [6].Figure 1
Frequency dependencies of the energy, amplitude, and effective radius ifh=0.99. Here and in all subsequent figures, the values of energy are denoted by circles, the amplitude by empty squares, the frequency and the inverse of the frequency by continuous line or full squares, the radius by crosses.Figure 2
The same frequency dependencies as in Figure1 for the reverse spin-flop transition, that is, at the decreasing magnetic field, if h=0.91.Probability of the spontaneous creation of PBS related to the fluctuations of energy at nonzero temperatures is proportional to probability of such fluctuations. For the metastable state of our system, we use the following expression for the probability of PBS creation with the energyEs near the bifurcation point (see [4, 5]):(22)Ps=Asexp⁡(-|Es|kBT).Here,As is the coefficient in general depending on ω,  Es and configuration of PBS. Generally, we have to use different configuration coefficients for positive and negative energies of PBS: As+ for Es>0 and As- for Es<0. Apparently that As+,As-≪1. In Figure 3, the temperature dependence of the probability of PBS near to the bifurcation point, where the energy of PBS is near zero, is shown.Figure 3
Energy dependence for probability of the spontaneous PBS creation near the point of bifurcation.Further evolution of PBS is described by (18). As it can be seen from this equation, subsequent change of PBS configuration depends on two factors: the spatial movement of it as a whole and dissipation of energy.In this part, we consider the change of PBS related to the dissipation only, that is, atk≡0. Let us assume that at the moment τ=0 the PBS of the q(r,0) form, corresponding to (19), have arisen. A subsequent change of PBS configuration proceeds in accordance with expression(23)∂q∂τ≅-Qk1ω(ω+h)q(2q2-1),
where ω=ω(τ).In conformity with (23), the character of PBS change is determined by a sign of the precession frequency and by its amplitude.In Figures 4 to 7, several examples are adduced ath<1 to illustrate the PBS change because of the dissipation for different field values. If the amplitude of PBS is big enough, we can consider it as consisting of two parts: the “bulk,” where value q>0.5, and the “corona,” where q<0.5.Ifωinit<0 and the amplitude of PBSqm,init>0.5, its bulk is increasing, but the corona is decreasing. As a result, the value of r corresponding to q=0.5 is increasing, the frequency ω is increasing in absolute value, the energy is decreasing, and PBS is growing and turning into the macroscopic domain of a high-field phase. Changes of PBS can be seen at ωinit<0 in Figure 4 and in Figure 5, at ωinit<-0.0011 in Figure 6, and at ωinit<-0.00419 in Figure 7. On the contrary, if ωinit>0 and qm,init<0.5, the PBS is decreasing in amplitude. Such change can be traced in Figure 4 in the range 0<ω<0.0058 and in Figure 5 in the range 0<ω<0.0008.Figure 4
The frequency dependencies of energy, amplitude, andω-1 value if h=0.99.Figure 5
The same frequency dependencies as in Figure4 if h=0.995.Figure 6
The same dependencies as in Figure4 if h=0.997.Figure 7
The same dependencies as in Figure4 if h=1.Ifωinit<0,qm,init<0.5, the q values in the bulk and corona of PBS are decreasing (see,  e.g., at 0>ω>-0.00119 in Figure 6, and at ω>-0.00419 in Figure 7). At the same time, the frequency ω is decreasing in absolute value, that is, ω→-0. In the last case, at h=1, the PBS disappear asymptotically, that is, qm→0,Es→0. Thus, only in cases ωinit<0,qm,init>0.5 PBS are transforming into macroscopic domains of the high-field phase.Equations (19) and (23) permit to obtain the time dependence of the frequency (and energy) due the dissipation, in accordance with the succession: Δω→Δqm→Δτ→ω(τ). For each change of the frequency Δω, from (19) we obtain the corresponding change of PBS amplitude Δqm; further from (23), we obtain the time interval Δτ corresponding to this change of amplitude. As a result, we obtain the ω(t) dependences for the cases of transformation of PBS into domains of a new phase. They are presented in Figures 8 to 11.Figure 8
The time dependency of the frequency ifh=0.99. Here the initial value of energy is Einit≅2.855eV (see in Figure 4). The dotted line shows the frequency value corresponding to zero energy.Figures8 and 9 correspond to the field h=0.99. Initial data for PBS in Figure 8 are ωinit=-0.0025,  Einit≅2.8553eV, and in Figure 9: ωinit=-0.0112,Einit≅0.0476eV. Note that at T=300K the average energy of the thermal fluctuations equals kBT=0.025eV. The frequencies corresponding to value Es≅0 are marked by dotted lines. Practically, the curve in Figure 9 is a part of the Figure 8 curve.Figure 9
Time dependency of the frequency ifh=0.99, but here Einit≅0.0476eV.In Figure10, it is shown the change of frequency for the field h=0.997, and in this case Einit=0.2919eV. At last, Figure 11 corresponds to h=1, Einit=0.3977eV. The four examples, in Figures 8 to 11, correspond to increase PBS and transformation of them into domains of high-field phase.Figure 10
The time dependency of the frequency ifh=0.997. Here Einit≅0.292eV (see in Figure 6).Figure 11
Time dependency of the frequency ifh=1 and the initial amplitude qm,init>0.5. Here the initial parameters of PBS are the following: Einit≅0.398eV, qm,init=0.747, ωinit=-0.005 (see in Figure 7).Sequence of change of the form and sizes for PBS ath=0.99, ωinit=-0.0025 has been shown in Figure 12 (note that the similar sequences presented in [5] are wrong, since the frequency change connected with dissipation has not been taken into consideration).Figure 12
A sequence of configurations of PBS ifh=0.99 and the initial amplitude qm,init>0.5 corresponding to Table 1.Asymptotical disappearance of PBS ath=1 is illustrated in Figure 13.Figure 13
Fading of PBS ifh=1 and the initial amplitude qm,init<0.5. Here the parameters of initial PBS are the following: ωinit=-0.0035,  qm,init≅0.666,  Einit=0.3996eV (maximum energy of PBS for h=1 equals to Esmax=0.405465eV, see in Figure 7).


