University of New Orleans SholarWorks@UNO Physis Faulty Publiations Department of Physis --2007 Physis of omplex transverse suseptibility of magneti partiulate systems Dorin Cimpoesu University of New Orleans Alexandru Stanu Leonard Spinu Universityof New Orleans Follow this and additional works at: http://sholarworks.uno.edu/phys_fapubs Part of the Physis Commons Reommended Citation Phys. Rev. B 76, 054409 (2007) This Artile is brought to you for free and open aess by the Department of Physis at SholarWorks@UNO. It has been aepted for inlusion in Physis Faulty Publiations by an authorized administrator of SholarWorks@UNO. For more information, please ontat sholarworks@uno.edu.
PHYSICAL REVIEW B 76, 054409 2007 Physis of omplex transverse suseptibility of magneti partiulate systems Dorin Cimpoesu* Advaned Materials Researh Institute (AMRI), University of New Orleans, New Orleans, Louisiana 7048, USA Alexandru Stanu Faulty of Physis, Al. I. Cuza University, Iasi 700506, Romania Leonard Spinu AMRI, University of New Orleans, New Orleans, Louisiana 7048, USA and Department of Physis, University of New Orleans, New Orleans, Louisiana 7048, USA Reeived 8 Marh 2007; published 6 August 2007 Complex transverse suseptibility is a reent proposed method for the determination of anisotropy and volume distributions in partiulate magneti media. So far, only thermal flutuations and rate-dependent damped dynamis of the magneti moment have been identified as reasons for the existene of the imaginary transverse suseptibility. In this paper, we apply a more general approah to derive the omplex transverse suseptibility, and we show that the hysteresis phenomenon is the most general onept behind the existene of omplex transverse suseptibility. In this paper, the physial origins of the imaginary part of transverse suseptibility are analyzed: rate-independent hysteresis, visous-type rate-dependent hysteresis, and thermal relaxation effet origin. The rate-independent origin is an intrinsi ontribution to the imaginary transverse suseptibility and annot be negleted beause it is a zero-temperature effet. DOI: 0.03/PhysRevB.76.054409 PACS numbers: 75.30.Gw, 75.40.Mg, 75.50.Ss, 75.50.Tt I. INTRODUCTION One of the important topis of ondensed matter physis is the magnetism of nanostrutured materials as ordered arrays of magneti nanopartiles and nanowires, nanolithographially patterned strutures, and multilayers. To reveal the properties of suh systems, espeially magneti anisotropy, one needs appropriate experimental tehniques. Transverse suseptibility TS is one of the most sensitive methods in determining the magneti anisotropy, and it was used with great suess to haraterize lose-paked arrays of monodisperse Fe nanopartiles, Fe00 square nanostrutures, 2 exhange-biased IrMn/FeCo multilayers, 3 epitaxial CrO 2 and CrO 2 /Cr 2 O 3 bilayer thin films, 4,5 and perpendiular reording media. Also, there is a great interest in desribing theoretially the transverse suseptibility in magneti nanosized systems. 6 However, after almost 00 years sine Gans 7 published his paper where the onept of transverse suseptibility was first disussed and in spite of many efforts dediated to its study, the phenomenon of transverse suseptibility in magneti systems is still not fully understood. The starting point in the way for experimental use of TS was the theoretial paper of Aharoni et al. 8 where the differential reversible suseptibility tensor of a magneti system subjet to oherent rotation magnetization proesses was first alulated. The diagonalized suseptibility tensor inludes the longitudinal and the two transverse omponents of the suseptibility with respet to the applied magneti field. Thirty more years passed until Pareti and Turilli 9 experimentally verified the theory of Aharoni et al. 8 whih predited that the plot of TS versus applied field presents harateristi peaks loated at the anisotropy and swithing fields. Subsequently, as TS was more frequently used as a method for magneti anisotropy investigations, the interest for TS grew exponentially and many studies ontributed to the improvement of both theory and experimental method. Thus, some of the restritions in the first theoretial approah of Aharoni et al. 8 were overome 0 5 and the inreased sensitivity of new experimental TS methods allowed the investigations of magneti systems with very low onentration of magneti moments. 3,4,6,7 A very reent development of TS onstitutes the omplex TS. It was first mentioned by Papusoi 8 in the ontext of magneti partiulate systems and it spurred a lot of attention as it was presented as a new way to determine the partile volume and anisotropy field distributions of a partiulate medium. 9 23 The omplex TS as alulated in Ref. 8 is a onsequene of thermal relaxation of system s magneti partiles, whose relaxation rate an be ontrolled by the externally applied d field and not by temperature as in the ase of a suseptibility. 24 In this work, we apply a more general approah to derive the omplex TS. We show that the imaginary part of TS an have three main origins whih inlude those of Refs. 4 and 8. The starting point of our approah is the observation that the existene of an out-of-phase omponent of TS is essentially assoiated with a lag of the output magnetization with respet to the input applied magneti field. Etymologially, this lag between the effet and its ause orresponds to the word hysteresis, and as it is shown in the following, the hysteresis phenomenon is the more general onept behind the existene of omplex TS. In studying the hysteresis in magnetism, Bertotti 25 showed that the hysteresis an be understood using different ideas as lag, dissipation, memory and branhing, or metastability. As defined in Ref. 25, the hysteresis is the whole set of intimately onneted phenomena arising from the simultaneous existene of metastable states, dissipation, mehanisms with harateristi time sales, and thermal relaxation. Analyzing the orrespondene between metastability and hysteresis, it an be easily 098-02/2007/765/0544095 054409-2007 The Amerian Physial Soiety
