Discussion Paper

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Discussion Paper
Geosci. Instrum. Method. Data Syst. Discuss., 4, 319–352, 2014
www.geosci-instrum-method-data-syst-discuss.net/4/319/2014/
doi:10.5194/gid-4-319-2014
© Author(s) 2014. CC Attribution 3.0 License.
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Discussion Paper
The origin of noise and magnetic
hysteresis in crystalline permalloy
ring-core fluxgate sensors
GID
4, 319–352, 2014
The origin of noise
and magnetic
hysteresis in
permalloy fluxgate
sensors
B. B. Narod
Title Page
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B. B. Narod1,2
Narod Geophysics Ltd., Vancouver, Canada
2
Department of Earth, Ocean and Atmospheric Sciences, University of British Columbia,
Vancouver, Canada
Received: 19 March 2014 – Accepted: 13 May 2014 – Published: 19 June 2014
Correspondence to: B. B. Narod ([email protected])
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Published by Copernicus Publications on behalf of the European Geosciences Union.
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4, 319–352, 2014
The origin of noise
and magnetic
hysteresis in
permalloy fluxgate
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B. B. Narod
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6-81.3 Mo permalloy, developed in the 1960s for use in high performance ringcore fluxgate sensors, remains the state-of-the-art for permalloy-cored fluxgate
magnetometers. The magnetic properties of 6-81.3, namely magnetocrystalline and
magnetoelastic anisotropies and saturation induction are all optimum in the Fe–Ni–Mo
system.
In such polycrystalline permalloy fluxgate sensors a single phenomenon may cause
both fluxgate noise and magnetic hysteresis, explain Barkhausen jumps, remanence
and coercivity, and avoid domain denucleation. The phenomenon, domain wall
reconnection, is presented as part of a theoretical model. In the unmagnetized state
a coarse-grain high-quality permalloy foil ideally forms stripe domains, which present
at the free surface as parallel, uniformly spaced domain walls that cross the entire
thickness of the foil. Leakage flux “in” and “out” of alternating domains is a requirement
of the random orientation, grain-by-grain, of magnetic easy axes’ angles with respect
to the foil free surface. Its magnetostatic energy together with domain wall energy
determines an energy budget to be minimized. Throughout the magnetization cycle the
free surface domain pattern remains essentially unchanged, due to the magnetostatic
energy cost such a change would elicit. Thus domain walls are “pinned” to free
surfaces.
Driven to saturation, domain walls first bulge then reconnect via Barkhausen
jumps to form a new domain configuration this author has called “channel domains”,
that are attached to free surfaces. The approach to saturation now continues as
reversible channel domain compression. Driving the permalloy deeper into saturation
compresses the channel domains to arbitrarily small thickness, but will not cause them
to denucleate. Returning from saturation the channel domain structure will survive
through zero H, thus explaining remanence. The Barkhausen jumps being irreversible
exothermic events are sources of fluxgate noise, powered by the energy available from
domain wall reconnection.
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The 1979 launch of the NASA spacecraft MAGSAT marked a milestone in the
development of magnetometry. In particular it terminated an effort by a group at Naval
Ordnance Laboratory, led by Daniel Israel Gordon, to create a low-noise magnetic alloy
now referred to as 6-81.3 Mo permalloy, “6-81” (Gordon et al., 1968). Answering the
question of why 6-81.3 Mo permalloy became the de facto standard for low noise ring
core fluxgates is the first motivation for this paper.
