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Generalized Permeability Tensor Model: Application to Barium Hexaferrite in a Remanent State for Self-Biased Circulators Philippe Gelin and Patrick Quéffélec LEST (UMR CNRS no 6165), ENST Bretagne, BP 832 29285 Brest Cedex, France University of Brest, CS 93832, 29238 Brest Cedex, France We describe a theoretical approach for determining the permeability tensor of polycrystalline ferrites regardless of their magnetization state. To take into account both the demagnetizing dynamic fields related to the magnetic domain and grain shapes and the magnetic interactions between adjoining domains and between adjoining grains, we transform the classical Landau–Lifschitz–Gilbert equation into a coupled two-equation system. We introduce statistical distribution laws for both the domain and grain demagnetizing coefficients into the calculation to take the domain and grain shape diversity into account and derive static vectorial quantities such as the internal magnetic dc field and magnetization in each domain that depends on the applied dc magnetic field from the Stoner–Wohlfarth hysteresis model. We compare results with those for existing models for various magnetized states. Then we apply the model to predict the permeability tensor behavior of barium hexaferrite thin film, especially in the remanent state, which is of great interest for the design of self-biased Y-junction circulators in the millimeter-wave range. Index Terms—Anisotropic media, ferrites.
I. INTRODUCTION ESIGNING ferrite microwave devices such as circulators, phase shifters, and isolators requires knowledge of the permeability tensor of the material. Assuming that a dc magnetic field is applied along the -direction, the permeability tensor is
D
where is the permittivity of vacuum and , , and are complex quantities. In most of these devices, the ferrite is assumed to be saturated, but in reality saturation is not reached due to the demagnetizing fields. For specific applications, the ferrite sample is partially magnetized either under weak static magnetic polarization or in a remanent state. This last case is currently a subject of interest, particularly for self-biased millimeter-wave hexaferrite circulators, where the high value of the magnetocrystalline field ensures the auto-polarization of the material. The permeability tensor of the material depends on the texture (mono or polycrystalline form) and, on the magnetic properties (saturation magnetization, anisotropy field, damping factor), but also on external parameters such as the shape of the sample and the applied dc magnetic field strength and direction. To minimize the manufacturing costs of the devices containing ferrite materials, they are generally used in a polycrystalline form. The internal structure of polycrystalline ferrite is composed of monocrystalline grains with random orientations of their anisotropy axes. Each grain is divided into domains to minimize the magnetic energy. The energy brought by an external dc magnetic field reorganizes the configuration of the magnetic domains inside the material. The mechanisms of this reorganization are magnetic wall displacement and magnetic moment rotations.
Digital Object Identifier 10.1109/TMAG.2007.909561
The dynamic response (permeability) of a polycrystalline ferrite excited by an electromagnetic wave is very different depending on its magnetization state. According to the applied dc field strength, a material can be demagnetized (zero or coercitive field), partially magnetized (moderate field or zero field at the remanent state), or saturated (high field). In the saturated state, all the magnetic moments in the ferrite are aligned along the applied dc field direction. The diagonal and off-diagonal component spectra of the permeability tensor exhibit resonant behavior. This resonance, due to the gyroscopic precession of the magnetic moments around the applied dc field direction, takes place at a well-defined frequency proportional to the dc field strength. The absorption peak which appears on the spectra of the imaginary parts of the permeability tensor components is relatively narrow, and its shape is well described by the Polder formulations [1]. In a demagnetized or partially magnetized state, the observed absorption peak is much broader. Among the physical phenomena which lead to this low-field loss region, one of the most important, described by Polder and Smit [2], has a maximal effect when magnetic domains with opposite magnetization directions are present in the material. That results in the spreading of the resonance frequencies and generates magnetic losses in a frequency band from the frequency , proportional to the anisotropy field, up to a frequency , proportional to the saturation magnetization of the medium ( is the gyromagnetic ratio). II. PERMEABILITY MODELS OF UNSATURATED FERRITES A. Description of the Existing Models The permeability of ferrite in a partially magnetized state has been treated by many authors. The first model was presented by Rado [3]. Considering noninteracting domains, this theory involves performing a spatial average of responses produced by all domains in the ferrite. This theory provides a good approximation for the off-diagonal term for frequencies above the gyroresonance frequencies, but gives inaccurate values for diagonal terms .