CIMPOESU, STANCU, AND SPINU observed that the metastability summarizes all the other interpretations of hysteresis. Various types of hysteresis an appear in a system with a ompliated free-energy landsape featuring many loal minima orresponding to metastable states. The main situations that an our are rateindependent hysteresis, visous-type rate-dependent hysteresis, and thermal relaxation. In the ase of TS, so far, only the last two approximations were used to explain the ourrene of an imaginary part, i.e., visous-type rate-dependent hysteresis in Ref. 4 and thermal relaxation in Ref. 8. These two situations will be revised in this paper in Ses. IV and V, where ertain larifiations and orretions will be made. Before this, in Se. III, the rate-independent hysteresis origin of the omplex TS will be presented and analyzed. This is an intrinsi ontribution to the omplex TS being a zerotemperature approximation. II. MAGNETIC SUSCEPTIBILITY TENSOR The magneti suseptibility is a quantity whih desribes the apability of a magneti material to magnetize in response to a magneti field. The magneti suseptibility is defined by the ratio between the indued magnetization of a magneti sample and the induing magneti field, desribing the material s response to an applied magneti field. Taking into aount that both magnetization and magneti field are vetors, the magneti suseptibility of a magneti sample is not a salar. Response is dependent on the state of the sample and an our in diretions other than that of the applied field. To aommodate this, a general definition using a seond rank tensor derived from derivatives of omponents of magnetization M with respet to omponents of the applied field H: ij = M i, H j alled the differential suseptibility, desribes ferromagneti materials, where i and j refer to the diretions e.g., x, y, and z in Cartesian oordinates of the magnetization and applied field, respetively. The tensor thus desribes the response of the magnetization in the ith diretion from an inremental hange in the jth diretion of the applied field. Generally, the suseptibility tensor should omprise nine omponents, but, in fat, a maximum of three an be independent, beause by appropriate hoie of the orientation of the body-fixed oordinate system, the suseptibility tensor an be redued to a diagonal form. The general tensorial harater of the suseptibility is, in fat, related to the anisotropy of magneti properties. The diagonalization of the suseptibility tensor takes plae when the privileged diretions of magnetization beome parallel with the oordinate system s axis. If the hange of the applied field provides only reversible hange of magnetization i.e., proesses whih involve no loss of energy, we have a reversible suseptibility. In a differential suseptibility experiment, it is required to apply two magneti fields: a d bias field H d, whih an be varied in a quite large range, and a small perturbing a field H a, and the magnetization s variation is measured. The suseptibility is usually measured as the amplitude ratio of the PHYSICAL REVIEW B 76, 054409 2007 FIG.. Color online Transverse suseptibility experiment for a uniaxial single-domain ferromagneti partile. first harmoni of the indued magnetization along a given diretion and that of the a field. If the Oz axis is in the diretion of the biasing d field, then the diagonal elements of the tensor are the parallel suseptibility zz measured in the field diretion and the two transverse suseptibilities xx and yy measured perpendiular to the field diretion. III. TS IMAGINARY PART: RATE-INDEPENDENT HYSTERESIS EFFECT Rate-independent hysteresis is a zero-temperature approximation where the system indefinitely remains in the loal free-energy minimum whih is initially oupied by the system. In this approximation, let us onsider that the d field H d ats along the Oz axis of a retangular oordinates system and the a field H a =H a,max sin t ats along the Ox axis see Fig.. Let us onsider a spherial partile whose magnetization M reverses by oherent rotation, M=M s by assumption and M s is the saturation magnetization, having the volume V and the uniaxial anisotropy of onstant K0, with the easy axis in the xoz plane making the angle k with the Oz axis. For low values of the a field frequeny, it may be assumed that the proesses leading to a ertain value of M have harateristi relaxation times muh shorter than the time sale over whih M varies signifiantly and the system approahes equilibrium; that is, the magnetization M lies in a minimum of the free energy at any moment, as in the theory of Aharoni et al. 8 Beause of symmetry reasons, M will lie in the xoz plane. If the orientation of the magnetization is defined by spherial oordinate, then the partile s free energy is E = 2KVh d os + h a sin + 2 os k 2, where h d =H d /H k, h a =H a /H k, and H k =2K/ 0 M s being the partile s anisotropy field. In the following, we will refer only to the behavior of the TS near the positive saturation, that is, /2,/2. The dependene of the partile s free energy on the magnetization s orientation is presented in Fig. 2 for two orientations of the easy axis. If the partile s easy axis is perpendiular to the diretion of the d field k =90, then in the absene of the a field, the free energy shows one minimum for h d, loated at =0, and two minima for h d, loated at 0 054409-2
PHYSICS OF COMPLEX TRANSVERSE SUSCEPTIBILITY PHYSICAL REVIEW B 76, 054409 2007 If de d =0, FIG. 2. Color online The partile s free energy Eq. for k =90 left and k =89.9 right. =os h d, referred hereafter as minimum, and 2 0 = os h d, referred hereafter as minimum 2 see Fig. 2. In the lassial theory in whih the amplitude of the a field tends to zero, the magneti moment osillates reversibly around one of the two minima. If h d =, the first nonvanishing derivative at the minimum point is of the order 4, else is of the order 2. So, for h d =, the minimum is the broadest, and any arbitrary a field, however small, will ause an osillation of the magneti moment and onsequently the theory of Aharoni et al. 8 predits a singular point of the TS at h d =. However, in any experiment, the amplitude of the a field is not zero in the mathematial sense and no singularity ours at h d =, and what is important to observe is that the peak of the TS ours at a field h d beause the magneti moment osillates around the Oz axis between the minimum orresponding to h a =h a,max and, respetively, h a = h a,max and not only around one of the two minima orresponding to h a =0 see Fig. 2. For a partile with k =89.9, the energy landsape is not symmetri with respet to the Oz axis, as in the previous ase. It an be seen in the Fig. 2 that for h a,max =0 3, the magneti moment osillates only on the side 0. A maximum