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Magnetoimpedance in amorphous wires and multifunctional applications: from sensors to tunable artificial microwave materials
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 272–276 (2004) 1452–1459
Magnetoimpedance in amorphous wires and multifunctional applications: from sensors to tunable artificial microwave materials Larissa V. Paninaa,*, Dmitriy P. Makhnovskiya, Kaneo Mohrib a
Department of Communication and Electrical Engineering, University of Plymouth, Drake Circus, Plymouth, Devon PL4 8AA, UK b Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan
Abstract This paper is aimed to give a generalized approach to magnetoimpedance (MI) in amorphous wires in relation to the applications in sensors and microwave materials, raising new ideas and concepts. The impedance change ratio per Oersted in Co–Fe wires runs into more than 100%/Oe at MHz frequencies. This makes wire elements unique for various sensing applications. The emphasis required here is on induced asymmetric and off-diagonal effects to produce highly sensitive and linear responses with respect to the sensed external forces (magnetic field, stress). The field sensitivity of the impedance in wires with a circumferential anisotropy remains very high even at the GHz range. The combination of microwave properties of wires with electrodynamic behavior of wire-composite materials is foreseen as resulting in completely new effects such as magnetic field-dependent dielectric response and microwave activity. Thus, the MI effect can be useful to design microwave composites with tunable properties including negative permittivity, band-gap structures and waveguides. r 2003 Elsevier B.V. All rights reserved. PACS: 75.30.Cr; 75.30.Gw; 75.40.Gb; 75.50.Cc; 75.50.Kj Keywords: Magneto-impedance; Surface impedance; Magnetic wires; Microwave materials; Tunability; Microwave activity
1. Introduction Interest in magnetic materials in the form of microfibers continues to expand owing to their strongly anisotropic electrical, magnetic and optical properties [1,2]. The applications range from using a tiny wire as a sensing element of sub nT magnetic field detectors or as a magnetic label to artificially structured wire-materials with unusual properties at microwave frequencies. A large number of these applications employ magnetoimpedance (MI) effect. The present paper gives a comprehensive analysis of MI in amorphous microwires, which is based on field-dependent surface impedance *Corresponding author. Tel.: +44-1752-232599; fax: +441752-232583. E-mail address: [email protected] (L.V. Panina).
tensor. This allows a variety of MI characteristics to be viewed from a generalized approach in a wide frequency range including microwave band. In fact, the concept of the surface impedance combines the micromagnetic analysis with special problems in electrodynamics representing a new branch of research. It also gives an accurate balance between physical principles and practical applications. Co-based amorphous wires with a circumferential (or helical) anisotropy exhibit large and sensitive magnetic response of a high-frequency voltage known as MI effect [3]. The sensitivity (impedance change ratio per Oersted) runs into more than 100%/Oe at MHz frequencies [4,5]. This makes wire elements unique for applications in magnetic sensors for low-field detection. A number of sensor prototypes based on self-oscillation or pulsed circuits have been developed, which operate at a
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.1285
ARTICLE IN PRESS L.V. Panina et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1452–1459
2. Family of MI characteristics Thin amorphous ferromagnetic wires having a negative magnetostriction are very popular for MI applications. In the outer layer of the wire, an internal stress from quenching coupled with the negative magnetostriction results in a circumferential anisotropy and an alternate left and right circular domain structure [19]. In this case, the circular magnetization processes determining the MI behavior are very sensitive to the axial magnetic field [20,21]. Along with this, special types of anisotropy as a helical one can be established by a corresponding annealing treatment, which results in unusual asymmetric MI behavior and the so-called microwave activity in wire-based composite materials. Many experimental results on MI and magnetic sensor application utilize Fe4:35 Co68:15 Si12:5 B15 amorphous wires having a small negative magnetostriction in the range of lE 107 : The wires of the composition
Co72:5 Si12:5 B15 having l ¼ 3 106 are used for stressimpedance (SI) [22] sensors showing the strain-gauge factor of more than 2000 which is more than 10 times larger than that of the semiconductor strain gauge. Currently, there are basically two methods of wire fabrication techniques. The first one utilizes in-waterspinning method for which as-cast wires have a diameter of 125 mm: The wires are then cold-drawn down to 20– 30 mm; and finally annealed with a tension to establish a moderate tensile stress. The other technique produces amorphous wires in a glass cover by modified Tailor method which is based on drawing a thin glass capillary with molten metal alloy. The diameter of the metal core is ranging between 5–50 mm: In this case, different temperature expansion coefficients of glass and metal alloy result in a longitudinal tensile stress, which is needed for the circumferential anisotropy. The value of this stress can be easily controlled by the ratio of the diameters of glass cover and metal core. This is a simple one-step process allowing a strict control of properties in as-cast state [23]. First experiments on MI dated to 1993 utilized a simple concept of measuring the AC voltage across the wire subjected to an AC current i ¼ i0 expðjotÞ and a DC magnetic field Hex applied in parallel with the current. It was found, that even in the case of a circumferential anisotropy the MI behavior can be completely different depending on the current amplitude i0 and frequency o [3]. The typical characteristics are given in Fig. 1. It is seen that with increasing frequency the field plots transfer from a single peak type to that having two symmetrical peaks situated at the anisotropy field. However, for small i0 less than 1 mA; only twopeak behavior is observed with a quite deep minimum at zero field [24,25]. The effect of DC current Ib also results in similar change to the two-peak plots as illustrated in Fig. 2. This suggests that the central maximum in Fig. 1 is due to domain wall processes which are suppressed by either applying Ib ; decreasing i0 or increasing frequency. Very high field sensitivities in the range of 100%/Oe are typical of glass-coated wires under very low-current
Re duce d Volta ge , V0 / VR
frequency of several tens MHz and give the field resolution of 106 Oe for a stationary AC field detection [6–8]. To improve the sensor linearity and temperature stability the asymmetric and off-diagonal MI effects are of a special importance [8–10]. The field sensitivity of the impedance in wires with a circumferential anisotropy remains very high at radio frequencies and even in the GHz range [11–13], which leads to new physical effects and applications. Thus, surface acoustic wave devices combined with MI wires as loading impedances can be used for wireless identification systems [14,15]. The combination of microwave properties of wires with electrodynamic behavior of wire-composite materials is foreseen as resulting in completely new effects such as magnetic field-dependent dielectric response [16,17]. The MI effect can be useful to design microwave composites with tunable properties including negative permittivity, band-gap structures and waveguides. The paper is organized as follows. Section 2 outlines the family of MI characteristics obtained at different frequencies and excitation conditions emphasizing their role in various sensor applications. Section 3 introduces into the surface impedance approach with the results applied to a single-domain wire with arbitrary anisotropy. The model limitations may seem to be quite rigid, nevertheless, it is demonstrated that this simplified description gives a clear picture of the MI behavior and agrees well with the experiment and applications suggesting further practical ideas. The last section is devoted to a new area of employing MI in tunable artificial materials characterized by the effective permittivity depending on the external magnetic field in the microwave frequency band. The possibility of microwave activity known for chiral structures[18] is further discussed.