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Starting from a coaxial configuration of alternatively parallel and antiparallel domains, Schloemann [4] has developed a theory based on the magnetostatic approximation which takes interactions between domains of opposite magnetization into account. In the completely demagnetized state, an average of the three diagonal tensor components yields an accurate value for the scalar permeability of isotropic ferrites
In these two approaches, the internal dc field in each domain is approximated by the anisotropic magnetocrystalline field regardless of the value of the external dc applied field , which is supposed to be almost entirely compensated for demagnetizing fields. Using experimental data, Green and Sandy [5] deduced empirical forms of the diagonal terms and as a function of the reduced magnetization which are compatible with Schlömann’s formula in the completely demagnetized state. Unfortunately, they do not fit with the Polder tensor for
(1) In a first paper, Igarashi and Naïto [6] presented a formula for the diagonal term in a partially demagnetized state, and they later gave an expression for the parallel diagonal term [7]. Unfortunately, experimental data must be used to adjust some of the parameters introduced in their formulations such as, for example, the internal dc field. Bouchaud and Zerah used the effective medium approximation (EMA) to determine the permeability tensor components in the demagnetized state [8]. This theory confirms the Schloemann prediction concerning the initial permeability. In the partially magnetized state, the results are only valid for weak applied dc fields because the internal dc field is assumed to be the anisotropy field as in [3] and [4]. Gelin and Berthou considered the polycrystalline ferrite in a demagnetized state as an assembly of randomly oriented and independent grains (crystallite) [9]. Each grain is divided into domains with magnetizations alternatively parallel and antiparallel to the easy axis of the grain. An applied dc field modifies the magnetization equilibrium direction (given by a unit vector ) as well as the internal dc field vector in each domain. From a dynamic point of view, each domain is coupled with its neighbor via the Polder Smit effect [2]. The dynamic interactions between adjoining domains are accounted for by simultaneously treating two coupled Landau–Lifschitz–Gilbert (LLG) equations. The permeability tensor components are computed using an average technique of the responses produced by all the domains in the ferrite. This model is predictive (only material provider data are necessary: , , and the effective resonance linewidth ), and it gives all tensor components as a function of the reduced magnetization . It is causal and therefore is compatible with the use of electromagnetic temporal simulators. Unfortunately, this model does not correctly describe the loss mechanism for low field bias. In addition, the
and , derived from a simple vecstatic quantities torial addition of the applied field and the anisotropic magnetocrystalline field and ignoring the hysteretic phenomenon, are too far from the physical reality to be valid. B. Critical Analysis of the Models Out of all the models [3]–[9], only the ratio characterizes the magnetization state of the ferrite, which is not sufficient to accurately describe its static magnetic properties (internal dc field, domain configuration, moment directions, etc.). Therefore, these models cannot be used to thoroughly determine the permeability, especially for hard materials which exhibit a rectangular hysteresis loop such as Barium hexaferrites. In addition, for models [4], [8], and [9], the domain configuration is too idealized to represent realistic situations. In [10], Gelin et al. proposed an evolution of the previous model described in [9]. First, the coupled Landau and Lifschitz equations were modified to take into account the demagnetizing effects relating to both the grain and domain shapes in accordance with the Polder–Smit and Smit and Wijn’s theories ([2], [11]), which describe the material response in the low-field loss region. Second, distribution functions of the demagnetizing coefficient are used to take the diversity of domain and grain shapes into account. For polycrystalline ferrites containing randomly oriented grains, this model gives results in good accordance with experiments. In particular, it predicts a spreading of gyromagnetic resonance frequencies as well as a behavior (losses) comof the permeability imaginary part spectrum parable to that experimentally observed for frequencies below . However, only the demagnetized state has been treated in [10]. To enforce the predictive character of the calculation, a permeability tensor model should be formulated in terms of the exinstead of the magnetization state ternal dc magnetic field , as was done in the previous work. Then, it is necessary to determine the internal static quantities in each domain of the material as a function of