of the TS ours for a field h d, but the distane between the two minima is onsiderably smaller than in the ase k =90 so that the amplitude of the TS signal is muh smaller. Consequently, for an ensemble of partiles with a distribution of the orientation, the most important ontribution to the TS omes from the partiles with the easy axis oriented exatly perpendiular to the d field, for whih the energy plot is symmetri. In this sense, the affirmation that the field dependene of TS presents a harateristi peak loated at the anisotropy field must be understood, and not that for any partile s orientations the TS presents a peak at h d =. Inreasing the amplitude of the a field, the magneti moment will have suffiient energy to osillate on both sides of the Oz axis and the maximum of the TS our at a field h d. d 2 E 2 0, d 2 then the magneti moment lies in a minimum of the free energy. The first ondition an be written as a polynomial equation of fourth order in tan/2 whih has expliit solutions, but it is rather diffiult to hoose the solutions whih satisfy the seond ondition. To simplify the disussion, we shall use that m a =sin is the normalized indued signal along the a field diretion, and then Eq. 2 an be written as h a = fm a, k, f 0, 3 m a where m a fm a, k = h d + m 2 a os 2 k 2m 2 a ma 2 sin 2 2 k. ma 4 The behavior of fm a, k as a funtion of m a for fixed k is shown in Fig. 3. Its profile an be deomposed into two stable branhes where f /m a 0, one for m a m a,2 and the other for m a m a,, and a entral unstable branh where f /m a 0. If h a, h a th a,2 for all t, then the hange of the magneti moment is reversible on one of the two stable branhes. Otherwise, the right branh is traversed when h a dereases from h a,max at point A =h a,,m a, where the right branh ends and the system jumps to point A =h a,,m a, on the left branh. A similar desription applies when h a inreases from h a,max. Plotting m a as a funtion of h a, we obtain the hysteresis loop shown in Fig. 4.We note that lim ha,max 0df/dm a gives the transverse suseptibility from Ref. 8. For k =90, we have h a, =h a,2 = h 2/3 d 3/2, and, onsequently, if h d h a,max 3/2 = h *, then the a field notation 2/3 provides suffiient energy to the moment to osillate around the Oz axis ases and 2 in Fig. 5, else, if h d h *, then the magneti moment osillates only around one of the two minima, as in the theory of Aharoni et al. 8 ase 3 in Fig. 5. For k =89.9, no sudden jumps our for h a,max =0 3 and the motion of the magneti moment is reversible. First we have alulated TS as m a,max m a,min = 0, 5 2h a,max where 0 =M s V/H k, so that the normalized suseptibility / 0 does not depend on the volume or anisotropy. The periodi indued signal m a has a phase shift with respet to h a, but its maximum takes plae at the same moment of time 054409-3
CIMPOESU, STANCU, AND SPINU PHYSICAL REVIEW B 76, 054409 2007 (a) FIG. 5. Color online Left TS as a funtion of the d field in the absene of the thermal flutuations for k =90, using extreme values Eq. 5 of the indued signal thin line and the first harmoni from the Fourier series thik line; the a field amplitude h a,max =0.00. Right Time evolution of the a field h a,ofthe indued signal m a thin line and its first harmoni thik line at the points indiated in the first graph time is normalized to the period of the a field. (b) denominator inreases muh more than the numerator. The experimental values reported in Refs. 9 22 are H a,max =5 80 Oe and H k =2 koe, namely, h a,max =4.2 0 4 6.60 3. So, we an easily see from Fig. 6 that ignoring the a field amplitude an lead to systemati errors FIG. 3. Color online The profile of the funtion f Eq. 4 and genesis of hysteresis loop under alternating field exitation: a k =90 and b k =89.9. When h a dereases from h a,max, the system jumps from point A =h a,,m a, to point A =h a,,m a,. When h a inreases from h a,max, the system jumps from point A 2 =h a,2,m a,2 to point A=h 2 a,2.,m a,2 with the maximum of the a field Fig. 5. Thus, relation 5 gives a real suseptibility. As the a field amplitude inreases, the position of the anisotropy peaks moves toward smaller field values, and the indued signal inreases Fig. 6a but the TS dereases Fig. 6b beause in Eq. 5 the (a) (b) FIG. 4. Color online Hysteresis loop under alternating field for k =90 and h d =0.99. The points A and A are the same as in Fig. 3. FIG. 6. a Indued signal and b TS as a funtion of the d field and the a field amplitude in the absene of the thermal flutuations, alulated using extreme values of the indued signal Eq. 5 for k =90. 054409-4
PHYSICS OF COMPLEX TRANSVERSE SUSCEPTIBILITY in the evaluation of the anisotropy using TS experiments. In Ref. 26, a perturbation approah was used to alulate the modifiation of magnetization due to a small a field. The problem with the perturbation tehnique is that near the TS peaks, the variation from Eq. A3 see the Appendix is not a small deviation from equilibrium position 0, and, for example, the hange in magnetization along the Oz axis annot be approximated by sin 0. Due to the retangular-like form of the signal, its maximum is very diffiult to be determined experimentally. So, we have analyzed the fitting with a sinusoidal funtion and, respetively, the alulus of the first harmoni from the Fourier series. Both methods give pratially the same results for k =90 and small differenes otherwise. If the a field is H a t = H a,max Ime it = H a,max sin t, where Im denotes the imaginary part of the omplex variable, then we define the magnetization Mt as a funtion of t by Mt = H a,max Im n e int V n= = H a,max V n sinnt + n osnt, n= where n = n +i n n=,2,3,..., and n and n an be alulated by n = M sv H k h a,max 2 n = M sv H k h a,max 2 T0 T0 Similar results are obtained if one uses T T m a tsinntdt, m a tosntdt. H a t = H a,max Ree it = H a,max os t, where Re denotes the real part. In Fig. 5, are presented the time evolution of the indued signal m a and its first harmoni for k =90 and h a,max =0.00. The first harmoni and the applied a field are out of phase. Thus, we obtain an imaginary part of TS similar to the one desribed in Ref. 8, but independent of thermal relaxation. The area of the loop desribed in the Fig. 4 measures the amount of energy dissipated during eah exitation yle the energy loss per a field yle. Losses are solely hystereti beause we have assumed that the magnetization M lies in a minimum of the free energy at any moment. Atually, the dissipation takes plae only in the Barkhausen jumps. The energy loss an be evaluated in terms of the imaginary part of the fundamental harmoni of suseptibility, as 2 E = 0 H a,max. By onstrution, the loop area is equal to E =2Em a,,h a, Em a,,h a,, where E is the partile s energy Eq.. Now, we have 2 h a,max = Em a,,h a, Em a,,h a,. 