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5 4
FeCoSiB-wire
I 0 = 5 mA
30 µm dia m. l = 5 mm
σ a =2 kg/mm 2
f = 10 MHz
3 3 MHz 2 1 MHz
1 -8
-4
0 4 Axial field, Oe
8
Fig. 1. Voltage amplitude in 30 mm diameter CoFeSiB wire vs. external field with a frequency as a parameter. i0 ¼ 5 mA:
ARTICLE IN PRESS L.V. Panina et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1452–1459
1454 3.0 2.5
f = 20 MHz l = 5 mm i < 0.8 mA
I =0 b
90%/Oe
Vw , mV
ac
2.0
10 mA
1.5
25 mA 46.5%/Oe
1.0
23.5%/Oe -20
-10
0 H ex , Oe
10
20
Fig. 2. Voltage amplitude vs. Hex with Ib as a parameter in glass-coated wires having a metal core diameter of 48 mm: i0 o0:8 mA: The sensitivity of 90%/Oe is seen for lb ¼ 0:
280
0.5 GHz
Impedance |Z|, [Ω]
260
1 GHz
240 220 200 180 160
2.2 GHz
140 -20
-10
0
10
20
Axial magnetic field [Oe]
Fig. 3. High-frequency impedance plots vs. Hex in glass-coated wires.
excitation. Recently, MI at much higher frequencies was investigated [11–13]. In this case, proper care has to be taken for calibrating the total measurement cell. Typical impedance plots are shown in Fig. 3. The two-peak characteristics are still seen for hundreds MHz. Their low-field behavior ðHex oHK Þ preserves high sensitivity, whereas the characteristics flatten in the high-field region ðHex > HK Þ and for frequency f > 2 GHz they remain constant for any Hex > HK where HK is the effective anisotropy field. This unusual behavior can be referred to as valve-like for which the impedance becomes insensitive to the field when the wire is axially magnetized. There is no correlation with ferromagnetic resonance for which much higher fields would be needed [26]. We can conclude that the MI effect remains very sensitive in a wide frequency range; also the field-plot appearance may change substantially. Megahertz frequency MI characteristics, for which extremely high sensitivity can be reached, are utilized in various magnetic, current and stress sensors. Microwave MI is very promising for remote sensing and tunable microwave materials. Substantial amount of works on MI has been devoted to the asymmetric effects [4,9,24,27–29]. In the case of
sensor application, its linearity is an important feature. On the other hand, the MI characteristics presented in Figs. 1–3 are not only non-linear, but also shaped in a way that the operation near zero-field point can present serious problems. Generally, a DC bias field is used to set properly the operating point on the MI characteristics, which can be regarded as producing asymmetry with respect to the sensed field Hex : Therefore, for linear sensing asymmetrical MI (AMI) is of great importance. There are mainly two ways to realize AMI in amorphous wires. The first one is related to an asymmetrical static magnetic structure and the other is due to a dynamic cross-magnetization process. In a wire with a helical anisotropy, the magnetization responds not the same way to positive and negative Hex in the presence of DC bias current Ib [30]. Then, the AC voltage will have asymmetry with respect to Hex : The helical anisotropy can be induced by applying an external torsion or by annealing under a torsion. In some cases, a spontaneous helical anisotropy may exist as in the case of as-cast CoSiB amorphous wires. The experimental results on AMI in wires annealed under torsion are given in Fig. 4 [27]. With increasing Ib ; the shifted maximum in MI plots becomes more pronounced, which is accompanied by enhanced sensitivity. In this particular case AMI plots do not exhibit a hysteresis which is due to nonlinear AC excitation. If the amplitude of the AC current i0 is small, the MI plots in wires with helical anisotropy exhibit a substantial hysteresis which disappears only under the effect of large Ib [28,29]. Figs. 5 and 6 illustrate the other way to produce asymmetry in a highfrequency voltage response. This method is related to the dynamic cross-magnetization process of inducing a circulatory magnetization mj (with respect to the current i) by AC magnetic field hex parallel to i (produced by the coil current ic ) [31]. Due to the rotational mechanism of AC dynamics, this process is possible even for circumferential anisotropy but the domain structure must be eliminated by a small DC current. Typically, in the experiments the wire and the coil are connected in series ði ¼ ic Þ and excited by the same AC source. As frequency is increased, the contribution to the voltage due to hex becomes in the region of that induced by the current i itself, and can considerably change the voltage behavior vs. Hex as shown in Fig. 6. It has been recently proposed to utilize for sensor applications the voltage response Vc detected in the coil mounted around the wire whilst the wire is still driven by the passing current i [10]. This operation is based on the other cross-magnetization process mz ðhj Þ which causes the current flow to induce an AC axial magnetization and hence voltage Vc : The ratio Zc ¼ Vc =i may be referred to as off-diagonal impedance. In single-domain wires with the circumferential anisotropy the function Zc ðHex Þ is antisymmetrical having a near-linear region
ARTICLE IN PRESS L.V. Panina et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1452–1459 400
CoFeSiB am. wire
i = 15 mA
I =0
0
f = 1 MHz
I = 4.5 mA b
300
0
Vo ltage, V (mV)
b
I = 9 mA
100
b
-6
-4
-2
0
2
4
6
Axial field, H ex (Oe)
Fig. 4. AMI plots in torsion annealed wires (280 turn/m, 500 C) with Ib as a parameter. i0 ¼ 15 mA; a ¼ 25 mm:
H ex
hex
Ib + i
ic
Vw
Fig. 5. Mixed AC excitation using both the wire current i and the coil current ic :
6
Coil with 30 turns f = 10 MHZ
inp w
V /V
2 f = 3 MHZ
-20
-10
0 H ex , (Oe)
10
20
Fig. 6. Normalized wire voltage Vw vs. Hex at mixed excitation by ði; ic Þ for different frequencies.