the applied field, including effects associated with hysteresis, the shape of the macroscopic sample, the texture of the material (preferential orientation or not), etc. To do so, hysteresis models of magnetization must be used to provide these internal static quantities. III. GENERALIZED PERMEABILITY TENSOR (GPT) A. GPT Framework The GPT model can be explained starting from Fig. 1. In the demagnetized state, the polycrystalline ferrite contains grains divided into domains. The grains are monocrystals with an easy . The domains are alteraxis characterized by the angles natively aligned either parallel or antiparallel to the easy axis. The vector indicates the equilibrium direction of the magnetization vector in domain 1 (2). The effective magnetostatic field in domain 1 (2) is parallel to the magnetization , where is the anisotropic magnetocrystalline field. In the partially magnetized state [see Fig. 1(b)], the equilibrium directions and of the local magnetizations vary as well as the local dc fields and , which depend not only on its current state (the anisotropic field , the external
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Fig. 2. Magnetization equilibrium direction in domain i.
where the quantity
Fig. 1. Moment directions in polycrystalline ferrite. (a) Demagnetized state. (b) Partially magnetized state.
applied field , the shape of the ferrite sample), but also on its past history (previous magnetic state). is applied, the When a radio-frequency (RF) excitation motion of the magnetization in domain ( ) is governed by the LLG equation
with
(2)
where represents the RF magnetization, is the local variable magnetic field, is the gyromagnetic ratio, and is the saturation magnetization. The resolution of (2) in domain requires the determination of the dc and the RF quantity . The first stage, from now on called the “dynamic problem,” consists in determining the RF quantity by taking into account the local couplings between neighboring domains and between neighboring grains and dynamic demagnetizing fields which depend on the domain and grain shapes. In the second stage, called the “static problem,” the dc quantities and , will be estimated in each domain of the material, i.e., for all the possible easy axis directions. Knowing all terms of the LLG equation in domain , solving (2) gives the relation , where is the local susceptibility tensor. The last stage consists in summing from a statistical approach all the local dynamic magnetizations over all directions to determine the effective permeability tensor of the material. B. Dynamic Problem We have shown in [10] that the motion of the magnetization vectors and relating to two interacting adjacent domains in a grain (crystallite), itself surrounded by other grains, under the action of an RF magnetic field can be described by the following system of coupled equations:
(3)
represents the dynamic demagnetizing field due to the shape of domain 1 including the Polder–Smit effect which couples the motion of the RF magnetization vectors of adjacent domains and is the demagnetizing coefficient related to the domain shapes. The quantity
is the dynamic demagnetizing field due to the grain immersed in an effective medium characterized by a mean RF magnetization and is the demagnetizing coefficient related to the grain shapes. C. Static Problem To solve (3), it is necessary to calculate the static quantities, i.e., the equilibrium direction of the magnetization and the local magnetic field in each domain of each grain of the ferrite sample. Among the different theories which are able to provide the static characteristics of ferrite as a function of an applied dc field, we have chosen the Stoner–Wohlfarth model [12]. Although this model considers a polycrystalline material as an assembly of noninteracting single-domain particles with uniaxial anisotropy and ignores the domain wall movement contribution to the material magnetization, it constitutes an interesting way of introducing the hysteresis phenomena and allows us to easily obtain the equilibrium direction of the magnetization and the local magnetic field in domain . 1) Magnetization Equilibrium Direction: Under the action , the total energy in domain is the of a dc magnetic field sum of the magnetocrystalline and the magnetostatic energies. The magnetization equilibrium direction is obtained by minimizing the energy . For domain , whose easy axis direction is defined by the angles compared to the direction [Fig. 1(a)], the equilibrium vector is in a direction defined by the angle (Fig. 2) related to by the relation
where is the domain crystal anisotropy ( is related to the anisotropy field by ). Note that the azimuthal angle does not intervene in this relation. When the applied field increases from zero, the changes of the energy minima are fully reversible. When the field reaches a critical value (depending on the values of and ), some minima become unstable and the magnetization vector jumps to a new equilibrium position: the process becomes irreversible.
