Dynamics of the magnetization of single domain particles having triaxial anisotropy subjected to a uniform dc magnetic field

Transcription

1 JOURNAL OF APPLIED PHYSICS 100, Dynamics of the magnetization of single domain particles having triaxial anisotropy subjected to a uniform dc magnetic field Bachir Ouari and Yury P. Kalmykov a Laboratoire de Mathématiques et Physique des Systèmes, Université de Perpignan, 52, Avenue Paul Alduy, Perpignan Cedex, France Received 29 June 2006; accepted 22 September 2006; published online 28 December 2006 Thermally induced relaxation of the magnetization of single domain ferromagnetic particles with triaxial orthorhombic anisotropy in the presence of a uniform external magnetic field H 0 is considered in the context of Brown s continuous diffusion model. Simple analytic equations, which allow one to describe qualitatively the field effects in the relaxation behavior of the system for wide ranges of the field strength and damping parameters are derived. It is shown that these formulas are in complete agreement with the exact matrix continued fraction solution of the infinite hierarchy of linear differential-recurrence equations for the statistical moments, which governs the magnetization dynamics of an individual particle this hierarchy is derived by averaging the underlying stochastic Landau-Lifshitz-Gilbert equation over its realizations. It is also demonstrated that in strong fields the longitudinal relaxation of the magnetization is essentially modified by the contribution of the high-frequency intrawell modes to the relaxation process. This effect discovered for uniaxial particles by Coffey et al. Phys. Rev. B 51, is the natural consequence of the depletion of population of the shallow potential well. However, in contrast to uniaxial anisotropy, for orthorhombic crystals there is an inherent geometric dependence of the complex magnetic susceptibility and the relaxation time on the damping parameter arising from the coupling of longitudinal and transverse relaxation modes American Institute of Physics. DOI: / I. INTRODUCTION The thermal fluctuations and relaxation of the magnetization of single domain ferromagnetic particles currently merit attention in view of their importance in the context of magnetic recording media and rock magnetism, as well as in connection with the observation of magnetization reversal in isolated ferromagnetic nanoparticles and nanowires. 1 The instability of the magnetization due to thermal agitation results in superparamagnetism 2 because each fine particle behaves like an enormous paramagnetic atom having a magnetic moment of 10 4 B 10 5 B. The theory of thermal fluctuations of the magnetization M of a single domain ferromagnetic particle due to Néel 3 was further developed by Brown 4,5 using the theory of the Brownian motion. In the context of the Brown continuous diffusion model, 4,5 the magnetization dynamics of magnetic nanoparticles is similar to the rotation of a Brownian particle in a liquid and is governed by a Fokker- Planck equation for the probability density function W of M. The Fokker-Planck equation is derived from Gilbert s equation 6 with a random field, which takes into account the thermal fluctuations of M in an individual particle. For the purpose of mathematical simplification, the magnetization relaxation has usually been considered for particles with uniaxial magnetic anisotropy for example, Refs. 3 and 7 15 and references cited therein. For such particles the free energy density V characterizing the magnetic anisotropy of the particle and Zeeman energy in a uniform external field H 0 is given by a Electronic mail: kalmykov@univ-perp.fr V = K sin 2 M H 0, where K is the anisotropy constant and is the polar angle. In order to estimate the characteristic time of the magnetization reversal over the internal anisotropy potential barrier, Brown 3,4 adapted to magnetic relaxation an ingenious method originally proposed by Kramers 16 see for a review Hänggi et al. 17 for thermally activated escape of Brownian particles from a potential well. Kramers idea is to calculate the prefactor A in an Arrhenius like equation for the escape rate over the potential barrier V reaction velocity in the case of chemical reactions, viz., = A a 2 ev/kt, 2 where the attempt frequency a is the angular frequency of a Brownian particle of mass m performing oscillatory motion at the bottom of a well for reviews of applications of Kramers theory as extended to magnetic problems, see Ref. 18. Referring to magnetic relaxation in uniaxial particles, Brown estimated the reversal time of the magnetization for the case when H 0 is applied along the easy axis of the magnetization as in our notation 4,5 N 1/2 3/2 1h 2 1 h2 1he 1 1he 1 h2 1, 3 where =K and h=/2 are the dimensionless barrier and field parameters, respectively; =M S H 0, =/kt, is the volume of the particle, T is the temperature, k is the Boltzmann constant, and M S is the saturation magnetization, /2006/10012/123912/10/$ , American Institute of Physics