around zero-field point. This characteristic can be used in linear sensing. A practical design of such a sensor is reported in Ref. [8] where the wire element is excited by a pulse current of C-MOS IC multivibrator. The output signal Vc is very small if no axial field Hex is applied. In the presence of the field, the voltage pulse rapidly increases and when the field is reversed the direction of the pulse is reversed as well. After rectification, antisymmetrical MI characteristic can be obtained with almost linear region within field interval 7HK :
3. Surface impedance in a magnetic wire The variety of MI characteristics discussed in previous section can be analyzed in terms of the generalized concept of surface impedance tensor [24]. The general voltage response ðVw ; Vc Þ from a magnetic material
ð1Þ
where r is a unit radial vector directed inside the wire. The field ht is set by the excitation conditions: the current i generates a circular magnetic field hj ¼ 2i=ca while the coil current ic produces an axial field hex ¼ 4pnic =c; where n is the number of turns per unit length in the coil and c is the velocity of light (Gaussian units are used). In ferromagnetic conductors, B# is a two-dimensional tensor even for the electrically isotropic case. In a wire with a uniform surface magnetization, the tensor B# is constant on the surface having Bzj ¼ Bjz (cylindrical coordinates are used). Using Eq. (1) and excitation conditions for the magnetic field, the voltage response is given by 2i Vw ¼ ez l ¼ Bzz Bzj hex l; ð2Þ ca Vc ¼ ej 2panl ¼
4
0
subjected to a high frequency excitation ði; ic Þ is given by the tangential components of electric field et taken at the wire surface. This field is related to the surface magnetic field ht via the surface impedance tensor B# et ¼ B# ðht rÞ;
200
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2i 2panl; Bjj hex þ Bjz ca
ð3Þ
where a is the wire radius, l is its length and c is the velocity of light. It is seen that only the first term in Eq. (2) corresponds to the usual case of Vw ¼ Zi where Z is the wire complex impedance. The second terms in Eqs. (2) and (3) are due to the AC cross magnetization processes mj hz and mz hj : Therefore, the impedance B# is the general characteristic describing the voltage response in the system excited by the external AC magnetic field h (of any origin). This approach is especially useful in understanding such effects as AC bias asymmetric MI (see Fig. 6) or microwave activity of wire-based composites. The calculation of B# is based on the solution of the Maxwell equations inside the conductor for the AC fields e and h changing in time as exp (jot) together with the equation of motion for the magnetization vector M: Taking a linear approximation with respect to the time-variable parameters e; h; m ¼ M M0 ; where M0 is the static magnetization and neglecting the exchange effects, the problem is simplified to find the solutions of the Maxwell equations with a given AC permeability matrix m# ¼ 1 þ 4p#w; m ¼ w# h: Further assumptions on dynamic magnetization processes are needed to determine m: # We use a single-domain model of a ferromagnetic wire where only the rotation of magnetization is taken into account. This model gives all the important features of the field dependence of B# : In general, the anisotropy axis nK has an angle c with the wire axis (z-axis) and the vector M0 is directed in a helical way having an angle y with the z-axis. Under these assumptions, the impedance matrix B# is expressed
ARTICLE IN PRESS L.V. Panina et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1452–1459
1456
in the analytical form in the limits of strong skin effect and relatively week skin effect. The details can be found in Ref. [24], and here only the final results are listed (i) ‘‘Strong’’ skin effect d=a51 cð1 jÞ pffiffiffi ð m# cos2 ðyÞ þ sin2 ðyÞÞ; 4psd0 cð1 jÞ pffiffiffi ¼ Bzj ¼ ð m# 1Þ sinðyÞcosðyÞ: 4psd0
Bzz ¼ Bjz
ð4Þ
(ii) ‘‘Weak’’ skin effect d0 =ab1; a=dB1; k1 c J0 ðk1 aÞ 1 a 4 cm23 þ ; ð5Þ 4ps J1 ðk1 aÞ 54 d0 psa 4 ao a Bjz ¼ Bzj ¼ j m3 3c d0 hm m m m i c : ð6Þ 1 3þ 2 3 60 30 psa pffiffiffiffiffiffiffiffiffiffiffiffi Here J0:1 are the Bessel functions, d0 ¼ c= 2pso; Bzz ¼
m* ¼ 1 þ 4pw; w ¼ w2 4pw2a =ð1 þ 4pw1 Þ; m1 ¼ 1 þ 4p cos2 ðyÞw; m2 ¼ 1 þ 4p sin2 ðyÞw: m3 ¼ 4p sinðyÞ cosðyÞw; k12 ¼ m1 ð4p jos=c2 Þ; w1 ; w2 and wa are diagonal and off-diagonal components of the susceptibility tensor w# written in the co-ordinate system with z0 jjM0 ; s is the conductivity and d is the magnetic skin depth. The intermediate case is obtained by the extrapolation. Eqs. (4)–(6) demonstrate that the tensor B# depends on both the effective susceptibility w and static magnetization angle y: Their field behaviors explain all the essential features of the MI plots. For circumferential anisotropy, the parameter w peaks at the anisotropy field, which leads to the two-peak MI characteristics. Applying the DC bias current eliminates the domain structure decreasing the total permeability down to its rotational part and the minimum at zero field becomes deeper (see Fig. 2). The parameter w has a very broad dispersion region. At high frequencies (higher than the frequency of the ferromagnetic resonance fFMR ) w becomes insensitive to Hex although preserving its relatively high values (jmjE10 at GHz range), as shown * in Fig. 7. In this case the field dependence of the impedance is entirely related with that for the magnetization angle y: This explains the valve-like behavior of the impedance observed at GHz frquencies (Fig. 3). The analysis of the surface impedance clearly demonstrates that the condition of the ferromagnetic resonance is not required for the MI effect, contrary to the widely expressed belief [26]. In the case of a helical anisotropy, the DC magnetization is represented by asymmetric hysteresis curves.
Fig. 7. Effective permeability spectra in the GHz range showing no change for moderate Hex ¼ 10 Oe; HK ¼ 2 Oe:
Because of this, the impedance plots show a progressive asymmetry with increasing Ib : Typically, the helical anisotropy results in a hysteresis in MI plots due to hysteresis in DC magnetization. However, in certain cases the MI characteristics measured at relatively low frequencies and large values of i0 do not have a hysteresis despite a hysteresis in M0 ðHex Þ (the case of Fig. 4). This is explained by considering nonlinear magnetization dynamics which is characterized by the averaged permeability over a circular magnetization cycle induced by the driving current i: The off-diagonal components of B# change sign if M0 is reversed. This property is useful to realize linear field characteristics by detecting the coil voltage Vc : However, in wires with the circumferential anisotropy this voltage is very small if the wire has a domain structure since Bjz averaged over domains with the opposite magnetization is zero. A small DC current is needed to eliminate the domain structure. The use of pulsed current which has both low- and high-frequency harmonics is then more preferable. Furthermore, asymmetric MI characteristics can be obtained using a mixed excitation by ði; ic Þ and measuring the wire voltage Vw : In this case, the voltage Vw involves both Bzz and Bzj components, one of which is symmetric and the other is antisymmetric with respect to Hex with the resulting response of the type shown in Fig. 6. Without Ib ; the effects related to the off-diagonal components are possible only in a wire with a helical anisotropy.
4. Field-dependent microwave permittivity of wire-based composite materials In this section we consider the MI effect in relation to composite materials containing ferromagnetic wires. In general, artificial material with conducting fibers present a considerable interest since their dielectric response can exhibit various dispersive behaviors having very large values of the effective permittivity even for diluted
ARTICLE IN PRESS L.V. Panina et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1452–1459