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When the field strength is decreased, the magnetization curve exhibits a hysteretic behavior. The hysteresis loop of a polycrystalline ferrite can be predicted from the Stoner–Wohlfarth model, pending the summation of the magnetization contribution of all the domains. 2) Local Effective DC Field: The second-order derivative of the energy provides the strength of the local dc magnetic in domain field
This field corresponds to the magnetocrystalline field when and . When a macroscopic magnetization appears in the ferrite sample, it generates a macroscopic demagnetizing magnetic is related to the external applied field. Then, the static field by the relation field
where is the demagnetizing coefficient along the applied field direction. This quantity depends on the sample shape. Even if the determination of magnetization orientation in the uniaxial domain particle is not appropriate to treat soft ferrites (multiaxial domain particle), the effect on the computed permeability is limited. This comes from the averaging process over all the possible directions of the individual anisotropy axes.
but for textured materials exhibiting a crystal preferential orientation of the easy axes, we propose the following distribution: (5) where is the angle between the easy axis of the Stoner–Wohlfarth particle (i.e., domain in our case) and the preferential orientation of the bulk material, induced for example during the manufacturing process for the realization of permanent magnets or preoriented hexaferrites. The parameter is deduced from the reduced remanent magnetization, given by the sample provider, using the following relation: (6) The main interest of (5) is its ability to take into account all the distributions that can be found in practice, from the to isotropic texture of bulk polycrystalline materials the strip configuration which characterizes the domain structure of monocrystalline thin film . To take the variety of domain and grain shapes into account, and statistical distributions of the demagnetizing factors must be introduced into (4). Considering the equal probability of shapes, the distribution law for both domain and grain demagnetizing coefficients is given by
Finally, using the relation
IV. EFFECTIVE PERMEABILITY TENSOR As all the static quantities are now determined in domain , we can solve (3) and calculate the dynamic magnetization as a function of the RF excitation for a given external dc field , a given domain shape and a given grain shape :
the effective permeability tensor obtained.
of the bulk material is
V. RESULTS The method of solving (3) is comparable to that already described in [9]. In order to calculate the effective permeability of the bulk sample, a statistical summation of the local dynamic magnetizations over all magnetic domain directions (defined by the variables , ) and over all domain and grain shapes is performed (4)
where
and
The distribution functions , , and are related to the easy axis orientations and to the grain and domain shapes, respectively. For an isotropic material with randomly oriented easy axes, the distribution function is given by
The variations of the real part of the off-diagonal permeability tensor component of an isotropic polycrystalline ferrite ( kA/m, kA/m, ) as a function of the ratio for two frequencies are depicted in Fig. 3, obtained from the present model, the Rado prediction, and the Green and Sandy measurements. The results obtained from the present model are in good agreement with the measurements and also with the Rado approximation. However, as Green and Sandy did not give all the characteristics of the ferrites under examination [5], i.e., the magnetocrystalline anisotropy field , the damping factor (or ), the coercitive field and the remanent magnetization, a comparison of the present model with their measurements does not allow for a definitive quantitative statement about the validity of the model. The GPT model predicts a very small difference between the values according to magnetization states with the same value on the hysteresis loop. We have also established that the off-diagonal term is weakly influenced by the strength of . Fig. 4 shows the variation of the real part of the diagonal term [Fig. 4(a)] and off-diagonal and parallel terms [Fig. 4(b)] of an isotropic polycrystalline ferrite ( kA/m, kA/m, ) as a function of frequency for two different operating points on the hysteresis loop with the same value .
