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Formalism to Optimize Magnetic Noise in Giant Magnetoimpedance-Based Devices Luiz G. C. Melo1 , D. Ménard1 , A. Yelon1 , Life Fellow, IEEE, L. Ding2 , S. Saez2 , and C. Dolabdjian2 Département de Génie Physique and Regroupement Québecois des Matériaux de Pointe, École Polytechnique de Montréal, Montréal, QC H3C3A7 Canada Groupe de Recherche en Informatique, Image, Automatique et Instrumentation de Caen, GREYC CNRS and University of Caen, 14050 Caen Cedex, France This paper describes a theoretical study of the sensitivity and intrinsic noise response in a longitudinal giant magnetoimpedance sensor of a magnetically soft microwire. We show that the ratio of the intrinsic magnetic noise voltage spectral density to the sensitivity is , where depends upon the internal field and frequency and is the angle between the equilibrium magnetization proportional to and the axial applied static field. The model allows us to calculate the influence of the easy axis angle and the applied fields on sensor Hz level. response and to obtain a working point. We show the conditions necessary to reduce the intrinsic noise level to the
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Index Terms—Giant magnetoimpedance (GMI), magnetic field sensors, magnetic noise, soft magnetic wires.
I. INTRODUCTION HE STUDY and development of sensor devices based on giant magnetoimpedance (GMI) in magnetically soft materials has been the subject of intense investigation since the first reports on the effect in the 1990s [1]. GMI magnetic field sensors measure the changes of the impedance when the sensing element is subjected to external quasi-static applied fields [2]. Low cost, easy implementation of a compact design, and the possibility of attaining high sensitivities are the main advantages of the GMI magnetometers, compared to other types of magnetometers such as Fluxgate, Induction coil, GMR, and SQUID [3]. These qualities are principally due to the unique physical properties of amorphous magnetic wires [4], notably low magnetostriction CoFeSiBNb wire. The low anisotropy of the wires results from the internal magnetoelastic coupling during rapid quench. Although considerable effort has been expended on optimizing the GMI signal and sensitivity [5], relatively little attention has been paid to the study of noise characteristics in GMI elements [6]. Miniaturization of a device yields thermal instabilities, leading to magnetization fluctuation noise in its electrical response. This will ultimately degrade its performance. We have analyzed the magnetic noise spectral density (MSD) of a GMI sensor consisting of a monodomain magnetic wire with a circumferential easy axis, in the quasi-static regime [7], and found that the magnetic noise level depends upon the anisotropy field as and that, under certain circumstances, the magnetic noise may contribute significantly to total sensor noise which also depends upon the thermal Johnson noise. However, it is known that a slight variation of the easy axis from the circumferential direction considerably changes the GMI response [5]. Here, we propose a generalization of the formalism for the calculation of the intrinsic sensitivity and magnetic noise of a GMI sensor, with a general helical anisotropy, operating in the megahertz range. The active element is a wire, treated as an anisotropic magnetic cylindrical conductor. Emphasis is put on
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Digital Object Identifier 10.1109/TMAG.2007.893791
the influence of static applied fields and the easy axis direction on MSD, which is the ratio of the magnetic noise voltage spectral density (VSD) to the sensitivity and has dimensions T/ Hz, which we evaluate in the white noise regime. II. THEORETICAL MODEL A. Magnetoimpedance Here, we briefly review the theory of magnetoimpedance (MI) in magnetic conductors [8]. We begin by defining the static equilibrium condition, followed by the analysis of the transverse permeability, whose dependence on field and frequency is the main mechanism leading to the GMI effect. The system under investigation is a cylindrical magnetic wire of radius , with a single domain magnetic structure and a uniform helical magnetization. Demagnetizing energies are neglected because the demagnetization factors along the axial and circumferential directions are negligible. The static magnetization has no radial component. and are the angles which and the easy axis make with the direction on the plane of a cylindrical coordinate system. Then, the uniaxial anisotropy energy of the wire can be described , where is the anisotropy constant. Static by fields, and , applied along the and directions, respecand tively, produce the magnetostatic energies , where is the modulus of , and is the is permeability of free space. The circumferential bias field the field at the surface of the wire produced by a dc current along the direction. Minimization of the total free energy density with respect to yields (1) where are the reduced static fields, is the anisotropy field. and In an MI measurement, in addition to the applied static fields, the wire is subjected to a dynamic magnetic field produced by a high-frequency longitudinal current. Assuming that the amplitude of the alternating signal is much smaller than the static fields, the solution of the Landau–Lifshitz or Gilbert and Maxwell equations satisfying the boundary conditions of the electromagnetic
