A Dynamic Calculation of the Responsivity
of Monodomain Fluxgate Magnetometers
J. G. DEAK and R. H. KOCH
IBM T. J. Watson Research Center, Yorktown Heights, NY
G. E. GUTHMILLER and R. E. FONTANA, Jr.
IBM Almaden Research Center, San Jose, CA
Abstract
A model describing the dynamic response of a single-domain fluxgate
magnetometer over a wide range of operating conditions in terms of a single
measurement of a hysteresis loop or the permeability of the fluxgate’s ferromagnetic core
is presented. The model is based on the Landau-Lifshitz-Gilbert equation, which
describes the dynamics of a coherently rotating ferromagnet. Measurements of the
response of a permalloy thin-film fluxgate and a molybdenum-permalloy tube fluxgate
demonstrate the accuracy and limitations of the model.
1
Fluxgate magnetometers are extremely sensitive room-temperature vector
magnetic-field sensors that are ideally suited for applications that require high sensitivity
and robust performance under extreme conditions. Typical applications for these sensors
have included submarine detection, surveillance, non-destructive testing, satellite
orientation, compasses for vehicle guidance systems, and precise measurement of
variations in the Earth’s magnetic field. In spite of a long history of successful
applications, a quantitative theory of the factors that limit the sensitivity of these devices
has not been developed. This is in part due to the nonlinear response of the fluxgate’s
ferromagnetic core to excitation fields and also to a wide variety of fluxgate
implementations with different core geometries and materials which can result in
different dominant noise producing mechanisms.
A fluxgate magnetometer is composed of a ferromagnetic core, which in the
simplest implementation, is wound with two coils. The two core geometries that we have
simulated are shown schematically in Figure 1. One coil, the drive coil, is used to
produce an alternating magnetic field, Hdrive, which periodically saturates the core first in
one and then in the opposite direction. The drive coil for the thin film fluxgate is not
shown in Figure 1. The second coil, the pick-up coil, is used to detect the change in
magnetic flux in the core. In the absence of an offset field, Hdc, the drive field saturates
the core symmetrically around zero field, so that the voltage induced in the pick-up coil
only contains odd harmonics of the drive frequency, ωο. When Hdc is nonzero, the core
remains in saturation longer for one direction of Hdrive, than the other. This asymmetric
saturation produces even harmonics, whose phase and amplitude are indicative of the
polarity and magnitude of Hdc. The second harmonic is typically used as a measure of
2
Hdc.1 The second harmonic is used, as opposed to the 0’th for instance, because of the
difficulty of inductively coupling to and measuring a signal at or near zero frequency.
Recently, improved noise performance on the order of 1 pT/rtHz at 1 Hz has been
demonstrated in fluxgates that keep the sensor’s ferromagnetic core saturated during the
entire drive cycle. 2,3 The noise reduction in these “single-domain” fluxgate
magnetometers is due to suppression of Barkhausen noise.2 The magnetization noise in
these single-domain fluxgates was found to be approximately independent of drive
conditions if Hdrive is large.4 This suggests that the noise is intrinsic to the core material
and not a result of residual domains. It is possible that the type of noise in a singledomain fluxgate magnetometer also limits the performance of other well-optimized
fluxgate designs. In order to develop a foundation for understanding the origin of this
residual noise, we present a dynamic model of the response of a single-domain fluxgate.
The model is based on the Landau-Lifshitz-Gilbert (LLG) equation, which describes the
dynamics of a coherently rotating ferromagnet.5
In SI units, the most general form of the LLG equation is
dM
= -γM x µoHeff ,
dt
(1)
where M is the magnetization of the material, γ the gyromagnetic constant, and Heff is an
effective magnetic field, which includes contributions due to the external field,
anisotropy, demagnetization, and a field describing the damping of the precession of M.