systems. They can be represented by a 3D-array of continuous wires or a composite with randomly oriented wire pieces. The former has characteristic features of a metallic response to radiation, but in the GHz range it has a negative permittivity eeff ðoÞ below the normalized plasma frequency [32]. In the second case, finite-wire inclusions behave as electric dipole scatterers. Then, the effective permittivity eeff can have a resonant dispersion seen near the antenna resonance for an individual wire [33]. If the losses are not very large, the real part of eeff has negative values past the resonance in this case as well. The possibility to have a negative permittivity is very important for a recent trend to create materials with a negative refraction index [34]. If the skin effect in wires is strong abd; the effective dielectric response is determined by wire geometry and the permittivity of host material. In a general case aEd; the dispersion characteristics of eeff depend on the surface impedance. Therefore, in composite containing ferromagnetic wires exhibiting MI effect, the effective permittivity may depend on a static magnetic field [16,17] via the corresponding dependence of B# ; as was discussed in the previous section. Furthermore, this special behavior of eeff requires the electric field of the external radiation to be directed along the wire. However, in a wire with a helical magnetization the external axial magnetic field will induce the axial electric field due to the off-diagonal impedance Bzj : Such composite will have the so-called microwave activity typical of chiral structures [18]. This can lead to completely new effects such as rotation of the polarization plane. The further analysis is restricted by the case of composite materials containing short-wire inclusions [17]. Such composite is similar in many respects to a dielectric material since the wire inclusions play a role of elementary dipole scatterers, which are polarized with an electric field e0 of the incident electromagnetic wave. In general, the local field eloc in the composite material at the wire location differs from e0 but for a diluted system this can be neglected. It is further assumed that e0 is directed along the wire. This field will excite the axial current iðzÞ which will have a nontrivial dependence on z when the wavelength l is of the order of the wire length. The current distribution is found by solving the scattering problem with the impedance boundary conditions (1). To describe the external scattered field, the current iðzÞ can be regarded as a linear current if the antenna conditions ða5l; a5lÞ are held. The current distribution inside the conductor is taken into account via the impedance of form (5). It will be also a nontrivial charge distribution qðzÞ along the wire in this case, which determines the induced dipole moment in the wire. Using the continuity equation @iðzÞ=@ðzÞ ¼ joqðzÞ and boundary conditions ið7l=2Þ ¼ 0; the wire dipole moment Pz is found by integrating the current along
the wire Z j l=2 Pz ¼ iðzÞ dz o l=2
1457
ð7Þ
The wire polarisability is defined as aII ¼ Pz =ðpa2 le0z Þ: The analysis of the scattering problem shows that the current iðzÞ can be approximated by a linear differential equation of the second-order involving a certain damping caused by the radiation as well as internal resistive and magnetic losses. The latter appears in this equation via the wire surface impedance. Thus, as in the case of a Lorentz oscillator the polarizability aII has the following form: X An iGn o; ð8Þ aII ¼ 2 2 ðo res;n o Þ n where the summation is carried out over thepantenna ffiffi resonance frequencies ores;n ¼ 2pc=lres;n ; ¼ 2l e=ð2n 1Þ is the resonance wavelengths, An and Gn are the amplitude and damping parameter respectively. The first resonance n ¼ 1 with the lowest frequency gives the main contribution to the polarizability. Each Gn can be decomposed into two parts related to the radiation and internal losses, respectively. The latter may depend on the external magnetic field Hex via the surface impedance Bzz : Thus, in the vicinity of the antenna resonance the polarisability aII will depend on Hex if the radiation losses are not very large (not very strong skin effect). In a ferromagnetic wire having nonzero Bzj ; the electromagnetic wave with magnetic field h0 along the wire will induce Pz as well. In this case eloc ¼ Bzj h0z implying that Pz is produced by a transverse electric field of the incident radiation. The corresponding polarisability a> will be of the form of (8) with An depending on Bzj : In fact, its dispersion will be different since Bzj is a function of frequency. Also, a> 5aII owing to Bzj 51: However, this cross-polarization process is principally important resulting in the constituent equations of the type adopted in chiral media. Let us now consider a thin composite slab parallel to the plane XY having randomly oriented wires. The effective permittivity matrix is obtained by averaging the local dipole moments over the orientation and is of the form 1 0 e2 0 e1 C B e#eff ¼ @ e2 e1 0 A; ð9Þ 0
0
e
where e1 ¼ e þ 2ppaII ; e2 ¼ 2ppa> and p is the volume concentration of wires. From Eq. (9) it follows that owing to the off-diagonal impedance of magnetic wires the composite system is characterized by the permittivity matrix having off-diagonal terms. In fact, this is a very unusual matrix with all the transverse components depending on a static magnetic field. For normal
ARTICLE IN PRESS L.V. Panina et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1452–1459
1458
incidence, the displacement vector D is of the form D ¼ e1 e0 þ kðk e0 Þ;
k ¼ e2 c=o
which is similar to that for medium with optical activity, such as spring-like, helical sort of structures. In our case, it can be referred to as microwave activity and its helical structure is supported by the DC helical magnetization in a wire. The existence of the off-diagonal components will result in such effects as the rotation of the polarization plane. They are at a very early stage of investigation. The spectrum of the diagonal component of e#eff is shown in Fig. 8 for two fields Hex ¼ 0 and Hex > HK : The field effect shows up in changing the character of the dispersion curves. In the absence of Hex the dispersion curves are of a resonance type. Applying a magnetic field Hex > HK ; the impedance is increased and, as a consequence, the internal losses in the inclusion, which results in the dispersion of a relaxation type. The field-dependent permittivity matrix e#ðHex Þ can be used in tunable microwave covers. The energy absorption in such kind of a composite is rather strong, therefore it has to be sufficiently thin to be used as a wave passage. In the case of a slab of 300 mm thick, the change in reflection and transmission coefficients is in the range of 15%. Another promising application suggested is employing these composites as an internal cover in the partially filled waveguides or layered thinfilm waveguides with a ‘‘dielectric/ferromagnetic’’ structure [35]. Such a waveguide system, by analogy to that already designed for the waveguides with ferromagnets,
28
ε = 16
22 eff
res
ex
24
ε/
~ 3.73 GHz )
H =0 (f
26
p = 0.001%
l = 10 mm
20 18
H >H (f ex
16 14
f
12
res
K
~ 4.32 GHz )
res
10 0
2
4
(a)
6
8
10
12
14
f ( GHz ) 16 14 ex
10
ε //
eff
ε = 16 l = 10 mm p = 0.001%
H =0
12 8 6
H >H
4
ex
K
8
10
2 0 0
(b)
2
4
6
12
14
f ( GHz )
Fig. 8. Effective permittivity of wire-composites as a function of frequency for Hex ¼ 0 and Hex > HK : The parameters used: p ¼ 0:001%; e ¼ 16; l ¼ 1 cm:
can be used for tunable filters and phase shifters operating up to tens GHz.
5. Conclusion The concept of magnetoimpedance (MI) has stimulated a new research combining micromagnetic and electrodynamic problems. Depending on magnetic structure, frequency and excitation method, various types of MI characteristics have been realized. The sensitivity remains very high even at the GHz frequency range. This suggests many potential applications in sensors, magnetic tags, microwave materials and devices.
References [1] I. Ogasawara, S. Ueno, IEEE Trans. Magn. 31 (1995) 1219. [2] M. Vazquez, A. Hernando, J. Phys. D Appl. Phys. 29 (1996) 939. [3] L.V. Panina, K. Mohri, K. Bushida, M. Noda, J. Appl. Phys. 76 (1994) 6198. [4] T. Kitoh, K. Mohri, T. Uchiyama, IEEE Trans. Magn. 31 (1995) 3137. [5] A.P. Zhukov, J. Gonzalez, J.M. Blanco, M. Vazquez, V.S. Larin, J. Mater. Res. 15 (2000) 2107. [6] K. Mohri, L.V. Panina, T. Uchiyama, K. Bushida, M. Noda, IEEE Trans. Magn. 31 (1995) 1266. [7] C.M. Sai, K. Mohri, H. Honkura, M. Yamamoto, IEEE Trans. Magn. 37 (2001) 2038. [8] K. Mohri, et al., IEEE Trans. Magn. 38 (2002) 3063. [9] L.V. Panina, J. Magn. Magn. Mater. 249 (2002) 278. [10] S. Sandacci, D.P. Makhnovskiy, L.V. Panina and K. Mohri, IEEE Trans. Magn. (2003), submitted for publication. [11] A.S. Antonov, A. Granovsky, A. Lagarkov, N. Perov, N. Usov, T. Furmanova, Physica A 241 (1997) 420. [12] M. Domingues, et al., J. Magn. Magn. Mater. 249 (2002) 117. [13] S. Sandacci, D.P. Makhnovskiy, L.V. Panina, J. Magn. Magn. Mater. (2003), submited for publication. [14] C. Hausleitner, R. Steindl, A. Pohl, H. Hauser, A.M.J. Goiser, F. Siefert, IEEE Trans. Microwave Theory Tech. 49 (2001) 817. [15] H. Hauser, R. Steindl, C. Hausleitner, J. Nicolics, A. Pol, Sensors and Materials 13 (2001) 25. [16] O. Reynet, A.-L. Adent, S. Deprot, O. Acher, M. Latrach, Phys. Rev. B 66 (2002) 94412. [17] D.P. Makhnovskiy, L.V. Panina, J. Appl. Phys. 93 (2003) 4120. [18] J. Petykiewicz, Wave Optics, Kluwer Academic Publishers, Dordrecht, 1992. [19] M. Takajo, J. Yamasaki, F.B. Humphrey, IEEE Trans. Magn. 29 (1993) 3484. [20] L.V. Panina, H. Katoh, K. Mohri, IEEE Trans. Magn. 29 (1993) 3484.