2 B. Ouari and Y. P. Kalmykov J. Appl. Phys. 100, N = M S 1 2 /2 is the free diffusion time of the magnetization, is the gyromagnetic ratio, and is the dimensionless damping dissipation parameter. Equation 3 is valid in the low temperature high barrier, 1 limit only. Cregg et al. 10 have derived an approximate equation for valid for all values of. Aharoni 8 and recently Coffey et al. 12 and Klik and Yao 13 have reconsidered this problem. They calculated numerically and demonstrated a good agreement of their results with the Brown equation Eq. 3. Equation 3 for is valid for all values of the damping parameter due to the axial symmetry of the potential V. By applying an uniform magnetic field H 0 at an oblique angle with respect to the easy axis, one can break the symmetry of the potential V, which will also depend on the azimuthal angle. In axially symmetric anisotropy Eq. 3 with H 0 parallel to the easy axis the energyscape is a uniform equatorial ridge zone separating two polar minima and has no saddle points, on the other hand, the external field H 0 generates azimuthally nonuniform energy distributions with a saddle point. Such a nonaxially symmetric energyscape leads to a new effect, viz., strong intrinsic dependence of magnetic characteristics such as the complex magnetic susceptibility and relaxation times on the value of the damping parameter arising from coupling of the longitudinal and transverse relaxation modes. A detailed treatment of the oblique field problem for a uniaxial superparamagnetic particle has been given by Coffey and co-workers They evaluated the dependence of the reversal time of the magnetization on the field strength, temperature, and angle and showed that the analytical calculations based on Kramer s theory 16,18 are in good agreement with their numerical results. Coffey and co-workers also demonstrated that in the presence of the field H 0, the relaxation behavior of the system above a certain critical value h c of the field parameters h is dramatically different from that in the absence of the field. As explained by Garanin 25 this is due to the fact that at hh c, the external field suppresses the low-frequency relaxation mode associated with the reversal of M and high-frequency relaxation modes come into play so that their contribution to the overall relaxation behavior of M cannot be ignored. Moreover, by using Kramers theory as extended to magnetic problems by Coffey et al., 18 Kalmykov 24 derived a universal analytic formula for valid for all the values of and for this oblique field problem. In the present paper, we study the effects of an external field on the relaxation dynamics of the magnetization of single domain particles with triaxial i.e., orthorhombic anisotropy, 26 where the free energy density is given by V =K 1 sin 2 cos 2 K 2 sin 2 sin 2 K 3 cos 2 M H 0, where K 1, K 2, and K 3 are the anisotropy constants. In spite of the practical importance of orthorhombic anisotropy, which may yield an essential contribution to the free energy density of magnetic nanoparticles, 27,28 the orthorhombic case in the presence of an external field has not yet been solved due to the mathematical difficulties encountered. The only available 4 5 appropriate formula for the reversal time has been given by Braun 31 in the low-temperature limit and intermediate-tohigh damping IHD 1 for the similar problem of the magnetization dynamics in elongated biaxial particles where the easy- and hard-axis anisotropy terms are present subjected to a strong dc magnetic field. Some quantum and field effects for magnetic relaxation of biaxial particles have been treated see, e.g., Refs. 32 and 33. In the absence of the field, a solution of the problem has been given in Refs. 29 and 30. Here we present the results of a study of the magnetization dynamics of single domain particles with triaxial anisotropy subjected to a dc magnetic field H 0 for wide ranges of the field strengths and anisotropy energy parameters; our solutions being given in a form valid for all values of damping parameter. Numerical results obtained with the help of matrix continued fractions are compared with asymptotic estimates based on Kramers escape rate theory and its further generalizations. 