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Fig. 3. Real part of the off-diagonal permeability tensor component for an isotropic polycrystalline ferrite (M = 139 kA/m, H = 1:59 kA/m, = 0:005) as a function of the ratio M=! for two frequencies. Comparison between the GPT model, the Rado prediction, and the Green and Sandy measurements.
The first operating point I ( ; A/m) belongs to the initial magnetization curve and the second one II ( ; A/m) belongs to the major hysteresis loop after saturation. The present model predicts different according to the magnetization state values for , , and on the hysteresis loop. For the first time, a theoretical approach shows the influence of the hysteresis phenomenon on the permeability tensor components, which has been experimentally established by Green and Sandy [5]. The next section illustrates a typical application justifying the interest of the present model aside from its generalized formulation. VI. APPLICATION TO BARIUM HEXAFERRITE Due to its very strong uniaxial anisotropy field and strong magnetization ( kA/m, kA/m), barium hexaferrite is widely studied [16], [17] due to its great interest for realizing integrated self-biased Y-junction circulators at millimeter wavelengths [18], [19]. In a typical Y-junction circulator, the ferrite material is magnetically biased by an external dc field normal to the plane of the ferrite chip. In the self-biased hexaferrite circulator, the magnetization of the hexaferrite is ensured by the very high value of the internal magnetocrystalline field, which behaves as a permanent magnet when the material is in the remanent state. At this operating point, the net magnetization is less than that exhibited by the material at saturation. In order to increase the remanent magnetization and therefore the efficiency of the device, it is necessary to study and manufacture oriented materials. At present, the design of an integrated self-biased circulator is performed from the Polder formulations, which suppose that all the magnetic moments in the ferrite are aligned in a unique direction. In that case, the internal field is the difference between the anisotropy field and the demagnetizing field. For materials exhibiting reduced remanent magnetization less than 1, this assumption is not satisfied in practice, and the Polder theory is no longer valid. It is certain that the optimization of the device response will be enhanced thanks to the knowledge of the dynamic behavior of the material via the permeability tensor.
Fig. 4. Permeability tensor components of an isotropic polycrystalline ferrite (M = 159 kA/m, H = 1:59 kA/m, = 0:005) as a function of frequency for two different operating points on the hysteresis loop with the same M=M value (M=M = 0:7), (a) real part of the diagonal term, (b) real parts and of the off-diagonal and parallel terms, respectively.
The permeability tensor of the barium hexaferrite sample depends on its magnetization state, its shape, and its texture (preferential or isotropic orientation). The aim of the following sections is to quantitatively evaluate the influence of these different parameters. A. Effect of the Magnetization State For an infinite material without interface implying demagnetizing effects, characterized by an isotropic texture , the Stoner–Wohlfarth theory predicts a remanent state [called B-state polarization in Fig. 5(a)], characterized by and . A single ratio corresponds to various internal magnetization states, all located inside the major loop. Among these states, we will study the permeability tensor related to B-state polarization and another one [called A-state polarization in Fig. 5(a)] located on the initial magnetization curve ( and kA/m). In the B-state, the domain structure no longer exists and the internal field strength is equal to kA/m in each grain. In the A-state, the Stoner–Wohlfarth theory predicts that only grains having initial domain orientation rad rad have a reversal of their magnetization. By taking into account the population law (5), one can see that most of the grains exibits a domain
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Fig. 6. Diagonal term spectra for hexaferrite (M H = 1350 kA/m, = 0:005).
=
382 kA/m,
Fig. 7. Influence of the sample shape and of the preferential crystal orientation on the hysteresis loop for barium hexaferrite.