Heff may be expressed as
Heff = H -
αeff dM
,
γµοMsat dt
(2a)
3
i
i
i i
H = Ho - n M ,
(2b)
where i = (x,y,z) is an index representing a vector component, Ho = Hext + Hdrive(t),
ni = ki + di the sum of anisotropy and demagnetization constants, Msat the saturation
magnetization, and αeff an effective damping parameter. Here the anisotropy is expressed
in a dimensionless form k = 2K/µoMsat2. K is the anisotropy constant in SI units (J/m3).
We define αeff as an effective damping parameter for a macroscopic ferromagnetic object.
In this paper we use the Gilbert and Kelly form of the LLG equations since the value of
αeff exceeds 1. To convert αeff and γ to the values for the Landau and Lifshitz form, both
be divided by (1+α2). The value of αeff is intended to reflect all loss mechanisms,
including eddy currents and nonlinear effects. It is therefore not intrinsic to the material,
and its value, which can be several orders of magnitude larger than the intrinsic values
measured by ferromagnetic resonance,6 depends on excitation conditions as well as
sample geometry. In order to model a fluxgate’s response, the LLG equation is integrated
to find the time-dependent magnetization, M(t), which is the field that induces the voltage
in the fluxgate’s pickup coil.
Here we model the response of single-domain fluxgate magnetometers. The
orientation of the fields used to obtain single-domain operation is shown in Figure 1. Our
single-domain fluxgates have an easy axis perpendicular to Hdrive, and thus an easy-axis
bias field is included in the model. There is, however, no fundamental reason that a
fluxgate model based on the LLG equation should be restricted to this geometry. The
responsivity of a single-domain fluxgate to a dc field, Hdc, may be defined in terms of the
magnetization as
4
M2
Hdc =
Lim
æ |µ2(Hdc,Hdrive,Heasy)| |Hdrive|ö
ç
÷,
Hdc→ 0 è
Hdc
ø
(3)
where M2 is the second harmonic of M(t), Heasy a dc easy-axis bias field, Hdrive the peak
amplitude of the sinusoidal drive field, and |µ2(Hdc,Hdrive,Heasy)| is the magnitude of the
second-harmonic complex permeability induced by Hdc. The second-harmonic
permeability is defined as follows
µ2(Hdc,Hdrive,Heasy) = µ 2’(Hdc,Hdrive,Heasy) + iµ2”(Hdc,Hdrive,Heasy) , (4a)
with
µ m’ =
τ
1
ó M(t) sin(mωοt)dt
τ õ
0
µo Hdrive
,
(4b)
and
µ m” =
τ
1
ó M(t) cos(mωοt)dt
τ õ
0
µo Hdrive
,
(4c)
where τ is the period of Hdrive and m = 2. The responsivity is defined in terms of
magnetization rather than voltage because the pick-up coil voltage is not an inherent
property of the core. In order to convert to a more familiar form, M2/Hdc can be scaled as
follows
M2
æ Volt ö
Responsivity çturn Tesla÷ = 2ωο . Acore . Geometric factor . H ,
è
ø
dc
(5)
5
where ωο is 2π times the drive frequency, Acore is the cross-sectional area of the
ferromagnetic core, and the geometric factor is constant less than one describing the
coupling between the pick-up coil and the core.
The key to modeling the fluxgate’s response using the LLG equation is to
accurately determine the anisotropy, demagnetization, and effective damping parameters
in Equation 2. The effective damping parameter can be determined from either a M(H)
loop recorded at one frequency or more simply as we show below from the first-harmonic
permeability. In most cases, a single determination of αeff is sufficient to describe a
single-domain fluxgate’s response over a wide range of easy axis bias fields, drive
frequency, and drive amplitude.
As a demonstration of the technique, we solve the LLG equation for hard-axis
permeability of a thin film as a function of a Hdrive, αeff, ωo, and dc easy-axis bias field.
We do this for clarity, since in practice, the LLG equation can be solved numerically for
the first-harmonic permeability in any arbitrary geometry and configuration of fields.