18 An accurate method of calculation of the reversal time due to thermal agitation of the magnetization of single domain nanoparticles subjected to a uniform external field H 0 is necessary for modeling experiments, e.g., to reproduce the angular variation of the switching field of individual particles and to deduce experimental values of the damping parameter, 27,34 for the purposes of evaluating the linear and nonlinear dynamic susceptibilities, 14,22 etc. II. BASIC EQUATIONS In the context of Brown s model, 4,5 the dynamics of the magnetization M of a single domain particle may be described by Gilbert s equation 6 augmented by a random field ht with white noise properties accounting for the thermal fluctuations of the magnetization, viz., 5 Ṁt = Mt ht V/M Ṁt, where is a phenomenological damping parameter which is related to as =M S. Brown derived from the Gilbert- Langevin equation Eq. 6 the Fokker-Planck equation for the distribution function WM,t of the orientations of the magnetization vector M 4 6 t W = L FPW = 1 1 u V W W V 2 N W, 7 where L FP is the Fokker-Planck operator, and are the gradient and Laplacian operators on the surface of unit sphere, and u is the unit vector directed along M. A detailed discussion of the assumptions made in the derivation of the Fokker-Planck and Gilbert equations is given elsewhere e.g., Refs. 35 and 36. A concise theoretical description of the magnetization dynamics in a superparamagnetic particle can be given by linear response theory Ref. 23, Chap. 2. Here it is supposed that a fine particle in the presence of a strong uniform magnetic field H 0 is subjected in addition to a small probe field H 1 M H 1 1 parallel to H 0. Then the decay of the

3 B. Ouari and Y. P. Kalmykov J. Appl. Phys. 100, longitudinal component of the averaged magnetization M t of the particle, when the field H 1 has been switched off at time t=0, is 23 M t M 0 = H 1 C t, where C t is the normalized relaxation correlation function of the longitudinal component of the magnetization defined as C t = M 0M t 0 M M M = c k e k t, k k are the eigenvalues of the Fokker-Planck operator L FP in Eq. 7, k c k =1, =M M is the static susceptibility of the particle, and the brackets and 0 designate the nonequilibrium and equilibrium ensemble averages, respectively. The equilibrium ensemble averages are defined as A 0 =Z A,e V, sin dd Z is the partition function. Having determined C t, one can calculate the longitudinal magnetic susceptibility of the particle = i given by 7 =1i0 e it C tdt = k c k 1i/ k According to Eq. 10, the behavior of in the frequency domain is completely determined by the time behavior of C t. In order to characterize quantitatively the time behavior of C t, one may formally introduce two time constants. These are the integral relaxation or correlation time defined as the area under C t, viz., 23 C tdt = c k / k, =0 k and the effective relaxation time ef in given by ef =1/Ċ 0 = c k k 1 k which yields precise information on the initial decay of ef C t in the time domain. The relaxation times and contain contributions from all the eigenvalues k of the Fokker-Planck operator L FP. The smallest nonvanishing eigenvalue 1 is associated with the slowest overbarrier relaxation mode and so with the long-time behavior of C t; the other eigenvalues k characterize high-frequency intrawell modes. In general, in order to evaluate and ef, a knowledge of all the k and c k is required. In the low temperature limit, 1 k and c 1 1c k k1 so that 1/ 1 13 provided the wells of the potential remain equivalent as for 1 the triaxial potential in the absence of the field. Thus 1 closely approximates the relaxation time in the low temperature limit for zero or very weak external fields. The dependences of the effective relaxation time ef on the model parameters external field, anisotropy constants may differ considerably from that of and as ef is not governed by 1. FIG. 1. Triaxial anisotropy potential in the presence of a dc field H 0 threedimensional 3D plot of V, from Eq. 14 with = and h=0.25. III. ESTIMATION OF THE LONGEST RELAXATION TIME The smallest nonvanishing eigenvalue 1 may be estimated with the help of Kramer s escape rate theory as extended to the magnetic problem by Brown, 3,4 Smith and de Rozario, 26 Klik and Gunther, 37,38 Braun, 31 and Coffey et al. 18 and Déjardin et al. 39 In the IHD limit, 1, appropriate formulas for the escape rate IHD of magnetic systems were derived by Smith and de Rozario, 26 Brown, 3,4 and Braun, 31 for an arbitrary nonaxially symmetric potential V,. Moreover, in 1990, Klik and Gunther 37,38 realized that the very low damping VLD regime, 1, also applied to magnetic relaxation of single domain ferromagnetic particles and derived the corresponding VLD formula for the escape rate VLD. The conditions of applicability of these IHD and VLD solutions for superparamagnets are defined by 1 and 1, respectively. For turnover values of damping, , Coffey et al. 18 and Déjardin et al. 39 have derived an universal formula for bridging the VLD and IHD escape rates