ARTICLE IN PRESS L.V. Panina et al. / Journal of Magnetism and Magnetic Materials 272–276 (2004) 1452–1459 [21] K. Mohri, K. Bushida, M. Noda, H. Yoshida, L.V. Panina, T. Uchiyama, IEEE Trans. Magn. 31 (1995) 2455. [22] L.P. Shen, T. Uchiyama, K. Mohri, K. Bushida, IEEE Trans. Magn. 33 (1997) 3355. [23] H. Chiriac, T.A. Ovari, Prog. Mater. Sci. 40 (1997) 333. [24] D.P. Makhnovskiy, L.V. Panina, D.J. Mapps, Phys. Rev. B 63 (2001) 144424. [25] A.P. Zhukov, J.M. Blanco, J. Gonzalez, M.J. Garcia Prieto, E. Pina, M. Vazquez, J. Appl. Phys. 87 (2000) 1402. [26] M.R. Britel, D. Menard, L.G. Melo, P. Ciureanu, A. Yelon, R.W. Cochrane, M. Rouabhi, B. Cornut, Appl. Phys. Lett. 77 (2000) 2737. [27] K. Kawashima, I. Ogasawara, S. Ueno, K. Mohri, IEEE Trans. Magn. 35 (1999) 3610.
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[28] V.A. Zhukova, A.B. Chizhik, J. Gonzalez, D.P. Makhnovskiy, L.V. Panina, D.J. Mapps, A.P. Zhukov, J. Magn. Magn. Mater. 249 (2002) 324. [29] D.P. Makhnovskiy, L.V. Panina, D.J. Mapps, J. Appl. Phys. 89 (2001) 7224. [30] L.V. Panina, K. Mohri, J. Magn. Magn. Mater. 157–158 (1996) 137. [31] D.P. Makhnovskiy, L.V. Panina, D.J. Mapps, Appl. Phys. Lett. 77 (2000) 121. [32] J.B. Pendry, A.J. Holden, D.J. Robbins, W.J. Stewart, IEEE Trans. Microwave Theory Tech. 47 (1999) 2075. [33] A.N. Lagarkov, A.K. Sarychev, Phys. Rev. B 53 (1996) 6318. [34] R.A. Shelby, D.R. Smith, S. Schultz, Science 292 (2001) 77. [35] R.J. Astalos, R.E. Camley, J. Appl. Phys. 83 (1998) 3744.
Journal of Magnetism and Magnetic Materials 272–276 (2004) 1452–1459
Magnetoimpedance in amorphous wires and multifunctional applications: from sensors to tunable artificial microwave materials Larissa V. Paninaa,*, Dmitriy P. Makhnovskiya, Kaneo Mohrib a
Department of Communication and Electrical Engineering, University of Plymouth, Drake Circus, Plymouth, Devon PL4 8AA, UK b Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan
Abstract This paper is aimed to give a generalized approach to magnetoimpedance (MI) in amorphous wires in relation to the applications in sensors and microwave materials, raising new ideas and concepts. The impedance change ratio per Oersted in Co–Fe wires runs into more than 100%/Oe at MHz frequencies. This makes wire elements unique for various sensing applications. The emphasis required here is on induced asymmetric and off-diagonal effects to produce highly sensitive and linear responses with respect to the sensed external forces (magnetic field, stress). The field sensitivity of the impedance in wires with a circumferential anisotropy remains very high even at the GHz range. The combination of microwave properties of wires with electrodynamic behavior of wire-composite materials is foreseen as resulting in completely new effects such as magnetic field-dependent dielectric response and microwave activity. Thus, the MI effect can be useful to design microwave composites with tunable properties including negative permittivity, band-gap structures and waveguides. r 2003 Elsevier B.V. All rights reserved. PACS: 75.30.Cr; 75.30.Gw; 75.40.Gb; 75.50.Cc; 75.50.Kj Keywords: Magneto-impedance; Surface impedance; Magnetic wires; Microwave materials; Tunability; Microwave activity
1. Introduction Interest in magnetic materials in the form of microfibers continues to expand owing to their strongly anisotropic electrical, magnetic and optical properties [1,2]. The applications range from using a tiny wire as a sensing element of sub nT magnetic field detectors or as a magnetic label to artificially structured wire-materials with unusual properties at microwave frequencies. A large number of these applications employ magnetoimpedance (MI) effect. The present paper gives a comprehensive analysis of MI in amorphous microwires, which is based on field-dependent surface impedance *Corresponding author. Tel.: +44-1752-232599; fax: +441752-232583. E-mail address: [email protected] (L.V. Panina).
tensor. This allows a variety of MI characteristics to be viewed from a generalized approach in a wide frequency range including microwave band. In fact, the concept of the surface impedance combines the micromagnetic analysis with special problems in electrodynamics representing a new branch of research. It also gives an accurate balance between physical principles and practical applications. Co-based amorphous wires with a circumferential (or helical) anisotropy exhibit large and sensitive magnetic response of a high-frequency voltage known as MI effect [3]. The sensitivity (impedance change ratio per Oersted) runs into more than 100%/Oe at MHz frequencies [4,5]. This makes wire elements unique for applications in magnetic sensors for low-field detection. A number of sensor prototypes based on self-oscillation or pulsed circuits have been developed, which operate at a
0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2003.12.1285
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2. Family of MI characteristics Thin amorphous ferromagnetic wires having a negative magnetostriction are very popular for MI applications. In the outer layer of the wire, an internal stress from quenching coupled with the negative magnetostriction results in a circumferential anisotropy and an alternate left and right circular domain structure [19]. In this case, the circular magnetization processes determining the MI behavior are very sensitive to the axial magnetic field [20,21]. Along with this, special types of anisotropy as a helical one can be established by a corresponding annealing treatment, which results in unusual asymmetric MI behavior and the so-called microwave activity in wire-based composite materials. Many experimental results on MI and magnetic sensor application utilize Fe4:35 Co68:15 Si12:5 B15 amorphous wires having a small negative magnetostriction in the range of lE 107 : The wires of the composition
Co72:5 Si12:5 B15 having l ¼ 3 106 are used for stressimpedance (SI) [22] sensors showing the strain-gauge factor of more than 2000 which is more than 10 times larger than that of the semiconductor strain gauge. Currently, there are basically two methods of wire fabrication techniques. The first one utilizes in-waterspinning method for which as-cast wires have a diameter of 125 mm: The wires are then cold-drawn down to 20– 30 mm; and finally annealed with a tension to establish a moderate tensile stress. The other technique produces amorphous wires in a glass cover by modified Tailor method which is based on drawing a thin glass capillary with molten metal alloy. The diameter of the metal core is ranging between 5–50 mm: In this case, different temperature expansion coefficients of glass and metal alloy result in a longitudinal tensile stress, which is needed for the circumferential anisotropy. The value of this stress can be easily controlled by the ratio of the diameters of glass cover and metal core. This is a simple one-step process allowing a strict control of properties in as-cast state [23]. First experiments on MI dated to 1993 utilized a simple concept of measuring the AC voltage across the wire subjected to an AC current i ¼ i0 expðjotÞ and a DC magnetic field Hex applied in parallel with the current. It was found, that even in the case of a circumferential anisotropy the MI behavior can be completely different depending on the current amplitude i0 and frequency o [3]. The typical characteristics are given in Fig. 1. It is seen that with increasing frequency the field plots transfer from a single peak type to that having two symmetrical peaks situated at the anisotropy field. However, for small i0 less than 1 mA; only twopeak behavior is observed with a quite deep minimum at zero field [24,25]. The effect of DC current Ib also results in similar change to the two-peak plots as illustrated in Fig. 2. This suggests that the central maximum in Fig. 1 is due to domain wall processes which are suppressed by either applying Ib ; decreasing i0 or increasing frequency. Very high field sensitivities in the range of 100%/Oe are typical of glass-coated wires under very low-current
Re duce d Volta ge , V0 / VR
frequency of several tens MHz and give the field resolution of 106 Oe for a stationary AC field detection [6–8]. To improve the sensor linearity and temperature stability the asymmetric and off-diagonal MI effects are of a special importance [8–10]. The field sensitivity of the impedance in wires with a circumferential anisotropy remains very high at radio frequencies and even in the GHz range [11–13], which leads to new physical effects and applications. Thus, surface acoustic wave devices combined with MI wires as loading impedances can be used for wireless identification systems [14,15]. The combination of microwave properties of wires with electrodynamic behavior of wire-composite materials is foreseen as resulting in completely new effects such as magnetic field-dependent dielectric response [16,17]. The MI effect can be useful to design microwave composites with tunable properties including negative permittivity, band-gap structures and waveguides. The paper is organized as follows. Section 2 outlines the family of MI characteristics obtained at different frequencies and excitation conditions emphasizing their role in various sensor applications. Section 3 introduces into the surface impedance approach with the results applied to a single-domain wire with arbitrary anisotropy. The model limitations may seem to be quite rigid, nevertheless, it is demonstrated that this simplified description gives a clear picture of the MI behavior and agrees well with the experiment and applications suggesting further practical ideas. The last section is devoted to a new area of employing MI in tunable artificial materials characterized by the effective permittivity depending on the external magnetic field in the microwave frequency band. The possibility of microwave activity known for chiral structures[18] is further discussed.