When a small oscillating field, Hzsin(ωot), is applied along the hard axis of a
ferromagnetic film, the LLG equation can be reduced to
αeff dMz(t)
æ
ö
+ µoçHeasy + (k+d) Msat ÷ Mz(t) = Msat µo Hz sin(ωot) .
dt
è
ø
γ
(6)
Here Mz(t) and Hz are the time-dependent magnetization and the amplitude of the drive
field in the hard direction. Note that Equation (6) is first order and is intended for the
overdamped limit where αeff is large. Equation (6) is easily solved for the complex
permeability, yielding
6
Msat
µstatic = (k+d) M + H
sat
easy
(7a)
-1
µstatic
µ1’ =
≈ µstatic
-2
æ ωoαeff ö2
µstatic + ç
÷
èγµοMsatø
(7b)
æ ωoαeff ö
-ç
÷
èγµοMsatø
æ ωoαeff ö 2
µ1" =
≈ -ç
÷µ
-2
æ ωoαeff ö2
èγµοMsatø static
µstatic + ç
÷
èγµοMsatø
(7c)
at low frequency. Here, µstatic is the permeability of the single-domain ferromagnetic core
in the limit of zero frequency and small Hz. Since Heasy is known, if necessary, Msat can
be determined from µ1’, so that αeff can be determined from µ1”.
Simulations of the magnitude of M2/Hdc for fluxgates made from permalloy thin
films and molybdenum permalloy tubes are presented below in order to demonstrate the
accuracy and limitations of the single-domain fluxgate model. The simulations are based
on αeff determined from simulations of the hard-axis M(H) loops of the tubes and from
measurements of the permeability of the thin films. All parameters used in the
simulations are summarized in Table 1. The films used in this study were composed of
82-18 permalloy that was sputtered onto glass substrates in the presence of a magnetic
field in order to induce an in-plane uniaxial anisotropy. The films were 4700 Å thick and
were patterned into a 0.025 m diameter circle. The tubes were machined from 4-80-15
molybdenum permalloy rods to a length of 0.05 m with an outer diameter of 0.0038 m
and a wall thickness 0f 0.00051 m. After machining, a circumferential uniaxial anisotropy
was induced in the tubes by vacuum annealing in the presence of a circumferential
7
magnetic field. When operated as a fluxgate, an easy-axis bias field large enough to keep
the samples saturated at all times during the drive cycle, was applied to both the films and
the tubes. This was done in order to keep them from breaking up into domains.2
Low-frequency hard-axis M(H) loops for the permalloy thin film are shown in
Figure 2(a) and those for the molybdenum permalloy tubes are shown in Figure 2(b). The
M(H) loops in Figure 2(a) are for two different orientations of Hdrive with µoHeasy = 0 T,
while those in Figure 2(b) are for two different values of Heasy when Hdrive is applied along
the hard-axis. When driven along the hard direction with µoHeasy = 0 T, both the films
and the tubes show linear response when unsaturated. This indicates that the
magnetization behaves as one would expect for coherent rotation,7 so that the singledomain fluxgate model should be applicable to both core geometries.
A measurement and simulation of the in and out-of-phase components of the
complex permeability, µ1’ and µ1”, of the permalloy thin film are shown in Figure 3.
Here, Hdrive is applied in the hard direction at a frequency of 22300 Hz and µoHeasy =
0.0001 T. This orientation of the fields and the amplitude of Heasy were chosen to insure
that the dynamic mode of the film during the permeability measurement is the same as
that in the fluxgate responsivity measurement presented below. Using permeability data
in Figure 3 and the saturation magnetization determined from Figure 2(a), equation (7)
gives αeff = 22.5 and k = 0.0009. In order to show that equation (7) is consistent with the
LLG equation, simulations of the permeability determined from equations (1), (2), and
(4), using the above αeff and k values are shown as dashed lines in Figure 3.