0 K 2 That is, the produt h a,max depends on the d field only and does not depend on the a field amplitude. When the dynamis is taken into aount, due to the damping, a phase shift ours between h a and m a, so an imaginary omponent of TS also appears. 4 Replaing sign with sign in Eqs. 2, we obtain h d os k + h a sin k = os 3 k, h d sin k h a os k = sin 3 k, from whih one further obtains h d os k + h a sin k 2/3 + h d sin k h a os k 2/3 =. 7 This equation gives the ritial fields h a, and h a,2 where the Barkhausen jumps our when h a dereases and inreases, respetively. When h a dereases, before the jump, m a, = sin = sin k h d os k + h a sin k /3 and after the jump, where PHYSICAL REVIEW B 76, 054409 2007 os k h d sin k h a os k /3, tan m a, = sin k = tan k + tan 2 k + tan 2 k, tan k = h d sin k h a, os k /3 h d os k + h a, sin k /3, tan k = tan k + tan 2 k + tan 2 k + tan 4 k. By expressing the energy Eq. in terms of tan k, we find 6 054409-5
CIMPOESU, STANCU, AND SPINU 2 + tan 2 k + tan 2 k = h a,max 2 + 0 if h d os k + h a,max sin k 2/3 + h d sin k h a,max os k 2/3 0 elsewhere. PHYSICAL REVIEW B 76, 054409 2007 + tan 2 k + tan 2 k + + tan2 k 8 Relations 6 and 7 an be better understood using the astroid formalism. 27,28 Assuming that the magnetization lies in a minimum of the free energy at any moment, its rotation is governed by the Stoner-Wohlfarth astroid see Fig. 7. The standard proedure in the ritial urve approah is to find the equilibrium orientation of the magnetization as the diretion of the tangent to the ritial urve passing through the tip of the total applied field vetor in our ase, the sum between the h d, with the origin in the astroid s enter, and h a. In Fig. 7, one sees in the ritial urve approah how the amplitude of the a field an indue the hystereti proesses desribed analytially for the partiular ase k =90. For h d, when h a dereases from h a,max point a in Fig. 7 to h a,max point b, the tangeny point shifts ontinuously from point a to point b; for h a =0, the tangeny point is in the astroid s usp, that is, the magneti moment swithes its diretion into the negative sense along with the a field. For h d h *, the vetor h a uts the astroid: when the extremity of the h a vetor shifts from point a to point d, the tangeny point moves ontinuously from a to d; dereasing in ontinuane, the a field from point d to point b, a jump will our and the tangeny point jumps into point d, from whih it ontinuously shifts to b. Inreasing the h a from point b to point a, a jump will our at point where the tangeny point will jump into point. For any arbitrary a field, however small, there is an interval of the d field for whih the vetor h a uts the astroid and, as a result, a jump of the magneti moment will our. The angular interval in whih the magneti moment osillates ab in Fig. 7 is maximum when the extremities of the vetor h d +h a are on 2/3 the astroid, that is, h d = h a,max 3/2. For lower values of the d field, h d h *, the vetor h a does not ut the astroid and the tangeny point shifts ontinuously between point a and point b, no sudden jump of the magneti moment ours, and the amplitude of the indued signal suddenly dereases giving rise to a peak in the TS urve. Inreasing the a field amplitude, the position of the TS peak moves toward smaller d field values. IV. TS IMAGINARY PART: RATE-DEPENDENT HYSTERESIS EFFECT FIG. 7. Color online The astroid urve and the applied fields with k =90, h a,max =0 3, and different values of the d field: h d =, i.e., the total applied field is in the exterior of the astroid and no sudden jump of the magneti moment ours; 2 h d 2/3 h a,max 3/2, i.e., the vetor h a ut the astroid; 3 h d = 2/3 h a,max 3/2, i.e., the extremities of the applied field are on the astroid, and the angular interval ab in whih the magneti moment 2/3 osillates is maximum; and 4 h d h a,max 3/2, i.e., the vetor h a does not ut the astroid and no sudden jump of the magneti moment ours. In the previous setion, the TS was derived without any referene to the rate at whih the external field is varied. The role of the external field was only to bring the system to the point in whih it beomes unstable and a spontaneous Barkhausen jump ours. The spontaneous jump time sale is muh shorter than the external field variation sale and, onsequently, the evolution of the system was independent of the external field rate. However, rate-independent hysteresis is an approximation and real magnetization proesses are rate dependent due to two main reasons. First of all, Barkhausen jumps have a ertain harateristi time sale determined by the proesses on whih the systems dissipates energy 25 as eddy urrents or magnetization damping. Thus, for high field rate values, the system eases to be desribed by spontaneous Barkhausen events and approahes a fored dynami regime driven by the external magneti field. In the present paper, we do not take into aount the effet of eddy urrents, and in Se. IV A, we will analyze the ourrene of imaginary TS due the magnetization damping as a visous-type rate-dependent hysteresis effet. Seond, as the temperature inreases, the assumption that the system stays indefinitely in a loal energy minima is not anymore valid and thermal triggered Barkhausen jumps an our. These thermal effets have a ertain time sale that 054409-6