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5 4
FeCoSiB-wire
I 0 = 5 mA
30 µm dia m. l = 5 mm
σ a =2 kg/mm 2
f = 10 MHz
3 3 MHz 2 1 MHz
1 -8
-4
0 4 Axial field, Oe
8
Fig. 1. Voltage amplitude in 30 mm diameter CoFeSiB wire vs. external field with a frequency as a parameter. i0 ¼ 5 mA:
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1454 3.0 2.5
f = 20 MHz l = 5 mm i < 0.8 mA
I =0 b
90%/Oe
Vw , mV
ac
2.0
10 mA
1.5
25 mA 46.5%/Oe
1.0
23.5%/Oe -20
-10
0 H ex , Oe
10
20
Fig. 2. Voltage amplitude vs. Hex with Ib as a parameter in glass-coated wires having a metal core diameter of 48 mm: i0 o0:8 mA: The sensitivity of 90%/Oe is seen for lb ¼ 0:
280
0.5 GHz
Impedance |Z|, [Ω]
260
1 GHz
240 220 200 180 160
2.2 GHz
140 -20
-10
0
10
20
Axial magnetic field [Oe]
Fig. 3. High-frequency impedance plots vs. Hex in glass-coated wires.
excitation. Recently, MI at much higher frequencies was investigated [11–13]. In this case, proper care has to be taken for calibrating the total measurement cell. Typical impedance plots are shown in Fig. 3. The two-peak characteristics are still seen for hundreds MHz. Their low-field behavior ðHex oHK Þ preserves high sensitivity, whereas the characteristics flatten in the high-field region ðHex > HK Þ and for frequency f > 2 GHz they remain constant for any Hex > HK where HK is the effective anisotropy field. This unusual behavior can be referred to as valve-like for which the impedance becomes insensitive to the field when the wire is axially magnetized. There is no correlation with ferromagnetic resonance for which much higher fields would be needed [26]. We can conclude that the MI effect remains very sensitive in a wide frequency range; also the field-plot appearance may change substantially. Megahertz frequency MI characteristics, for which extremely high sensitivity can be reached, are utilized in various magnetic, current and stress sensors. Microwave MI is very promising for remote sensing and tunable microwave materials. Substantial amount of works on MI has been devoted to the asymmetric effects [4,9,24,27–29]. In the case of
sensor application, its linearity is an important feature. On the other hand, the MI characteristics presented in Figs. 1–3 are not only non-linear, but also shaped in a way that the operation near zero-field point can present serious problems. Generally, a DC bias field is used to set properly the operating point on the MI characteristics, which can be regarded as producing asymmetry with respect to the sensed field Hex : Therefore, for linear sensing asymmetrical MI (AMI) is of great importance. There are mainly two ways to realize AMI in amorphous wires. The first one is related to an asymmetrical static magnetic structure and the other is due to a dynamic cross-magnetization process. In a wire with a helical anisotropy, the magnetization responds not the same way to positive and negative Hex in the presence of DC bias current Ib [30]. Then, the AC voltage will have asymmetry with respect to Hex : The helical anisotropy can be induced by applying an external torsion or by annealing under a torsion. In some cases, a spontaneous helical anisotropy may exist as in the case of as-cast CoSiB amorphous wires. The experimental results on AMI in wires annealed under torsion are given in Fig. 4 [27]. With increasing Ib ; the shifted maximum in MI plots becomes more pronounced, which is accompanied by enhanced sensitivity. In this particular case AMI plots do not exhibit a hysteresis which is due to nonlinear AC excitation. If the amplitude of the AC current i0 is small, the MI plots in wires with helical anisotropy exhibit a substantial hysteresis which disappears only under the effect of large Ib [28,29]. Figs. 5 and 6 illustrate the other way to produce asymmetry in a highfrequency voltage response. This method is related to the dynamic cross-magnetization process of inducing a circulatory magnetization mj (with respect to the current i) by AC magnetic field hex parallel to i (produced by the coil current ic ) [31]. Due to the rotational mechanism of AC dynamics, this process is possible even for circumferential anisotropy but the domain structure must be eliminated by a small DC current. Typically, in the experiments the wire and the coil are connected in series ði ¼ ic Þ and excited by the same AC source. As frequency is increased, the contribution to the voltage due to hex becomes in the region of that induced by the current i itself, and can considerably change the voltage behavior vs. Hex as shown in Fig. 6. It has been recently proposed to utilize for sensor applications the voltage response Vc detected in the coil mounted around the wire whilst the wire is still driven by the passing current i [10]. This operation is based on the other cross-magnetization process mz ðhj Þ which causes the current flow to induce an AC axial magnetization and hence voltage Vc : The ratio Zc ¼ Vc =i may be referred to as off-diagonal impedance. In single-domain wires with the circumferential anisotropy the function Zc ðHex Þ is antisymmetrical having a near-linear region
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CoFeSiB am. wire
i = 15 mA
I =0
0
f = 1 MHz
I = 4.5 mA b
300
0
Vo ltage, V (mV)
b
I = 9 mA
100
b
-6
-4
-2
0
2
4
6
Axial field, H ex (Oe)
Fig. 4. AMI plots in torsion annealed wires (280 turn/m, 500 C) with Ib as a parameter. i0 ¼ 15 mA; a ¼ 25 mm:
H ex
hex
Ib + i
ic
Vw
Fig. 5. Mixed AC excitation using both the wire current i and the coil current ic :
6
Coil with 30 turns f = 10 MHZ
inp w
V /V
2 f = 3 MHZ
-20
-10
0 H ex , (Oe)
10
20
Fig. 6. Normalized wire voltage Vw vs. Hex at mixed excitation by ði; ic Þ for different frequencies.