Figure 4(a) shows the simulated and measured magnitude of the responsivity of
the permalloy thin film fluxgate using αeff and k determined from Figure 3. The symbols
8
are actual data from the thin-film fluxgate measured at four different Heasy values and a
frequency of 22420 Hz. The dashed lines are a simulation of equation (3). In spite of the
fact that αeff was determined at one Heasy field, the simulation is capable of describing
M2/Hdc over a wide range of fields.
Figure 5 is a plot of the simulated magnitude of M2/Hdc and the frequency, fpeak =
γµοMsat/(2παeffµstatic), where µ1”, determined from Equation (7c), is a maximum as a
function of αeff. The phase of M2/Hdc with respect to Hdrive shifts strongly in the vicinity
of fpeak. As long as one is only interested in the magnitude of M2/Hdc, however, an
accurate value of M2/Hdc may be determined for any value of αeff chosen so that fpeak is
much greater than the drive frequency. It is generally a good idea to operate a fluxgate at
a frequency lower than fpeak, since the magnitude of M2/Hdc tends to become small at
higher frequency.
Figure 4(b) shows the simulated and measured magnitude of M2/Hdc for the
molybdenum permalloy tube fluxgate as a function of Hdrive for two values of Heasy. In
this case, αeff, k, and Msat were determined by simulating the M(H) loop with µoHeasy = 0
T, shown in Figure 2(b). In Figure 4(b), Hdrive was applied as a 260 Hz triangle wave.
Because in this case, the change in permeability of the tube during the drive cycle is large
enough to alter the drive waveform, the model was modified to include the inductance of
the drive coil and the capacitance of a capacitor connected in series between the drive
amplifier and the drive coil. With these modifications, the agreement between
measurement and simulation is good enough to be useful, but it is not as good as that for
the thin film fluxgate. There are several reasons why this might be the case. First, note
that in the case of the thin film, d is the same in all directions in the plane of the
9
magnetization. It is therefore not likely that the distribution of the magnetization changes
much during the fluxgate’s drive cycle. In the case of the tube, d differs by several orders
of magnitude between the easy and hard directions, so that as the magnetization
configuration is changed, the single domain approximation for the core of this fluxgate is
not as good as in the case of the thin film core. A second consideration deals with the
difference between feddy and fpeak shown in Table 1. The frequency, feddy ≈ ρ/(2πµot2) for
the tube and feddy ≈ 2ρ/(πµot2) for the film, is the frequency where Hdrive would just fully
penetrate the core if it were composed of a normal metal (µstatic = 1). This is the frequency
where µ1” would exhibit a maximum if all dissipation was due to eddy currents.8 Here, ρ
is the resistivity, and t is the sample thickness. If fpeak is near feddy, then damping is
dominated by eddy currents. Since eddy current damping is dependent on µ(Hdrive), αeff
will depend on Hdrive. fpeak for the tube is orders of magnitude closer to feddy than it is for
the film, indicating that eddy currents play a much more important role in the tubes than
in the films. In summary, it is clear that any process which leads to nonlinear damping is
difficult to simulate. In particular, one should be careful when simulating fluxgates
where the d values relevant to the magnetization during the drive cycle are orders of
magnitude different, and where eddy current damping dominates.
A simple extension of the single-domain fluxgate model would be to use the
stochastic LLG equation, which includes thermal noise as a Langevin term.9 Although a
model of this type cannot describe non-equilibrium noise, preliminary work with the
stochastic model has suggested a resolution limit of several hundred fT/rtHz for
permalloy fluxgates due to thermal noise if all non-equilibrium noise sources are
suppressed.4
10
A dynamic model based on the LLG equation describing the response of a singledomain fluxgate magnetometer over a wide range of operating conditions in terms of a
single measurement of a hysteresis loop or the permeability of a fluxgate’s ferromagnetic
core was presented. Measurements of the response of a thin-film permalloy fluxgate and a
molybdenum permalloy tube fluxgate demonstrate the model’s accuracy and utility in
studying fluxgate response. We have found a characteristic frequency, fpeak, above which
a fluxgate’s responsivity decreases. The single-domain fluxgate model breaks down
when αeff is dependent on drive conditions, and we show evidence suggesting that this is
likely to occur when damping becomes dominated by eddy currents or when the
demagnetizing fields are strongly dependent on relevant orientations of the
magnetization. In either of these cases, it is probably necessary to determine the
dependence of αeff and di on drive conditions. Finally, we suggest that although the
model was intended for single-domain fluxgates, it should provide a good approximation
to the response of a fluxgate made from a multidomain core, if the Hdrive dependence of
αeff and di are included in equation (2).