as a function of the dissipation parameter for singledomain ferromagnetic particles having a nonaxially symmetric free-energy density for a review of applications of Kramer s method to magnetic problems, see Refs. 13, 18, and 23. Here we apply the approach of Coffey et al. 18 and Déjardin et al. 39 to estimate the reversal time of the magnetization when the field H 0 is applied along the easy axis of the triaxial anisotropy potential. For this case the anisotropy potential is given by V = sin 2 sin 2 sin 2 2h cos const, 14 where =K 2 K 1 0 and =K 3 K 2 0 are the dimensionless anisotropy and barrier height parameters, respectively it is assumed that K 1 K 2 K 3. For 0h1, the potential equation Eq. 14 has two nonequivalent wells and two equivalent saddle points see Fig. 1. For h1, the potential equation Eq. 14 has only one well; the second well disappears at h=h s =1. In the absence of the field, h =0, the potential equation Eq. 14 has two equivalent wells and two equivalent saddle points. Thus according to the approach of Coffey et al. 18 and Déjardin et al. 39 as applied to the problem in question, the reversal time of the magnetization is given by the universal formula which is universal in the sense that it is valid for all values of damp-

4 B. Ouari and Y. P. Kalmykov J. Appl. Phys. 100, ing including IHD, VLD, and turnover regions, 1 1 IHD AS 1 S 2 AS 1 AS 2, 15 where IHD is the longest relaxation time in the IHD limit 1, function A is defined as AS = exp 1 0 1/4 1 dx, ln1 exp Sx 2 1/4x 2 and the dimensionless action variables S 1 and S 2 are defined by the contour integrals, 18,37 S 1,2 = Vp 1,2,=V 01p 2 p Vd 1 1p 2 Vdp. 16 Here V 0 =1h 2 / is the value of V at saddle points and p=cos. The integrals in Eq. 16 are taken along the critical trajectories or separatrixes, which are parametric curves p 1,2 passing between saddle points. The separatrixes are determined by the equation V,=V 0. Noting Eq. 14 for the problem under consideration this equation becomes p 2 1 cos 2 ±2hp h 2 cos 2 =0, where =/. Appropriate solutions of the above equation yield two separatrixes given by p 1,2 V=V0 = 1 1 cos 2 h cos 1h 2 2 cos 2. Thus we can evaluate the contour integral S 1 and S 2 in Eq. 16 analytically, /2 S 1,2 =21h 2 /2 1h2 1h 2 cos 2 cos d ±2h cos 1h 2 cos 2 1 cos 2 2 = 41h2 1 3/2 11/1h 2 h arctan h 1h 2 11/ ± h The development of S 1,2 into a Taylor series at small h yields S 1,2 =4 1 oh h2 ± h 2 11/ 1 1 h2 Estimation of IHD for a multiwell potential requires analysis of the possible escape routes using the discrete orientation model. 5 Such analysis indicates that the mean magnetization of orthorhombic crystals decays with time constant IHD given by IHD IHD 12 IHD 21 1, 18 where IHD 12 is the escape rate over two saddle points from deeper well 1 to well 2 and IHD 21 is the escape rate from well 2 to well 1. The escape rates IHD 12 and IHD 21 for the problem in question can readily be evaluated from general equation given, e.g., in Refs. 5, 18, and 30. However, here we can simply use the result of Braun s calculation 31 of the escape rate from the shallow well IHD 21, which reads in our notation IHD h = e1 h N 1h 1h 1h2 1h h 2 / The escape rate from the deeper well IHD 12 is then given by 12 h = 21 h. 20 We remark that the characteristic time N appearing in Eq. 19 depends on the damping constant as defined by Eq. 4. Equations allow one to evaluate from Eq. 15 for all values of damping constant. Noting that for h=0, Eq. 15 reduces to the known equation for the reversal time of particles having triaxial anisotropy in the absence of the external field, viz., 29,30 Ne A8 11/A / 2 1. IV. ANALYTICS EQUATIONS FOR C T AND 21 The dynamics of the magnetization in fine magnetic particles can be understood by using a bimodal approximation. 23,29,40 According to this approximation, the

5 B. Ouari and Y. P. Kalmykov J. Appl. Phys. 100, correlation function C t which in general comprises an infinite number of decaying exponentials, see Eq. 9 may be approximated in the IHD limit by two exponentials only, viz., C t 1 e 1 t 1 1 e t/ W, 22 where the amplitude 1 and the relaxation time W are expressed in terms of, ef, and 1 see Refs. 23 and 40 for details. Whence the dynamic susceptibility given as an infinite series of Lorenzians may be approximated by a sum of two Lorentzians only, 23,40 1 1i/ i W. 23 The parameters 1 and W in Eq. 23 are determined in such way as to guarantee the exact asymptotic behavior of in the extreme cases of very low and very high frequencies, 23,40 1i, 0 i/ ef 24,. In particular, this approximation was used in the interpretation of the relaxation of the magnetization in single domain particles with uniaxial anisotropy in the presence of a dc magnetic field and with triaxial and cubic anisotropies in the absence of the external field. 