around zero-field point. This characteristic can be used in linear sensing. A practical design of such a sensor is reported in Ref. [8] where the wire element is excited by a pulse current of C-MOS IC multivibrator. The output signal Vc is very small if no axial field Hex is applied. In the presence of the field, the voltage pulse rapidly increases and when the field is reversed the direction of the pulse is reversed as well. After rectification, antisymmetrical MI characteristic can be obtained with almost linear region within field interval 7HK :
3. Surface impedance in a magnetic wire The variety of MI characteristics discussed in previous section can be analyzed in terms of the generalized concept of surface impedance tensor [24]. The general voltage response ðVw ; Vc Þ from a magnetic material
ð1Þ
where r is a unit radial vector directed inside the wire. The field ht is set by the excitation conditions: the current i generates a circular magnetic field hj ¼ 2i=ca while the coil current ic produces an axial field hex ¼ 4pnic =c; where n is the number of turns per unit length in the coil and c is the velocity of light (Gaussian units are used). In ferromagnetic conductors, B# is a two-dimensional tensor even for the electrically isotropic case. In a wire with a uniform surface magnetization, the tensor B# is constant on the surface having Bzj ¼ Bjz (cylindrical coordinates are used). Using Eq. (1) and excitation conditions for the magnetic field, the voltage response is given by 2i Vw ¼ ez l ¼ Bzz Bzj hex l; ð2Þ ca Vc ¼ ej 2panl ¼
4
0
subjected to a high frequency excitation ði; ic Þ is given by the tangential components of electric field et taken at the wire surface. This field is related to the surface magnetic field ht via the surface impedance tensor B# et ¼ B# ðht rÞ;
200
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2i 2panl; Bjj hex þ Bjz ca
ð3Þ
where a is the wire radius, l is its length and c is the velocity of light. It is seen that only the first term in Eq. (2) corresponds to the usual case of Vw ¼ Zi where Z is the wire complex impedance. The second terms in Eqs. (2) and (3) are due to the AC cross magnetization processes mj hz and mz hj : Therefore, the impedance B# is the general characteristic describing the voltage response in the system excited by the external AC magnetic field h (of any origin). This approach is especially useful in understanding such effects as AC bias asymmetric MI (see Fig. 6) or microwave activity of wire-based composites. The calculation of B# is based on the solution of the Maxwell equations inside the conductor for the AC fields e and h changing in time as exp (jot) together with the equation of motion for the magnetization vector M: Taking a linear approximation with respect to the time-variable parameters e; h; m ¼ M M0 ; where M0 is the static magnetization and neglecting the exchange effects, the problem is simplified to find the solutions of the Maxwell equations with a given AC permeability matrix m# ¼ 1 þ 4p#w; m ¼ w# h: Further assumptions on dynamic magnetization processes are needed to determine m: # We use a single-domain model of a ferromagnetic wire where only the rotation of magnetization is taken into account. This model gives all the important features of the field dependence of B# : In general, the anisotropy axis nK has an angle c with the wire axis (z-axis) and the vector M0 is directed in a helical way having an angle y with the z-axis. Under these assumptions, the impedance matrix B# is expressed
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in the analytical form in the limits of strong skin effect and relatively week skin effect. The details can be found in Ref. [24], and here only the final results are listed (i) ‘‘Strong’’ skin effect d=a51 cð1 jÞ pffiffiffi ð m# cos2 ðyÞ þ sin2 ðyÞÞ; 4psd0 cð1 jÞ pffiffiffi ¼ Bzj ¼ ð m# 1Þ sinðyÞcosðyÞ: 4psd0
Bzz ¼ Bjz
ð4Þ
(ii) ‘‘Weak’’ skin effect d0 =ab1; a=dB1; k1 c J0 ðk1 aÞ 1 a 4 cm23 þ ; ð5Þ 4ps J1 ðk1 aÞ 54 d0 psa 4 ao a Bjz ¼ Bzj ¼ j m3 3c d0 hm m m m i c : ð6Þ 1 3þ 2 3 60 30 psa pffiffiffiffiffiffiffiffiffiffiffiffi Here J0:1 are the Bessel functions, d0 ¼ c= 2pso; Bzz ¼
m* ¼ 1 þ 4pw; w ¼ w2 4pw2a =ð1 þ 4pw1 Þ; m1 ¼ 1 þ 4p cos2 ðyÞw; m2 ¼ 1 þ 4p sin2 ðyÞw: m3 ¼ 4p sinðyÞ cosðyÞw; k12 ¼ m1 ð4p jos=c2 Þ; w1 ; w2 and wa are diagonal and off-diagonal components of the susceptibility tensor w# written in the co-ordinate system with z0 jjM0 ; s is the conductivity and d is the magnetic skin depth. The intermediate case is obtained by the extrapolation. Eqs. (4)–(6) demonstrate that the tensor B# depends on both the effective susceptibility w and static magnetization angle y: Their field behaviors explain all the essential features of the MI plots. For circumferential anisotropy, the parameter w peaks at the anisotropy field, which leads to the two-peak MI characteristics. Applying the DC bias current eliminates the domain structure decreasing the total permeability down to its rotational part and the minimum at zero field becomes deeper (see Fig. 2). The parameter w has a very broad dispersion region. At high frequencies (higher than the frequency of the ferromagnetic resonance fFMR ) w becomes insensitive to Hex although preserving its relatively high values (jmjE10 at GHz range), as shown * in Fig. 7. In this case the field dependence of the impedance is entirely related with that for the magnetization angle y: This explains the valve-like behavior of the impedance observed at GHz frquencies (Fig. 3). The analysis of the surface impedance clearly demonstrates that the condition of the ferromagnetic resonance is not required for the MI effect, contrary to the widely expressed belief [26]. In the case of a helical anisotropy, the DC magnetization is represented by asymmetric hysteresis curves.
Fig. 7. Effective permeability spectra in the GHz range showing no change for moderate Hex ¼ 10 Oe; HK ¼ 2 Oe:
Because of this, the impedance plots show a progressive asymmetry with increasing Ib : Typically, the helical anisotropy results in a hysteresis in MI plots due to hysteresis in DC magnetization. However, in certain cases the MI characteristics measured at relatively low frequencies and large values of i0 do not have a hysteresis despite a hysteresis in M0 ðHex Þ (the case of Fig. 4). This is explained by considering nonlinear magnetization dynamics which is characterized by the averaged permeability over a circular magnetization cycle induced by the driving current i: The off-diagonal components of B# change sign if M0 is reversed. This property is useful to realize linear field characteristics by detecting the coil voltage Vc : However, in wires with the circumferential anisotropy this voltage is very small if the wire has a domain structure since Bjz averaged over domains with the opposite magnetization is zero. A small DC current is needed to eliminate the domain structure. The use of pulsed current which has both low- and high-frequency harmonics is then more preferable. Furthermore, asymmetric MI characteristics can be obtained using a mixed excitation by ði; ic Þ and measuring the wire voltage Vw : In this case, the voltage Vw involves both Bzz and Bzj components, one of which is symmetric and the other is antisymmetric with respect to Hex with the resulting response of the type shown in Fig. 6. Without Ib ; the effects related to the off-diagonal components are possible only in a wire with a helical anisotropy.