PHYSICS OF COMPLEX TRANSVERSE SUSCEPTIBILITY when orrelated to the external field rate an determine an imaginary TS. This made the objet of Ref. 8 and will be revised in Se. IV B. A. Visous-type rate-dependent hysteresis In the model presented previously, the TS is evaluated by employing a free-energy minimization method, namely, the model is frequeny independent beause it is assumed that in eah moment the magnetization reahes an equilibrium state independent of the a field rate. In real magneti systems, the dynamis is assoiated with energy dissipation, determining a lag between the magnetization s response and the external exitation, whih is strongly dependent on the exitation frequeny. Consequently, knowing the frequeny dependene of the TS beomes ritial. The frequeny used in the latest TS experimental setups has inreased from kilohertz to megahertz 4 or even to gigahertz range, 29 so, it is important to have a model in whih the frequeny dependene of TS is taken into aount. Moreover, owing to the damping at high frequeny, a phase shift ours between the applied a field and the alternating magnetization; so, an imaginary omponent of TS appears. A study of TS by fousing on magnetization dynamis is presented in Refs. 3 and 4. This model allows one to find a general formula for TS for any magneti system if an expression for the magneti free-energy density is known. The starting point of the model, as in the ase of ferromagneti resonane, 30 is the magnetization s equation of motion Landau-Lifshitz-Gilbert LLG: 3 dm dt = + 2M H eff + 2 M M H eff, M s where =2.20 5 rad/s/a/m is the gyromagneti ratio, is the Gilbert damping onstant, and H eff is the deterministi effetive field and it inorporates the applied fields and the effets of different ontributions in the free energy; the first term on the right-hand side of the LLG equation desribes the gyromagneti preession, and the seond term desribes the damping. The dynamis of the magnetization is governed by the relation that exists between the frequeny of the external exitation and the frequeny of resonane of the magneti moment, whih depends on the phenomenologial damping term in the LLG equation. In order to preserve the magnetization s magnitude, the LLG equation an be expressed better in spherial oordinates,, and the singularity of the spherial oordinates along the polar axis an be avoided by an adequate hoie of the Oz axis. The dynami suseptibility tensor in spherial oordinates is 4 9 = 2 r 2 + i2 + 2 0 where PHYSICAL REVIEW B 76, 054409 2007 F sin 2 F sin F sin F + im s, 0 r = + 2 2 F F F 0 M s sin and = + F 0 M sf sin 2, with F, F, and F being the seond derivatives of the free-energy density F at the equilibrium position, in the absene of the a field. The suseptibility tensor in the laboratory system is given by xyz = T t T, where os os os sin sin T =. sin os 0 From Eq. 0, we an see that the suseptibility tensor is related to the seond derivative of the free-energy density F and is a measure of the urvature of F. We an also observe that the omponents of the suseptibility tensor are omplex, the apparition of the imaginary part of TS being a diret onsequene of the damping, in the absene of the damping the imaginary part being zero. So, an imaginary part of TS an our even in the absene of thermal relaxation due only to the damping. Atually, thermal flutuations and dissipation are related manifestations of one and the same interation of the magneti moment with its environment. We note that, however, this model does not take into aount the amplitude of the a field. B. Thermal relaxation effet Thermal relaxation is taken into aount using a two-level model in Refs. 0 and where only a real suseptibility is predited, and it is shown that the temperature an hange the position of the oerivity peak while the anisotropy peaks maintain their position, though their shape is altered. Monte Carlo simulations of the TS for a hexagonal array of dipolar interating nanopartiles with random anisotropy are presented in Ref. 6, where only a real suseptibility is analyzed. However, the effet of the a field amplitude is not disussed, even if in simulations h a,max =0 2 is taken and the obtained shift of the anisotropy peak toward lower field values is assigned only to thermal flutuations. On the other hand, Papusoi 8 used also the two-level model, but an imaginary 054409-7
CIMPOESU, STANCU, AND SPINU PHYSICAL REVIEW B 76, 054409 2007 r = 2 8 2 hd 2 +4 2 + h d os2 a, h 2 d 2 os 2 a + sin 2 a, h d h d h d, FIG. 8. Color online Sketh of the geometry used to alulate the omplex TS. TS is predited beause during a TS experiment, the inreasing d field shifts the partiles relaxation time toward lower values, making the ourrene of the thermal relaxation on the time sale of the a field period possible. This mehanism is similar to that giving rise to a omplex a suseptibility, 24 where the role of the transverse field is played by the temperature. For a system of idential partiles having the easy axis perpendiular to the d field, the thermal relaxation gives rise to a seondary peak in the real part of the TS and to a peak in its imaginary part. The omplex suseptibility is now used to separate volume and anisotropy field dispersions 9 22 by fitting the experimental data with the expressions obtained in Ref. 8. In Ref. 23, the stohasti Landau-Lifshitz-Gilbert SLLG equation 32 is used in order to desribe an array of 400 Voronoi ells with intergranular exhange and magnetostati interations, with different average diameters and dispersions, and a Gaussian anisotropy field distribution. It is shown that the volume dispersion only weakly affets the peak shape, aording the model from Ref. 8, while the dependene on both anisotropy field dispersion and magnetization is substantial. The magnetostati interations distort the theory from Ref. 8, but the most substantial hange is due to intergranular exhange. Reently, our miromagneti analysis based on the SLLG equation showed a omplex behavior of the real and imaginary parts of TS for a system of interating partiles. 33 35 In the model presented in Ref. 8, the d field ats along the Oz axis of a retangular oordinates system and the a field ats in the plane xoy, with the azimuth a Fig. 8. Initially, a system of idential partile is taken into aount. Eah partile has the volume V and a uniaxial anisotropy of onstant K, the easy axis being onfined in the xoz plane, with the polar oordinate k. The partile magnetization s spherial oordinates are denoted by,. In order to better larify the method used in Ref. 8 and beause in Ref. 8 some typographial errors seem to be present, a detailed derivation of TS formula is exposed in the Appendix. Unfortunately, these errors propagated in the subsequent papers 9 23 dealing with the determination of volume and anisotropy field distributions in reording media using the model from Ref. 8. The omplex TS for a system of partiles with the easy axis oriented perpendiular to the d field diretion k =90 is given by see the Appendix i = 4 h 2 d 2 +4 2 os2 a, h d 2 0, h d, where = f 0 exp h d 2, f 0 is the attempt frequeny and it is of the order of 0 9 0 3 Hz, and =KV/k B T is the thermal fator. In Ref. 8, Eq. is approximated by = h 2 82 d r 2 +4 2 os2 a + sin 2 a, h d h d os 2 a, h d, where is the Dira delta funtion. However, this approximation does not give the orret variation of r h d in the proximate neighborhood of the point h d =. For a partile system with volume and anisotropy field distributions, r0 =0 h d0 +h d0 os 2 a Gh k dh k h k h d0 h k 0 8 h 2 d0 h k v 2 0 2 +4 2v2 Fvdvos 2 a h 2 d0 + h k h k h 2 d0 h k os2 a + sin 2 agh k dh k, h k i0 = 4 0 v os2 ah d0 0 2 +4 2v2 Fvdvdh k, h d0 h k 2Gh k 3 4 where h a0 =h a /H k0, h d0 =h d /H k0, h k =H k /H k0, 0 =m a /h a0, 0 =K 0 V 0 /k B T, and H k0 =2K 0 / 0 M s being the most probable value of the anisotropy field distribution, Gh k is the normalized anisotropy field distribution, v =V/V 0, v is the mean normalized volume, V 0 is the most probable value of the partile volume distribution, Fv is the normalized partile volume distribution. 054409-8