23,29,40 In practical calculations from Eq. 24, one can use an approach of Garanin, 25 which was developed for uniaxial particles and which we have applied for triaxial anisotropy in the absence of the field in Ref. 29. Here the relaxation time W can be evaluated as W 1 well = N /2h 1, 25 where well is the oscillation angular frequency in the deeper well. Noting Eqs. 23 and 24, one can estimate 1 as W / ef. 26 The effective relaxation time ef is given by an exact analytic equation, 23 ef cos cos 0 =2 N 1cos Finally, the inverse of the smallest eigenvalue 1 can readily be evaluated from Eq. 15. Equations 15 and allow one readily to calculate from Eq. 23. The domain of applicability of Eq. 23 is 1 in its derivation precessional effects have been neglected. However, if we are interested in the low frequency behavior of only or long time behavior of C t in the time domain, this equation may be used for all. V. MATRIX CONTINUED FRACTION SOLUTION We can also calculate numerically the dynamic susceptibility and the relaxation times and by using the matrix continued fraction approach developed in Refs. 23 and 41. Moreover, just as for uniaxial particles, a solution can be given for an arbitrary orientation of H 0 with respect to the easy axis of the particle. Using general formulas derived in Refs. 23 and 41 and generalizing our solution for the field free case, 29 one can obtain from Eq. 6 the differentialrecurrence equations for the statistical moments expectation values of the spherical harmonics Y l,m, c l,m t=y l,m ty l,m 0 governing the dynamics of the magnetization, viz., d N dt c n,mt = v n,m c n2,m t w n,m c n1,m t x n,m c n,m t y n,m c n1,m t z n,m c n2,m t v n,m c n2,m2 t w n,m c n1,m2 t x n,m c n,m2 t y n,m c n1,m2 t z n,m c n2,m2 t v n,m c n2,m2 t w n,m c n1,m2 t x n,m c n,m2 t y n,m c n1,m2 z n,m c n2,m2 t x n,m c n,m1 t x n,m c n,m1 t y n,m c n1,m1 t y n,m c n1,m1 t w n,m c n1,m1 t w n,m c n1,m1 t 28 the definition of all parameters are given in the Appendix. Equation 28 can be solved exactly for the one-sided Fourier transforms c l,m i= 0 c l,m te it dt by matrix continued fractions see Appendix. Having determined c l,m i, one may evaluate the spectrum C i= 0 e it C tdt of the relaxation function C t, the integral relaxation time =C 0, and the complex susceptibility from Eq. 10; moreover, by using matrix continued fractions, one can also evaluate the smallest nonvanishing eigenvalue 1 and consequently the reversal time see Appendix. The main advantage of the matrix continued fraction method is that it allows us to evaluate 1,, and for all values of the thermal and anisotropy energies including low barriers where asymptotic approaches such as Kramers one are not applicable. Moreover, it provides us with a complementary tool for the purpose of estimating the accuracy of asymptotic solutions at high barriers. VI. RESULTS AND DISCUSSION The inverse of the greatest relaxation time predicted by the universal equation Eq. 15 and the smallest nonvanishing eigenvalue 1 calculated numerically by the matrix continued fraction method from Eq. A5 of the Appendix are shown in Fig. 2 as functions of the barrier height parameter

6 B. Ouari and Y. P. Kalmykov J. Appl. Phys. 100, FIG. 2. The normalized smallest nonvanishing eigenvalue 1 N vs the barrier height parameter 1/T for various values of the field parameter h. Values of the dissipation and anisotropy parameters: =0.01 and =5. Solid lines 1 7: matrix continued fraction solution Eq. A5. Symbols: the inverse of the longest relaxation time N / calculated from asymptotic equation Eq. 15., Fig. 3 as functions of the damping parameter, and Fig. 4 as functions of the field parameter h for the case when the field H 0 is applied along the easy axis of the triaxial anisotropy potential. Apparently, at high barriers, 2, the universal equation Eq. 15 provides a good approximation of 1 1 for all values of see Fig. 3 and 0 h0.7 see Fig. 4. On the other hand, the asymptotic solution deviates considerably from the numerical results at 0.7h1 Fig. 4. This is due to the fact that the parabolic approximation of the potential V in the vicinity of the saddle points used in the derivation of Eq. 15 is unjustified for such values of h. Furthermore, we emphasize that Eq. 15 is not valid for =/ 0 corresponding to uniaxial anisotropy; here N 1 is given by Brown s formula Eq. 3. 