4. Field-dependent microwave permittivity of wire-based composite materials In this section we consider the MI effect in relation to composite materials containing ferromagnetic wires. In general, artificial material with conducting fibers present a considerable interest since their dielectric response can exhibit various dispersive behaviors having very large values of the effective permittivity even for diluted
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systems. They can be represented by a 3D-array of continuous wires or a composite with randomly oriented wire pieces. The former has characteristic features of a metallic response to radiation, but in the GHz range it has a negative permittivity eeff ðoÞ below the normalized plasma frequency [32]. In the second case, finite-wire inclusions behave as electric dipole scatterers. Then, the effective permittivity eeff can have a resonant dispersion seen near the antenna resonance for an individual wire [33]. If the losses are not very large, the real part of eeff has negative values past the resonance in this case as well. The possibility to have a negative permittivity is very important for a recent trend to create materials with a negative refraction index [34]. If the skin effect in wires is strong abd; the effective dielectric response is determined by wire geometry and the permittivity of host material. In a general case aEd; the dispersion characteristics of eeff depend on the surface impedance. Therefore, in composite containing ferromagnetic wires exhibiting MI effect, the effective permittivity may depend on a static magnetic field [16,17] via the corresponding dependence of B# ; as was discussed in the previous section. Furthermore, this special behavior of eeff requires the electric field of the external radiation to be directed along the wire. However, in a wire with a helical magnetization the external axial magnetic field will induce the axial electric field due to the off-diagonal impedance Bzj : Such composite will have the so-called microwave activity typical of chiral structures [18]. This can lead to completely new effects such as rotation of the polarization plane. The further analysis is restricted by the case of composite materials containing short-wire inclusions [17]. Such composite is similar in many respects to a dielectric material since the wire inclusions play a role of elementary dipole scatterers, which are polarized with an electric field e0 of the incident electromagnetic wave. In general, the local field eloc in the composite material at the wire location differs from e0 but for a diluted system this can be neglected. It is further assumed that e0 is directed along the wire. This field will excite the axial current iðzÞ which will have a nontrivial dependence on z when the wavelength l is of the order of the wire length. The current distribution is found by solving the scattering problem with the impedance boundary conditions (1). To describe the external scattered field, the current iðzÞ can be regarded as a linear current if the antenna conditions ða5l; a5lÞ are held. The current distribution inside the conductor is taken into account via the impedance of form (5). It will be also a nontrivial charge distribution qðzÞ along the wire in this case, which determines the induced dipole moment in the wire. Using the continuity equation @iðzÞ=@ðzÞ ¼ joqðzÞ and boundary conditions ið7l=2Þ ¼ 0; the wire dipole moment Pz is found by integrating the current along
the wire Z j l=2 Pz ¼ iðzÞ dz o l=2
1457
ð7Þ
The wire polarisability is defined as aII ¼ Pz =ðpa2 le0z Þ: The analysis of the scattering problem shows that the current iðzÞ can be approximated by a linear differential equation of the second-order involving a certain damping caused by the radiation as well as internal resistive and magnetic losses. The latter appears in this equation via the wire surface impedance. Thus, as in the case of a Lorentz oscillator the polarizability aII has the following form: X An iGn o; ð8Þ aII ¼ 2 2 ðo res;n o Þ n where the summation is carried out over thepantenna ffiffi resonance frequencies ores;n ¼ 2pc=lres;n ; ¼ 2l e=ð2n 1Þ is the resonance wavelengths, An and Gn are the amplitude and damping parameter respectively. The first resonance n ¼ 1 with the lowest frequency gives the main contribution to the polarizability. Each Gn can be decomposed into two parts related to the radiation and internal losses, respectively. The latter may depend on the external magnetic field Hex via the surface impedance Bzz : Thus, in the vicinity of the antenna resonance the polarisability aII will depend on Hex if the radiation losses are not very large (not very strong skin effect). In a ferromagnetic wire having nonzero Bzj ; the electromagnetic wave with magnetic field h0 along the wire will induce Pz as well. In this case eloc ¼ Bzj h0z implying that Pz is produced by a transverse electric field of the incident radiation. The corresponding polarisability a> will be of the form of (8) with An depending on Bzj : In fact, its dispersion will be different since Bzj is a function of frequency. Also, a> 5aII owing to Bzj 51: However, this cross-polarization process is principally important resulting in the constituent equations of the type adopted in chiral media. Let us now consider a thin composite slab parallel to the plane XY having randomly oriented wires. The effective permittivity matrix is obtained by averaging the local dipole moments over the orientation and is of the form 1 0 e2 0 e1 C B e#eff ¼ @ e2 e1 0 A; ð9Þ 0
0
e
where e1 ¼ e þ 2ppaII ; e2 ¼ 2ppa> and p is the volume concentration of wires. From Eq. (9) it follows that owing to the off-diagonal impedance of magnetic wires the composite system is characterized by the permittivity matrix having off-diagonal terms. In fact, this is a very unusual matrix with all the transverse components depending on a static magnetic field. For normal
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incidence, the displacement vector D is of the form D ¼ e1 e0 þ kðk e0 Þ;
k ¼ e2 c=o
which is similar to that for medium with optical activity, such as spring-like, helical sort of structures. In our case, it can be referred to as microwave activity and its helical structure is supported by the DC helical magnetization in a wire. The existence of the off-diagonal components will result in such effects as the rotation of the polarization plane. They are at a very early stage of investigation. The spectrum of the diagonal component of e#eff is shown in Fig. 8 for two fields Hex ¼ 0 and Hex > HK : The field effect shows up in changing the character of the dispersion curves. In the absence of Hex the dispersion curves are of a resonance type. Applying a magnetic field Hex > HK ; the impedance is increased and, as a consequence, the internal losses in the inclusion, which results in the dispersion of a relaxation type. The field-dependent permittivity matrix e#ðHex Þ can be used in tunable microwave covers. The energy absorption in such kind of a composite is rather strong, therefore it has to be sufficiently thin to be used as a wave passage. In the case of a slab of 300 mm thick, the change in reflection and transmission coefficients is in the range of 15%. Another promising application suggested is employing these composites as an internal cover in the partially filled waveguides or layered thinfilm waveguides with a ‘‘dielectric/ferromagnetic’’ structure [35]. Such a waveguide system, by analogy to that already designed for the waveguides with ferromagnets,
28
ε = 16
22 eff
res
ex
24
ε/
~ 3.73 GHz )
H =0 (f
26
p = 0.001%
l = 10 mm
20 18
H >H (f ex
16 14
f
12
res
K
~ 4.32 GHz )
res
10 0
2
4
(a)
6
8
10
12
14
f ( GHz ) 16 14 ex
10
ε //
eff
ε = 16 l = 10 mm p = 0.001%
H =0
12 8 6
H >H
4
ex
K
8
10
2 0 0
(b)
2
4
6
12
14
f ( GHz )
Fig. 8. Effective permittivity of wire-composites as a function of frequency for Hex ¼ 0 and Hex > HK : The parameters used: p ¼ 0:001%; e ¼ 16; l ¼ 1 cm:
can be used for tunable filters and phase shifters operating up to tens GHz.
5. Conclusion The concept of magnetoimpedance (MI) has stimulated a new research combining micromagnetic and electrodynamic problems. Depending on magnetic structure, frequency and excitation method, various types of MI characteristics have been realized. The sensitivity remains very high even at the GHz frequency range. This suggests many potential applications in sensors, magnetic tags, microwave materials and devices.
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