We would like to thank J. R. Rozen and P. R. Duncombe for help with the
preparation of the molybdenum permalloy tubes and F. Milliken for reviewing the first
draft of this paper.
11
References
1. F. Primdahl, IEEE Trans. Magn. MAG-6, 376 (1970); P. Ripka, Sensors and Actuators
33, 129 (1992).
2. J. Deak, A. H. Miklich, J. Slonczewski, and R. H. Koch, Appl. Phys. Lett. 69, 1157
(1996).
3. Deak, Koch unpublished data on 6-82 Molypermalloy sensors with 1.5 pT/rtHz at 1
Hz.
4. R. H. Koch and J. Deak (unpublished).
5. L. Landau and E. Lifshitz, Phyzik. Z. Sowjetunion 8, 153 (1935); T. L. Gilbert and J.
M. Kelly, Proceedings of the Pittsburg Conference on Magnetism and Magnetic Materials
(American Institute of Electrical Engineers: New York) p. 253 (1955); T. L. Gilbert,
Phys. Rev. 100, 1243 (1955).
6. R. L. Conger and F. C. Essig, Phys. Rev. 104, 915 (1956).
7. B. D. Cullity, Introduction to Magnetic Materials, (Addison-Wesley, Reading, MA,
1972).
8. J. R. Clem in Magnetic Susceptibility of Superconductors and Other Spin Systems,
(Plenum:New York) p. 177 (1992).
9. W. F. Brown, Phys. Rev. 130, 1677 (1963); D. A. Garanin, Phys. Rev. B. 55, 3050
(1997).
12
Tables
Table 1. A summary of the parameters used in simulations of the responsivity of fluxgates
made from the thin films and tubes. Here µoMsat is the saturation magnetization, K the
uniaxial anisotropy constant, k the uniaxial anisotropy constant in dimensionless form, d
the demagnetization factor along the drive field direction, αeff the effective damping
parameter, feddy the frequency where the magnetic field penetration depth along the drive
direction for a saturated core is equal to the core thickness, fpeak the frequency where
Equation (7c) is a maximum, and fdrive the drive frequency.
13
Figures
Figure 1. Schematic drawing of a tube core and thin film fluxgate magnetometers. The
drive coil is not shown for the thin film geometry.
Figure 2 (a) M(H) loops measured with the applied drive field along the hard and easy
axes of a permalloy film, and (b) M(H) loops measured in the axial direction of the
molybdenum permalloy tubes as a function of an easy-axis bias.
Figure 3. Measured and simulated amplitude dependence of µ1’ and µ1” for the
permalloy thin film. This simulation is used to determine damping and anisotropy
parameters.
Figure 4. (a) Hdrive dependence of the fluxgate responsivity measured and simulated with
µoHeasy = 0.0029, 0.0047, 0.0086, and 0.00102 T for the permalloy thin film and (b) with
µoHeasy = 0 and 0.0004 T for the molybdenum permalloy tubes.
Figure 5. A plot of the a dependence of the magnitude of M2/Hdc and the calculated
frequency of the peak in µ”.