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MELO et al.: FORMALISM TO OPTIMIZE MAGNETIC NOISE IN GIANT MAGNETOIMPEDANCE-BASED DEVICES
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fields allows us to calculate the relative transverse permeability . For applied fields that are smaller than the saturation magis approximately given by [8] netization (2) where
is the angular frequency of the driving current, is the gyromagnetic ratio, is the Gilbert damping , where , constant, and (3)
is the internal stiffness field resisting rotation produced by the external fields. For a given equilibrium configuration of the static magnetization satisfying (1), the internal field given by (3) modifies the permeability as a function of frequency through (2). As a result, will change the penetration depth of the propagating electromagnetic fields, thus changing the impedance of the conductor , leading to the magnetoimpedance effect. For an unsaturated anisotropic wire, is given by [8]
p
(3 )
Fig. 1. Modulus of nondimensional intrinsic magnetic noise, j j as a function of modulus of intrinsic GMI sensitivity j j, for easy axis angles of ; ; , and . Applied field h is an implicit variable in curves and it is varied from saturation to zero, as indicated by arrows. Points a; b; c, and dpshow the point where sensitivity is maximum. In above curves, point where j and j j simultaneously tend to zero corresponds to value of h j where Z=Z is maximum (zero derivative).
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(3 ) 1
3
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3
(4) , and , and are Bessel functions of the first kind, and The propagation constant is related to the penein the conductor of conductration depth tivity , as . In (4), is the is the dc resistance of the wire) nonmagnetic impedance ( or the impedance of a nonmagnetic wire of the same radius and conductivity as the magnetic wire [9]. Using (4), the magne, which is toimpedance ratio commonly used to indicate the relative magnitude of the GMI . effect [5], is
where Hz, where mension is the temperature and In (7), the derivative
B. Intrinsic GMI Sensitivity
is the nondimensional intrinsic magnetic noise. In (8), , which, according to (2) and (3), depends upon the internal field and the frequency of the driving current. From (5), (7), and (8), the ratio of the intrinsic magnetic noise voltage spectral density to the sensitivity is . Therefore, the MSD, in Hz is given by
where where
(V/T), where The GMI sensitivity is the rms amplitude of the alternate current, in units of A, indicates the rate of change of with respect to the applied field . can be written as (5) where
is a constant of diis the Boltzmann constant, is the volume of the wire. can be written as
, where is given by (6). The term is calculated by differentiating (1) with respect to and . Thus, using (3) to produce , where (8)
(9)
, and (6)
is a nondimensional quantity whose modulus may be viewed as the nondimensional intrinsic sensitivity of the GMI sensor. C. Intrinsic GMI Magnetic Noise For a GMI-based sensor, the magnetic noise voltage spectral density (VSD), in Hz, due to the thermal fluctuations in magnetization, may be calculated using the fluctuation-dissipation theorem, yielding [7] (7)
III. DISCUSSION As described in Section II-A, for a given and , and for a given set of , the solution of (1) yields the magnetization equilibrium angles . These are subsequently used to calculate through (3). Next, for a given frequency the internal field of the alternate current , the permeability (2), the impedance (4), and magnetoimpedance ratio are evaluated. From (5), the sensitivity is then calculated as a function of , permitting the subsequent computation of the intrinsic magnetic noise , through (8). Fig. 1 shows as a function at 10 MHz, for , and . The field of
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 6, JUNE 2007
jj j
Fig. 2. Ratio of intrinsic magnetic noise and maximum sensitivity (3 ) = 3 and applied field where sensitivity is maximum, h , as a function of easy axis angle.