3,4 If the departures from axially symmetry become small 0.1 the nonaxially symmetric asymptotic formulas for the escape rate such as Eq. 15 may be smoothly connected to the axially symmetric results by means of suitable bridging integrals. This procedure is described in Refs. 18 and 42 for the particular case of a uniform field transversally applied to the easy axis of the magnetization for a particle with uniaxial anisotropy. As seen in Fig. 3, for triaxial anisotropy, the ratio N / depends on. This effect does not exist for uniaxial particles see Eq. 3. As is apparent from Fig. 5 the integral relaxation time FIG. 4. The normalized smallest nonvanishing eigenvalue 1 N vs the field parameter h for various values of also here =10 and. Solid lines 1 5: matrix continued fraction solution Eq. A5. Symbols: N / from asymptotic equation Eq. 15. may have a behavior dramatically different from that of 1 1 above certain critical values of the parameters h. This effect of the magnetic field H 0 over the integral relaxation time is qualitatively similar to that for uniaxial particles 12,23,25 and can be understood as follows. According to Eq. 22, the relaxation time can be evaluated as 23, W. 29 At high barriers, the inverse of the smallest eigenvalue 1 1 has Arrhenius exponential dependence versus temperature, i.e., 1 1 e V 0V 2 1, while the amplitude 1 is estimated as 25 1 e V 1V 2 1 V 0, V 1, and V 2 are the values of V in the saddle point, deep well, and shallow well, respectively. Noting that V 1,2 = 2h and V 0 =1h 2, one has e V 0 V 1 2V 2 = e 16hh2. Here the argument of the exponential 16hh 2 may change its sign at a critical value of h=h c = Thus if hh c, the term in the right hand side of Eq. 29 increases exponentially as the temperature decreases and so determines completely the temperature dependence of the intrawell relaxation time W has a weak temperature dependence 25 and so its contribution to the temperature dependence of may here be ignored. On the other hand, if hh c, the term becomes exponentially small due to the depletion of population in the upper shallow well, thus FIG. 3. The normalized smallest nonvanishing eigenvalue 1 N vs the dissipation parameter for various values of the field parameter h also here =10 and =10. Solid lines 1 5: matrix continued fraction solution Eq. A5. Symbols: N / from asymptotic equation Eq. 15. FIG. 5. The integral relaxation time / N and 1 N 1,vs for various values of the field parameter h here =5 and. Solid lines 1 3: matrix continued fraction solution of / N Eq. A4. Symbols: the longest relaxation time / N from asymptotic equation Eq. 15.

7 B. Ouari and Y. P. Kalmykov J. Appl. Phys. 100, FIG. 6. The imaginary part of the complex magnetic susceptibility Im vs N for =10, =10, =0.005, and various values of the field parameter. Solid lines 1 5: matrix continued fraction solution Eqs. 10 and A3. Symbols: the two mode approximation Eq. 23. Dotted and dashed lines: the low- and high-frequency asymptotes Eq. 24. FIG. 8. The imaginary part of the complex magnetic susceptibility Im vs N for =10, =0.005, =5, and various values of. Solid lines 1 3: matrix continued fraction solution Eqs. 10 and 33. Symbols: the two mode approximation Eq. 23. Dotted and dashed lines: the lowand high-frequency asymptotes Eq. 24. the integral relaxation time has no longer Arrhenius-type behavior; its temperature dependence is mainly determined by the second term 1 1 W in the right hand side of Eq. 29. Thus at the critical value of h, the relaxation switches from being dominated by the slowest overbarrier mode to being dominated by the fast intrawell relaxation modes. The critical value h c 0.17 so obtained coincides with that found by Garanin 25 for uniaxial particles. The explanation of this coincidence is simple: the energy density of triaxial anisotropy for =/2 where the saddle points are located coincides with that of uniaxial anisotropy. In Figs. 6 8 we have compared the results of the calculation of the imaginary part of the normalized susceptibility M S 2 =1 from Eq. 23 and those obtained using matrix continued fractions. Here is plotted for typical values of the model parameters,,, and. The results indicate that a marked dependence of on exists and that three distinct dispersion bands appear in the spectrum of. The characteristic frequency 1 and half-width 1 of the low-frequency band are determined by the smallest nonvanishing eigenvalue 1. Thus the low frequency behavior of is characterized by the barrier crossing mode. With increasing field strength, the amplitude of this band 1 decreases see Fig. 6; in the strong fields h1, the band disappears. In addition, a second relaxation peak appears at FIG. 7. The imaginary part of the complex magnetic susceptibility Im vs N for =10, =0.005, =5, and various values of. Solid lines 1 4: matrix continued fraction solution Eqs. 10 and 33. Symbols: the two mode approximation Eq. 23. Dotted and dashed lines: the lowand high-frequency asymptotes Eq. 24. high frequencies. This relaxation band is due to the intrawell modes. The characteristic frequency 2 and half-width 2 of this band are equal to 1 W. At low fields, the amplitude of