## 4. The Change of PBS during Their Movement

Now, we consider the influence of movement on soliton form. Using (7), (9), and (10) for the Qω=0 case, we obtain the following expression:(24)∂q∂τ=k12k(mz∂q∂x+lz∂p∂x)≅2k1k(ω+h)(3q2-1)∂q∂x.Let us present theq(x,y,z,τ) function as q(x,y,z,τ)=qs(xs,y,z,τ), where xs=(x-v0τ). In such a case, we have (25)-v0∂qs∂xs+∂qs∂τ=-2k1k(ω+h)∂qs∂xs+6k1k(ω+h)q2∂qs∂xs.If we designate the value(26)v0=2k1(ω+h)k
as the velocity of movement of a soliton as a whole, then the expression ∂qs/∂τ=6k1k(ω+h)q2∂qs/∂xs or, in spherical coordinates:
(27)∂qs∂τ=3v0q2∂qs∂rssinθcos⁡φ,
(here rs is the radial coordinate in the system of moving soliton) describes the deformation of a soliton because of its movement along the x-axis. For considering solitons, the derivative ∂qs/∂r<0. Consequently, the PBS frontal side is decreasing, that is, it becomes steeper, but the back side is increasing, that is, it becomes more sloped in the same extent. Thereby, “the centre of gravity” of soliton is displaced in a direction opposite to a direction of the main movement. Thus, the form of moving soliton differs from spherical, but if Q=0 its energy does not change. However, the dissipation (Q≠0) results not only in change of frequency, but also in decrease of the velocity and, accordingly, of kinetic energy:(28)Ekinet=πM0Bk1(ω+h)2(αK1)1.5v02∫q2r2dr.In turn, as a result of the velocity reduction, PBS asymptotically becomes more spherical in the form (more precisely, owing to anisotropy, ellipsoid of rotation).


## 5. Influence of the Demagnetizing Fields and a More Correct Equation for PBS

To estimate approximately the influence of demagnetizing field in the case of PBS, let’s use the formula that concerns homogeneity magnetize sphere only:(29)Hdemag=-4π3Mz.Corresponding to (12) and (13), magnetization of PBS equals(30)Ms=2M0k1(ω+h)q2.If to us the expression (30) for the magnetization in (29), we receive the following: (31)hdemag≅-8πM03B(ω+h)q2.We can use this expression to estimate the value of demagnetizing field. With constants that were used in our examples, we receivehdemag=-0.9×10-3q2, that is, the relative change of magnetic field Δ<10-3 (in our cases h≅1). Such correction does not change qualitatively the characteristics of PBS and changes them quantitatively very little. Therefore, we can neglect demagnetizing fields.It is possible to write down a more correct equation for PBS, in comparison to (19), taking into account the additions proportional to k1:(32)(1+k1(ω+h)2q2)(d2qdr2+2rdqdr)+q1-q2×[1+2k1(ω+h)2-k1(ω+h)2q2](dqdr)2=q(1-q2)[(1-k2k1q2)(1+k1(ω+h)21(1-q2))×(1-k1(ω+h)2q2)-(ω+h)2(1-k2k1q2)].
However, using the value k1=1.43×10-4 corresponding to Cr2O3 [6], the solutions of (32) differ by less than 1% from the solutions of (19).


## 6. Conclusions

(1) It is shown that dissipation of energy for precessing ball solitons is accompanied by the change of the precession frequency and by the deceleration of spatial movement.(2) Kinetics of the PBS is defined by the sign of initial precession frequency and the amplitude of the originated PBS. PBS transforms into the domains of the new phase if the initial frequency of precession is negative (ωinit<0) and the initial amplitude qm,init>0.5. In the paper, the whole process of such transformation of PBS has been analyzed.(3) Ifωinit<0,qm,init<0.5, the q values in the PBS are decreasing and frequency ω is decreasing in absolute value, ω→-0.(4) At spatial movement of PBS, its form is deformed, but it does not change the size and the amplitude. The dissipation results not only in change of precession frequency, but also in reduction of the velocity of movement. As a result of the velocity reduction, the shape of PBS is approaching to spherical.Table 1No.t (μs)ωEs (eV)R0.5 (Å´)10−0.00252.789263527.6896−0.010.851376538.0899−0.0145−3.671488548.2097−0.02−20.08108058.2236−0.0233−48.8129068.2243−0.027−130.7161078.2263−0.03−321.2320008−0.032−791.82580



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