14
µoMsat (T)
K (J/m3)
k
d
αeff
feddy (Hz)
NiFe thin film
0.95
320
9.0e-4
1e-5
22.5
2.2e11
1.0e6
2.2e4
MoNiFe tube
0.8
74
1.5e-4
4.3e-3
1092
2.7e5
8.8e4
260
fpeak (Hz) fdrive (Hz)
Table 1
A Dynamic Calculation of the Responsivity of Monodomain Fluxgate Magnetometers
J. Deak, R. H. Koch, G. E. Guthmiller, and R. E. Fontana, jr.
Drive Coil
Pickup Coil
Hdrive + Hdc
Heasy
permalloy tube
tube fluxgate
Hdrive + Hdc
Pickup Coil
Heasy
permalloy film
thin film fluxgate
Figure 1
A Dynamic Calculation of the Responsivity of Monodomain Fluxgate Magnetometers
J. Deak, R. H. Koch, G. E. Guthmiller, and R. E. Fontana, jr.
20
10
82-18 permalloy film
f = 10 Hz
dc bias Beasy = 0 T
0
Flux (nWb)Bdrive || hard axis
-10
Bdrive || easy axis
-20
-0.0010
-0.0005
0.0000
moHdrive (T)
0.0005
0.0010
Figure 2(a)
A Dynamic Calculation of the Responsivity of Monodomain Fluxgate Magnetometers
J. Deak, R. H. Koch, G. E. Guthmiller, and R. E. Fontana, jr.
0.8
0.6
0.4
molybdenum permalloy tube
10 Hz drive
0.2
0.0
M
(T)
mo-0.2
-0.4
moHeasy = 0.0004 T
-0.6
moHeasy = 0 T
-0.8
-0.004
-0.002
0.000
0.002
0.004
moH (T)
Figure 2(b)
A Dynamic Calculation of the Responsivity of Monodomain Fluxgate Magnetometers
J. Deak, R. H. Koch, G. E. Guthmiller, and R. E. Fontana, jr.
1200
100
a = 22.5
kx = 0.0009
80
moMsat = 0.95 T
1000
800
' 1 600
m
400
200
4700 Å thick
82-18 permalloy film
f = 22300 Hz
moHeasy = 0.0001 T
60
m
"
1
40
20
0
0
0.0000 0.0003 0.0006 0.0008 0.0011 0.0014 0.0017 0.0020
moHdrive (Tpeak)
Figure 3
A Dynamic Calculation of the Responsivity of Monodomain Fluxgate Magnetometers
J. Deak, R. H. Koch, G. E. Guthmiller, and R. E. Fontana, jr.
250
moHeasy = 0.00029 T
200
150
dc
/H
2
M100
moHeasy = 0.00102 T
50
0
0.000
thin-film permalloy fluxgate
f = 22420 Hz
0.001
0.002
0.003
moHdrive (Tpeak)
0.004
Figure 4(a)
A Dynamic Calculation of the Responsivity of Monodomain Fluxgate Magnetometers
J. Deak, R. H. Koch, G. E. Guthmiller, and R. E. Fontana, jr.
80
60
molybdenum permalloy
tube fluxgate
260 Hz
simulated
moHeasy = 0.0002 T
moHeasy = 0 T
dc
40
/H
2
M
measured
moHeasy = 0.0002 T
moHeasy = 0 T
20
0
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
moHdrive (T peak)
Figure 4(b)
A Dynamic Calculation of the Responsivity of Monodomain Fluxgate Magnetometers
J. Deak, R. H. Koch, G. E. Guthmiller, and R. E. Fontana, jr.
200
8
10
characteristic
frequency
150
100
/H
2
M
7
10
6
10
responsivity
50
0
5 f
10 peak
(Hz)
4
10
2wo
dc
simulated
thin film fluxgate
driven at 22420 Hz
moHeasy = 0.00102 T
3
10
2
10
1
1
10
100
1000
10000
100000
10
aeff
Figure 5
A Dynamic Calculation of the Responsivity of Monodomain Fluxgate Magnetometers
J. Deak, R. H. Koch, G. E. Guthmiller, and R. E. Fontana, jr.