PHYSICS OF COMPLEX TRANSVERSE SUSCEPTIBILITY PHYSICAL REVIEW B 76, 054409 2007 We note that in Ref. 8, the lower and upper limits of integrations in Eq. 3 are 0 and, respetively, for both integral over anisotropy field distribution. Also, in Ref. 8, both lower limits of integration are equal to zero in Eq. 4 and is used instead of 0. In the next setion, these results are ompared with the results obtained by numerial integration of stohasti Landau-Lifshitz-Gilbert equation, and an analysis of the influene of the a field amplitude and of the interations between partiles on the omplex TS method for the determination of volume and anisotropy field distributions in perpendiular reording media 9 22 is also presented. V. LANGEVIN SIMULATION OF TS In the presene of thermal agitation, it is supposed that the dissipative term from the LLG equation desribes only the statistial ensemble average of rapidly flutuating random fores and that for an individual partile the effetive field must be augmented by a stohasti thermal field H th whih is assumed to be a normal Gaussian random proess with the following statistial properties: 32 H th,i t =0, H th,i th th,j t =2D ij t t, D = k BT 0 M s V, where means the statistial average over different realizations of the flutuating field, and i and j are the Cartesian indies. The Kroneker ij expresses the assumption that the different omponents of thermal field are unorrelated and the Dira expresses that H th,i t and H th,j t are orrelated only for time intervals t t muh shorter than the time required for an appreiable hange of M aording to the equation LLG i.e., the random thermal fores are redued to a purely random proess, with a white spetrum. The onstant D measures the strength of the thermal flutuations and it is determined on the grounds of statistial-mehanial onsiderations. For a partile system, it is also assumed that the flutuating fields ating on different magneti moments are independent. The stohasti field hanges the deterministi motion of the magnetization into a random walk. As a result, the LLG equation is onverted into a stohasti differential equation of Langevin type with multipliative noise: 32 dm dt = + 2M H eff + H th + 2 M M H eff + H th. 5 M s In this paper, we integrate the stohasti Landau-Lifshitz- Gilbert SLLG equation numerially by means of the Heun numerial integration sheme. 36 No temperature dependenes of the anisotropy onstant and saturation magnetization are taken. The magneti properties follow from averages FIG. 9. Color online TS urves at T=300 K for two values of the partiles diameter, d=50 nm left and d=00 nm right, obtained with the CP model symbols and SLLG with h a,max =0 3 line. over many numerial realizations of the dynami proess. In order to test our program and to determine the time step size in numerial integration, we have onsidered a system of idential, noninterating Stoner-Wohlfarth partiles and we have ompared our numerial results with the expliit solutions available in this ase both for T=0 K and T0 K. The results obtained in this way are, in fat, idential to those alulated for one partile, beause time average and average over the statistial ensemble are the same ergodi hypothesis. In our simulation, we have onsidered a system of 4096 noninterating spherial partiles, and the desired quantities are averaged over 0 30 yles of the a field, after the attenuation of the transitory effets whih initially appeared. The results obtained by dupliation of the number of partiles pratially oinide with the former, but the omputation time inreases signifiantly. A. Idential partiles In Fig. 9, we ompare our numerial results for a system of idential partiles, obtained for h a,max =0 3 and =0., with relations and 2 obtained in Ref. 8 referred hereafter as the CP model. The parameters values used in the simulations are M s =300 ka/m, K=200 kj/m 3, T =300 K, f = MHz, and a =0. The results presented in Fig. 9 onfirm the results of the CP model with ertain differenes, whih an be explained in experimental onditions. However, as the SLLG method generates the stohasti trajetories for eah individual magneti moment, it provides muh more insight into the dynamis of the system. The prinipal differene is that we do not obtain two peaks for the real part of TS due to no-vanishing amplitude of the a field, while Eq. always predits two peaks, one of them being the anisotropy peak from the theory of Aharoni et al. 8, at h d =. In our results, both the position and the amplitude of the TS peaks depend on h a,max, so that the omparison of these quantities between the two models is, in some ways, superfluous. However, we an see that for k =90 and h a,max =0 3, as the partiles diameter d inreases, the real TS peak is always at higher values than that given by Eq. 054409-9