, which is an implicit variable in Fig. 1, was varied from positive saturation to zero, as indicated by the arrows in the . We have used the following paramecurve for ters in the calculations of Fig. 1: A/m (1 Oe), ( m) kA/m, kHz/(A/m), and m, which are typical of the CoFeSiBNb amorphous wires. The , and in Fig. 1 indicate the maximum sensitivity points , related to each . For , the point corresponds to which yields a relative sensitivity of % , and, from (9), an expected intrinsic K, of cm Hz, MSD, at where is the length of the wire in centimeters. For example, a 0.5-cm-long wire of 18- m radius ( mm ) Hz, corresponding yields an MSD of to a very low noise magnetometer [6]. The SQUID intrinsic noise level at cryogenic working temperature (4.2 K or 77 K) is Hz (1 cm area) [10]. around 1 to 10 For simplicity, the working point of the sensor is taken . as the applied field where the sensitivity is maximum, The corresponding working points of the curves of Fig. 1 are , and . At these points, we notice that and decrease when decreases. Since both the noise decreases more rapidly than the sensitivity, the ratio will decrease asymptotically, as shown in Fig. 2. Fig. 2 also shows the variation of as a function of . For the range of easy axis angles investigated, the above results as the easy axis angle which optimizes and Fig. 2 yield the sensor response. For which, from (9), leads to an intrinsic MSD of , or cm Hz, occurring at (Fig. 2). At this point, the maximum relative sensitivity is % .
The intrinsic MSD can be further reduced by the ap. Following plication of a circumferential dc bias field increases, the vector will move towards (9), as the circumferential direction ( tends to 90 ), decreasing . For (corresponding to the field at the surface of the wire produced by a dc current of 9 mA) cm Hz, at and a maximum sensitivity of % , ob. Although the above is 50% served at cm Hz), the lower than that for sensitivity is drastically reduced from that obtained with % . The reduction of noise by the application of is likely to be more effective for easy axis . Further investigation angles in the range is necessary to confirm these results. Moreover, it is worth noticing that the conditioning sensor electronics is presently an important restriction to the achievement of the theoretical MSD levels. The best experimental results so far are within the cm Hz level [11]. 1/ We have analyzed the behavior of the magnetic noise associated with the GMI response of an anisotropic magnetic wire. By taking the working point as the field producing the maximum sensitivity, we have found that the lowest MSD occurs at . Different choices of the working point and their impact upon the noise and sensitivity response will be discussed elsewhere. ACKNOWLEDGMENT This work was supported in part by NSERC, Canada. REFERENCES [1] K. Mohri, K. Bushida, M. Noda, H. Yoshida, L. V. Panina, and T. Uchiyama, IEEE Trans. Magn., vol. 31, pp. 2455–2460, 1995. [2] K. Mohri, T. Uchiyama, L. P. Shen, C. M. Cai, L. V. Panina, Y. Honkura, and M. Yamamoto, IEEE Trans. Magn., vol. 38, pp. 3063–3068, 2002. [3] D. Robbes, C. Dolabdjian, and Y. Monfort, J. Mag. Magn. Mat., vol. 249, pp. 393–397, 2002. [4] H. Hauser, L. Kraus, and P. Ripka, IEEE Instr. Meas. Mag., pp. 28–32, 2001. [5] Knobel, M. Vazquez, and L. Kraus, Handbook of Magnetic Materials, K. H. J. Buschow, Ed. Amsterdam, The Netherlands: Elsevier, 2003, vol. 15, pp. 497–563. [6] D. Robbes, C. Dolabdjian, S. Saez, Y. Monfort, G. Kaiser, and P. Ciureanu, IEEE Trans. Appl. Supercond., vol. 11, pp. 629–634, 2001. [7] D. Ménard, G. Rudkowska, L. Clime, P. Ciureanu, A. Yelon, S. Saez, C. Dolabdjian, and D. Robbes, Sens. Actuators A, vol. 129, pp. 6–9, 2006. [8] Menard and Yelon, J. Appl. Phys., vol. 88, pp. 379–393, 2000. [9] In the limit of high applied fields, (2) yields 1. Also, k 1, so that (4) produces Z = Z . k ; = [10] J. Clarke, “SQUID fundamentals,” in SQUID Sensors: Fundamentals, Fabrication and Applications, H. Weinstock, Ed. New York: NATO ASI series E: Applied Sciences, 1996, vol. 329, pp. 1–63. [11] L. Ding, S. Saez, C. Dolabdjian, L. G. C. Melo, D. Menard, and A. Yelon, Sensor Lett., to be published.
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Manuscript received October 31, 2006 (e-mail:
[email protected]).
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