this band is far weaker then that of the first band. However, in strong magnetic fields 1, this band can dominate in the spectrum see Figs Just as in the absence of the bias field, 29 there is an inherent geometric dependence of on the value of the damping parameter arising from the coupling of the longitudinal and transverse relaxation modes. This coupling appears at 0 in the dynamic equation of motion Eq. 28, where the longitudinal component of the magnetization c 1,0 t is coupled with the moments c 1,±1 t the latter characterize the precession of the transverse components of M in the field of anisotropy superimposed with the external field H 0, and results in the appearance of the third ferromagnetic resonance FMR peak in the spectrum of with characteristic frequencies close to H the precession frequency of the magnetization 0 pr =H 0 H 0 =0 H pr, where 0 =0 pr is the precession frequency at H 0 =0. The FMR peak appears only at 1 and 0 and strongly manifests itself at high frequencies. As decreases, the FMR peak shifts to higher frequencies and its half-width decreases in our normalized units see Fig. 6. As one can see in Figs. 6 8, the agreement between the numerical calculation and Eq. 23 is very good in the low-frequency band region even for low damping because the low-frequency response is mainly determined by the overbarrier relaxation mode. However, Eq. 23 does not allow one to describe the FMR peak which appears at low damping. All the above results have been given for the field H 0 applied along the easy axis of the magnetization. However, our matrix continued fraction solution allows us to calculate all quantities of interest for an arbitrary orientation of H 0.As an example, the smallest nonvanishing eigenvalue 1 as a function of the oblique angle for the orientation of H 0 in the xz plane so the direction cosines of the field H 0 are 1 =sin, 2 =0, and 1 =cos is shown in Fig. 9. Such a behavior of 1 is very similar to that for uniaxial particles. 19 The oblique angle dependence of the greatest relaxation time 1/ 1 in the presence of a uniform external field H 0 of arbitrary orientation can be used for reproducing the angular

8 B. Ouari and Y. P. Kalmykov J. Appl. Phys. 100, FIG. 9. a The normalized smallest nonvanishing eigenvalue 1 N vs the oblique angle for =0.01, =10, =10, 1 =sin, 2 =0, 1 =cos, and various values of the field parameter h. Solid lines 1 5: matrix continued fraction solution Eq. A5. b The imaginary part of the complex magnetic susceptibility Im vs N for =7, =1, =5, 1 =0.6, 2 =0.6, , and various values of. Solid lines 1 3: matrix continued fraction solution Eqs. 10 and 33. Symbols: the two mode approximation Eq. 23. Dotted and dashed lines: the low- and high-frequency asymptotes Eq. 24. variation of the switching field in individual particles and for deducing experimental values of the damping parameter just as for uniaxial particles. 27,34 By using the method suggested in Refs. 27 and 34, here it should be possible to determine the damping coefficient by fitting theoretical and experimental switching curves astroids with only one fitting parameter. Furthermore, can be determined at different temperatures. This is of importance because of its implications in the search for other mechanisms of magnetization reversal of M e.g., macroscopic quantum tunneling 2, as a knowledge of and its T dependence allow the separation of the various relaxation mechanisms. For an arbitrary orientation of H 0, the smallest nonvanishing eigenvalue 1 can also be evaluated analytically just as for the uniaxial particles 24 using the analytical approach described in Sec. III. Furthermore, noting that here the effective relaxation time ef is given by an exact analytic equation, ef =2 N U 2 0 U 0 2 1U 2 0, 30 where U 1 = 1 cos sin 2 sin sin 3 cos, and evaluating the intrawell relaxation time W, one can calculate the complex magnetic susceptibility from Eq A typical example of such calculations from Ref. 43 is shown in Fig. 9b, where the imaginary part of the normalized complex susceptibility M S 2 =1 evaluated from Eq. 23 is compared with that obtained using matrix continued fractions. A detailed analytical treatment of the oblique field case will be published elsewhere. To conclude we have shown how external field effects on the dynamics of the magnetization of single domain ferromagnetic particles with triaxial orthorhombic anisotropy can be treated using Brown s model. 