CIMPOESU, STANCU, AND SPINU PHYSICAL REVIEW B 76, 054409 2007 (a) (b) FIG. 0. Color online TS urves at T=300 K for different values of the partiles diameter d for k =90 left and k =89.9 right. The urves for T=0 K are also presented for omparison., while the imaginary TS peak is moving from higher values to smaller values than that given by Eq. 2. In both models, an inrease of the diameter d sharpens the TS peaks and shifts them to higher values of h d, but in our model, the amplitude of real TS as a funtion of partiles diameter has a maximum see Figs. 0 and, while in Ref. 8, itis inreasing toward an infinite value given by the theory of Aharoni et al. 8 When k dereases from 90, both real and imaginary parts of the TS undergo a derease see Fig. 0 where the sale of the vertial axes for k =90 is twie larger than that for k =89.9. This derease is very abrupt for the biggest partiles but is smaller for the smallest partiles beause for these partiles the effet of thermal relaxation is more important. Therefore, the TS is dominated by the partiles having the easy axis perpendiular to the d field k =90 only if the partiles are suffiiently large. In order to estimate the role of the temperature in the dynamis of the magneti moment, we also present in Fig. 2 some typial time evolutions of the projetion m a of the magneti moment on the diretion of the a field. Due to the random harater of the thermal flutuations, the magneti moment does not exatly follow a ertain path on the energy surfae, from one minimum to another one passing through the saddle point with the smallest energy barrier, like in the FIG.. The dependene of the maximum of the real part up and the minimum of the imaginary part down of TS on partiles diameter d. () FIG. 2. Time evolution of the indued signal m a time is normalized to the period of the a field for one partile with d =50 nm at T=300 K for h a =0.00, =0., and for different values of d field: a h d =0.97, b h d =0.954, i.e., Re is maximum, h d =0.948, i.e., Im is minimum, and d h d =0.942, i.e., the phase shift between m a and h a is maximum. two-level model. In fat, there is always a probability for the magneti moment to follow any other path, the path with the lowest energy barrier being the most probable. As we showed in Se. III, if h * h d and T=0 K, then the a field provides suffiient energy to the magneti moment to osillate around the Oz axis. For T0 K, the magneti moment effetuates random osillations by thermal ativation lose to the bottom of the energy minima, but also a number of energy barrier rossings followed by a return to the original energy minima, from one part of the Oz axis to other, during a yle of the a field an be observed even for h a, h a h a,2 if the dimension of the partile is suffiiently small. Consequently, the indued signal m a does not have a retangular-like form as in the ase T=0 K and it is approximately in phase with the a field, and its mean value over many yles of the a field is less than that obtained for T=0 K see Fig. 2a. Dereasing the dimension of the partile, the magneti moment effetuates a onsiderable number of overbarrier rotations during a yle of the a field. Similar features are enountered for k =89.9, but in this ase, beause the energy landsape is not symmetri, the higher energy minimum is less frequented by the magneti moment and the indued signal is more distorted from its ideal sinusoidal form. For h d h * and T=0 K, the magneti moment osillates only around minimum or minimum 2. However, for T 0 K, in ontinuane, the magneti moment an pass from one part of the Oz axis to the other by means of the energy gained from the thermal field, the indued signal inreases with the derease of the d field, and, onsequently, the position of the anisotropy peaks moves toward smaller field values; that is, the effet of no-vanishing amplitude of the a (d) 054409-0
PHYSICS OF COMPLEX TRANSVERSE SUSCEPTIBILITY field is amplified by the thermal agitation. When the d field dereases, the energy minima beome narrower and the amplitude of the random osillations around the two minima dereases, the energy barriers inrease, and the probability to pass from one minimum to other dereases, so that indued signal shifts into a retangular-like form, and its amplitude inreases exhibiting a maximum in the amplitude and, subsequently, in the phase shift behind the a field see Figs. 2b 2d. At low values of h d, the magneti moment osillates lose to the bottom of the energy minima, with the overbarrier relaxation mehanism being bloked and the indued signal is very small and in phase with the a field. The minimum of the imaginary TS Eq. 2 for the partiles with the easy axis perpendiular to the d field takes plae when 2, that is, when the energy barrier in the absene of the a field is E 0 /k B Tln2f 0 /. From this relation, it an be seen that inreasing the frequeny of the exitatory a field, the position of the imaginary TS peak moves toward higher d field values, beause inreasing the frequeny dereases the probability of surmounting the energy barrier. This inrease is onfirmed by our simulations. If =2 is the relaxation time, then the above relation an be written as T a /2, where T a is the period of h a ; that is, the minimum of the imaginary part Eq. 2 takes plae when the relaxation time is 2 times smaller than the period T a of the a field. In order to test this relation, we have ounted in our numerial simulations the number of rossings of the magneti moment of one partile from one part of the Oz axis to the other, and we have normalized it to the number of the yles of the a field, so that it does not depend on the number of yles over whih m a is averaged. For T=0 K, when h d dereases, the number of rossings n is equal to until the norm of TS reahes its maximum value and is equal to 0 after that. Taking into aount the temperature, the magnetization reversal an take plae by thermal ativation over the energy barriers, and when h d dereases, the number of rossing n dereases ontinuously from an infinite value for h d = when there is no energy barrier to 0 when the energy barrier is very high and the probability of rossing is pratially zero. In Fig. 3, presented are the number of rossings n r when the real part of TS reahes its maximum value and the number of rossings n i when the imaginary part reahes its minimum value as a funtion of partile s diameter. We an see that n i inreases from to a limit value of approximately 2.2 as the partile s dimension dereases, while in the ase of n r, the inrease is more abrupt. So, the miromagneti analysis showed the omplex behavior of the dynamis of the magneti moment when the energy barrier is smaller than k B T, the domain in whih the two-level model is not valid. 