Fig. 5. Permeability tensor component spectra for hexaferrite (M 382 kA/m, H = 1350 kA/m, = 0:005).
=
structure in this magnetization state. The coupling between domains in each grain is different, that leads to a spreading of the internal field strengths. The present model provides the three permeability tensor components in a single calculation procedure. All the permeability tensor component spectra are given in Fig. 5 for barium hexaferrite with the following magnetic properties: kA/m, kA/m, . This figure shows a very different dynamic behavior between the states denoted A and B corresponding to the same ratio .The spreading of the internal field strengths in the A state leads to a spreading of the resonance frequencies, which is reinforced by a decrease the
permeability tensor component magnitudes and a broadening of the resonance curves (Fig. 5). Fig. 6 compares the diagonal term of the pemeability tensor obtained for the demagnetized state and the remanent state (B-state). Note that the off-diagonal term is equal to zero in the demagnetized state contrary to the remanent state [see Fig. 5(b)]. As would be expected, a shift of the gyromagnetic frequency and a broadening of the absorption peak are predicted by our model. These results show that the GPT approach permits the differentiation of the dynamic responses of the ferrite for different states on the hysteresis loop corresponding to the same magnetization value ( constant). The GPT approach constitutes a significant step in the study of the microwave behavior of unsaturated ferrites, because in all previously published models ([3], [6], [9]), the tensor components only depended on the quantity . B. Effect of the Sample Shape and of the Preferential Crystal Orientation The permeability tensor components of the ferrite material, integrated in a Y-junction circulator, depend on its shape (see Section III-C2) and on its texture (see Section IV). Figs. 7–9 show the influence of these two parameters on both static and dynamic properties. For an isotropic distribution of grains (isotropic texture ), the reduced remanent magnetization of an infinite ferrite medium predicted by the Stoner–Wohlfarth model is equal
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Fig. 8. Influence of the sample shape and of the preferential crystal orientation on the static internal magnetic fields as a function of the orientation # of the local easy axes for barium hexaferrite.
to 0.5. Fig. 7 shows that the demagnetizing macroscopic field related to the sample shape (demagnetizing factor in the external dc field direction ) reduces the remanent macroscopic magnetization (thin line curve in Fig. 7, see hysteresis loop in Fig. 5(a) for comparison with the infinite medium). Another parameter which influences the hysteresis loop is the material texture. For example, for a thin sample, a preferential orientation of the magnetic moments perpendicular to the film plane, achieved during the manufacturing process, increases the remanent macroscopic magnetization (thick line curve in Fig. 7). The internal dc magnetic field strengths in each domain in the two remanent C- and D-magnetization states are given in Fig. 8. The mean value of these fields decreases, dropping to and to 1132 kA/m 1268 kA/m for an isotropic material for the preoriented material , compared to 1352 kA/m for an infinite medium (without the demagnetizing effect). The diagonal and off-diagonal components of the permeability tensor as a function of frequency for the two magnetization states C (isotropic texture) and D (preoriented material) are depicted in Fig. 9. The comparison of Fig. 9(a) and (b) shows the influence of the factor on the resonance linewidth of the ferrite. One observes a broadening of about 1 kOe between the two cases, even if the corresponding ratios are not very different [ and , see (6)]. For higher values of (or ) the difference of resonance linewidth between the preoriented and isotropic materials will be much larger. The factor traduces the misalignment of the individual anisotropy axes of domains and predictes the variation of the resonance linewidth as function of this misalignment. These theoretical results, which are the first published concerning the permeability tensor of hexaferrites in a magnetic remanent state, demonstrate that the barium hexaferrite presents an interesting ratio that can be used for millimeter-wave circulators. For isotropic textured materials, theavailable frequency band isfound beyond 60 GHz to avoid high magnetic losses. For preoriented materials, the situation is more interesting since the narrowness of the absorption curves and allows the circulator designer to use the material in a frequency band close to the gyromagnetic resonance frequency to achieve a greater ratio. VII. CONCLUSION The permeability tensor model of polycrystalline ferrites as a function of the dc bias-field is, for the first time, derived from a self-consistent theoretical approach.