5 Simple analytic equations, which allow one to understand the qualitative behavior of the system and to accurately predict the reversal time of the magnetization and spectrum of the complex susceptibility in wide ranges of the barrier height, the external field strength, and dissipation parameters, are proposed for the case when the external magnetic field H 0 is applied along the easy axis of the triaxial anisotropy potential. The accuracy of these equations is demonstrated by comparing their predictions with the exact solutions obtained in terms of matrix continued fractions. We have noticed common features of the external field effects in the relaxation behavior of the magnetization in particles with triaxial and uniaxial anisotropies. Furthermore, we have demonstrated a main difference in the relaxational behavior of two systems. Namely, in contrast to uniaxial particles, where the damping enters only in the diffusion time N, for particles with triaxial anisotropy there is an explicit dependence of the susceptibility and the normalized reversal time / N on the damping parameter arising from the coupling of the longitudinal and transverse relaxation modes. The results we have presented pertain to the memoryless white noise limit Ohmic damping of the magnetization reversal. Furthermore, surface/defect properties which are heavily influenced by sample preparation technique and interactions between particles have been ignored. These effects present formidable mathematical difficulties in the formulation of a realistic theoretical model. 37 However, they can, in principle, be controlled in the interpretation of experimental data by varying the particle volumes and their concentration. Furthermore, the description of the relaxation processes in the context of Eq. 6 neglects quantum effects such as macroscopic quantum tunneling a mechanism of magnetization reversal suggested in Ref. 2. These effects are important at very low temperatures and necessitate an appropriate quantum mechanical treatment. A detailed discussion of these problems is given, e.g., in Refs ACKNOWLEDGMENTS The authors thank Professor W. T. Coffey, Dr. H. Kachkachi, Dr. A. Thiaville, and Dr. S. V. Titov for stimulating discussions and useful comments. One of the authors B.O. is grateful to Dr. H. Al-Awadhi, Dr. N. M. Jisrawi, and Dr. A. A. Aldouri for a discussion of the results and a welcome reception at the University of Sharjah. APPENDIX: MATRIX CONTINUED FRACTION SOLUTION OF EQUATION 28 Equation 28 can be transformed into the three-term vector recurrence equation,

9 B. Ouari and Y. P. Kalmykov J. Appl. Phys. 100, d N dt C nt = Q n C n1 t Q n C n t Q n C n1 t, A1 where C n t are the column vectors arranged in an appropriate way from c n,m t, viz., C 0 t = 0, C n t = c2n,2nt ] c 2n,2n t c 2n,2n1 t c 2n1,2n1 c 2n1,2n2 ] c 2n1,2n1 t, and the matrices Q n, Q n, and Q n are defined as X 2n W 2n Q n = Q Y 2n1 X n 2n1, = = V 2n 0 W 2n1 V 2n1. n 1, Z 2n Y 2n 0 Z 2n1, Q n In turn, the matrices Q n, Q n, and Q n consist of submatrices V l, W l, X l, Y l, and Z l which have the dimensions 2l1 2l3, 2l12l1, 2l12l1, 2l12l 3, and 2l12l5, respectively. The elements of these submatrices are given by V l n,m = n4,m v l,lm3 W l n,m = n3,m w l,lm2 n,m w l,lm1 X l n,m = n2,m x l,lm1 n1,m x l,lm2 Y l n,m = n1,m y l,lm n2,m y l,lm3 Z l n,m = n,m z l,lm1 where nn 1 x n,m = 2 x n,m =x n,m n2,m v l,lm1 n,m v l,lm1, n2,m w l,lm1 n1,m w l,lm n1,m w l,lm2, n1,m x l,lm n,m x l,lm1 n2,m x l,lm3, n,m y l,lm1 n1,m y l,lm2 n3,m y l,lm4, n2,m z l,lm3 n4,m z l,lm5, i m nn 1 3m2 2n 12n 3, * =i 1 i 2 1nmn m, 4 y n,m = 3 n 2 i 2 m n 12 m 2 2n 12n 3, y n,m =y n,m * = n 4 1 i 2 1nm2nm, 12n32n w n,m = w n,m 3n 1 2 =w n,m * = 1 n 1 i 2 4 i 2 m n2 m 2 4n 2 1, n mn m 1 4n 2, 1 and 1, 2, and 3 are the direction cosines of H 0. The coefficients with superscripts and v n,m,v n,m,w n,m,w n,m,x n,m,x n,m,y n,m,y n,m,z n,m,z n,m,x n,m do not depend on the field parameter and were given in Ref. 29 there these elements have been defined with the single superscripts and. The exact solution of Eq. 28 for the Laplace transform C 1s= 0 C 1 te st dt can be given in terms of matrix continued fractions, 23,41 C 1s = N 1 sc 1 0 n=2 n k=2 Q k1 k sc n 0, A2 where the infinite matrix continued fraction n s is defined by the recurrence equation, n s = N si Q n Q n n1 sq n1 1. The initial value vectors C n 0 can be evaluated in terms of n 0. Here we may apply with small modifications the algorithm developed for uniaxial anisotropy. 21,23 As shown in Ref. 21, Sec , the initial vectors C n 0 are given by C n 0 = 1 Kˆ n K n Kˆ n1 H S n1 S n S n1...s 1, 4 where 1 =M S H 1, S n = n 0Q n, the superscript H designed the Hermitian i.e., transposition and complex conjugate, and K n = F 2n D 2n H Kˆ n F 2n1, = 0 0 Kˆ 1 D 2n1 = 0, 0 D 1. D 2n The matrix K n and Kˆ n are constituted by the diagonal submatrix F l and three diagonal submatrix D l with the matrix elements defined as with F l n,m = 4 3 Re 3Y 1, i 2 Y 1,1 0 n,m, D l n,m = n2,m d l,lm1 d n,m = 3 n2 m 2 4n 2 1, n1,m d l,lm n,m d l,lm1,

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