37 We note that the minimum values of the imaginary part and the phase shift do not take plae for the same value of h d. The lag behind h a reahes the extreme value when the magneti moment osillates mainly around one minimum, but there is a small probability to pass from one minimum to another during a number of yles of the a field, so that n Fig. 2d. So, the temperature has two antagonisti effets on the phase shift: for h * h d, thermal flutuations derease the phase shift, while for h d h *, thermal flutuations inrease the phase shift. PHYSICAL REVIEW B 76, 054409 2007 FIG. 3. Color online The number of rossings of the magneti moment of one partile from one part of the Oz axis to the other, normalized to the number of the yles of the a field, as a funtion of partile s diameter: when the real part of TS reahes its maximum value n r and when the imaginary part reahes its minimum value n i. B. Partile distributions From this point on, we are disussing the properties of a system of aligned partiles lying in a regular network formed within one layer, the system being generated using lognormal distributions for volume and for the anisotropy onstant. The results obtained for a noninterating system are qualitatively similar of those obtained using the CP model. However, our results show the importane of the a field amplitude. An inrease of the most probable volume V 0 shifts TS s peaks to higher fields and sharpens the peaks. An inrease of the most probable anisotropy onstant K 0 also shifts the peaks to higher fields but their width inreases. However, the width s relative variation is not so important in both ases. This fat is an important soure of errors for the fitting proedure desribed in Refs. 9 22. The flattening of the volume distribution determines a derease of imaginary TS and shifts its peak toward higher fields while its width inreases see Fig. 4. The most important fator in Eqs. 3 and 4 is k : both the amplitude and the width of the TS peak strongly depend on it see Fig. 4. As a onsequene, k is determined with the smallest errors by fitting proedure. By sys- FIG. 4. Color online TS urves for different volume dispersions V left and anisotropy dispersions k right for T=300 K and h max a0 =0.00. 054409-
CIMPOESU, STANCU, AND SPINU PHYSICAL REVIEW B 76, 054409 2007 FIG. 5. Color online TS urves for V =0., k =0.0, and different values of the a field amplitude for T=50 K left and T =300 K right; with symbols, the urves predited by the CP model are also presented. temati simulations, it was observed that a small angular dispersion of the easy axis affets the signal s amplitude only. Figure 5 shows the a field amplitude dependene of TS for two temperatures. The urves given by the CP model are also presented. We observe that only at low temperatures, as the a field amplitude dereases, the CP model gives a better desription of the results obtained with the SLLG equation. The effet of the inrease of a field amplitude is similar to a derease of V and V 0. In order to see how the magnetostati interations between partiles influene the TS urves, we have generated samples with inreasing paking fration. The magnetostati interations between partiles are ontrolled by the network harateristi length D and periodi boundary onditions are used. The fast Fourier transform tehnique is applied in order to ompute the field due to all partiles. As the intensity of the interations inreases, the position of the imaginary TS peak moves toward smaller fields values Fig. 6 and its amplitude dereases; that is, the effet of the interations is somehow similar to the a field amplitude. In Ref. 34, a systemati quantitative analysis of the influene of the a field amplitude and of the interations between partiles on the omplex TS method for the determination of volume and anisotropy field distributions in perpendiular reording media is presented. Following a similar methodology as desribed in Refs. 9 22, we have fitted the experimental data produed by our miromagneti model with the relation from the CP model. Beause experimentally a signal proportional to the magnetization and not its real value is measured and beause, as we mentioned in previous setions, the measured suseptibility is not equal to the differential suseptibility given by CP model, we have fitted the normalized imaginary TS urves even if in this way important information ontained in the TS amplitude is lost. The most affeted by the a field amplitude and the interations between partiles are the volume distribution s parameters. In this ase, the error generated by not taking into aount the finite value of the a field amplitude and by ignoring the interpartile interations an be up to 55%. Similar results FIG. 6. Color online Imaginary TS urves for different values of the paking fration and different values of the a field amplitude temperature T=300 K; with symbols, the urve predited by the CP model is also presented. have been obtained for wider distributions, but the errors with whih V is obtained derease if V inreases. However, for too large volume and anisotropy distributions, the peaks in the TS signal disappear. Assuming that the values of V 0 and K 0 are available through another measurement, we also tried to fit only V and k. In this ase, the error with whih V is obtained inreases beause its effet is overed by the effet of the a field amplitude. VI. CONCLUSIONS We have analyzed the origins of the imaginary part of TS: rate-independent hysteresis, visous-type rate-dependent hysteresis, and thermal relaxation effet origin. The rateindependent origin is an intrinsi ontribution to the imaginary TS and annot be negleted beause it is a zerotemperature effet. We have shown that beause in the experimental proedure the amplitude of the a field is not zero in the mathematial sense, a sinusoidal exitation gives rise to a distorted output, an irreversible suseptibility is measured, and the lag of the magnetization with respet to the applied field an be assoiated with a omplex suseptibility. Starting from the energy loss per a field yle, an expression for the imaginary TS is obtained at T=0 K, assuming that in eah moment the system lies in a minimum of the free energy. Also, using the ritial urve formalism, we have shown how the a field an indue a hysteresis and, onsequently, imaginary TS. The magnetization dynamis in the framework of the LLG equation was used to analyze the ourrene of imaginary TS as a onsequene of magnetization damping, whih determines a lag between the magnetization s response and the applied exitation. This lag is strongly dependent on the exitation frequeny. The model developed in Ref. 8, using a two-level approximation to take into aount the temperature effets, was revised and ertain larifiations and orretions have been made. Ther- 054409-2