Fig. 9. Diagonal and off-diagonal components of the permeability tensor for barium hexaferrite as a function of frequency for the two magnetization states C (isotropic texture) and D (preoriented material).
From a dynamic point of view, demagnetizing effects at the grain and domain scale levels as well as dynamic interactions between domains and between grains are taken into account. Mathematically, these interactions lead to a coupling between Landau–Lifschitz–Gilbert equations related to neighboring domains. To solve these equations, the static quantities, i.e., the equilibrium direction of the local magnetization and the local magnetic field in each domain of each grain of the ferrite sample, are determined from the Stoner–Wohlfarth theory. In order to obtain the effective permeability tensor of the ferrite sample, an average of local dynamic responses over all the initial magnetic domain orientations and over all domain and grain shapes can be performed by using a statistical distribution of the domain and grain demagnetizing factors. The GPT model ensures the causality of the permeability tensor components, which is a property required for use in timedomain electromagnetic methods. Comparisons with the Rado extradiagonal term and with the Green & Sandy empirical expressions for the diagonal terms of the permeability tensor show the ability of the model to reproduce the second order hysteresis effect observed experimentally by J. Green and F. Sandy. Finally, to illustrate the interest of this theoretical approach for practical applications, an example of tensor component spectra of barium titanate hard ferrite is discussed, which is of great interest for fabricating integrated self-biased Y-junction circulators at millimeter wavelengths.
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[16] M. N. Afsar, D. Lisjak, A. Bahadoor, and Y. Wang, “Microwave ferromagnetic resonance of cobalt and nickel substituted U-type hexaferrites,” IEEE Trans. Magn., vol. 41, no. 10, pp. 3472–3474, Oct. 2005. [17] M. C. Dimri, S. C. Kashyap, and D. C. Dube, “"Complex permittivity and permeability of Co U (Ba Co Fe O ) hexaferrite bulk and composite thick films at radio and microwave frequencies,” IEEE Trans. Magn., vol. 42, no. 11, pp. 3635–3640, Nov. 2006. [18] N. Zeina, H. How, C. Vittoria, and R. West, “Self biasing circulators operating at Ka-band utilizing M-type hexagonal ferrites,” IEEE Trans. Magn., vol. 28, no. 5, pp. 3219–3221, Sep. 1992. [19] S. A. Oliver, P. Shi, W. Hu, H. How, S. W. McKnight, N. E. McGruer, P. M. Zavracky, and C. Vittoria, “Integrated self biased hexaferrite microstrip circulators for millimeter-wavelength applications,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 2, pp. 385–387, Feb. 2001.
Manuscript received March 28, 2007; revised September 25, 2007. Corresponding author: P. Queffelec (e-mail:
[email protected]). Philippe Gelin received the Ph.D. degree in electronics from the University of Lille, France, in 1981. Presently, he is Professor at the “Ecole Nationale Supérieure des Télécommunication de Bretagne” in the Laboratory for Electronics and Communication Systems (LEST, UMR CNRS no 6165). His research interests are in the area of electromagnetic theory, electromagnetic wave propagation in complex materials, and modeling of magnetic materials.
Patrick Quéffélec received the Ph.D. degree in electronics from the University of Brest, France, in 1994. Presently, he is Professor at the University of Brest in the Laboratory for Electronics and Communication Systems (LEST, UMR CNRS no 6165). His research activities deal with electromagnetic wave propagation in magnetic materials and the analysis of measurement methods for the microwave characterization of materials. Motivated by the applications of magnetic materials in nonreciprocal or tunable devices, he investigates the fundamental properties of ferrites and ferromagnetic composites using microwaves.