I begin with an examination of data that relate fluxgate noise to various physical
parameters and provide evidence that the geometric configurations of a fluxgate’s
magnetic material are significantly more important than Gordon’s investigators may
have thought. As surface area, either free surface or intergrain, of the magnetic material
increases, so does the fluxgate noise power. This suggests there is a causal link
4, 319–352, 2014
The origin of noise
and magnetic
hysteresis in
permalloy fluxgate
sensors
B. B. Narod
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Introduction
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A simplified domain energy model can then provide a predictive relation between
ring core magnetic properties and fluxgate sensor noise power. Four properties
are predicted to affect noise power, two of which, are well known: saturation total
magnetic flux density and magnetic anisotropy. The two additional properties are
easy axes alignment and foil thickness. Flux density and magnetic anisotropy are
primary magnetic properties determined by an alloy’s chemistry and crystalline
lattice properties. Easy axes alignment and foil thickness are secondary, geometrical
properties related to an alloy’s polycrystalline fabric and manufacture. Improvements
to fluxgate noise performance can in principle be achieved by optimizing any of these
four properties in such a way as to minimize magnetostatic energy.
Fluxgate signal power is proportional to B–H loop curvature (d2 B/dH 2 ). The degree
to which Barkhausen jumps coincide with loop curvature is a measure of noise that
accompanies fluxgate signal. B–H loops with significant curvature beyond the open
hysteresis loop may be used to advantage to acquire fluxgate signal with reduced
noise.
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4, 319–352, 2014
The origin of noise
and magnetic
hysteresis in
permalloy fluxgate
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B. B. Narod
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between fluxgate noise and magnetostatic energy by way of domain wall energy,
and that all are fundamentally dependent on the properties magnetic moment and
anisotropy.
Amos et al. (2008) using magnetic force microscopy (MFM) have shown that
sputtered permalloy films assume configurations called “stripe domains.” Thus it should
be beneficial to examine the causes of stripe domains. Since fluxgate sensor function
requires periodic core saturation in order to generate a signal, it is also beneficial to
study the evolution of stripe domains throughout the magnetization cycle.
Recently Coïsson et al. (2009) also using MFM, have determined the surface
expression of stripe domains in sputtered films throughout the magnetization cycle,
confirming constraints on magnetostatic energy which can be derived from simple
principles. I propose that on approach to saturation each stripe domain undergoes
a well-defined, exothermic and irreversible transition to a saturated state, with domain
wall loss supplying the energy. It further follows that these transitions could be used
to explain many aspects of irreversible and reversible magnetization processes in
magnetically soft materials, in particular DC coercivity, remanence and hysteresis.
Fluxgate noise and DC hysteresis are thus arguably two facets of the same
phenomenon, and that fluxgate sensor noise is causally tied to domain wall energy
by a simple relationship: fluxgate noise is proportional to available domain wall energy,
which itself is a function of domain size and domain wall energy density.
Magnetocrystalline anisotropy complicates the story: in any polycrystalline permalloy
each crystal assumes an easy axis from a distribution, an axis that in general will not
lie parallel to the magnetizing field, nor to the free surface. This affects domain size and
local magnetization processes. Saturation transitions in one crystal will not in general
coincide with those in any other crystal. Fluxgate noise and a ferromagnet’s path to
saturation will be an aggregate of all the individual crystals’ processes. But the basic
phenomena can be understood by examining processes in a single, ideal crystal.
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4, 319–352, 2014
The origin of noise
and magnetic
hysteresis in
permalloy fluxgate
sensors
B. B. Narod
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The original choice of “6-81” followed on the work of Pfeifer (1966; Pfeifer and Boll,
1969) who presented data which showed that “6-81” has near zero polycrystalline
magnetostriction and magneto-crystalline anisotropy for very slow cooling. The higher
molybdenum content of 6-81 has additional useful aspects, the first of these being
a lower saturation total magnetic flux density Bs , now known to directly affect the noise
power level of a magnetometer. The first indications that lower Bs led to lower fluxgate
noise were provided by Shirae (1984) and Narod et al. (1985), both using selections of
amorphous alloys.
Another useful aspect to using “6-81” is its need for very slow cooling rates to achieve
low anisotropy. While the added furnace operating time is a significant processing cost,
it results in lower average noise power levels, as compared with those for a rapidly
cooled alloy such as 4-79 permalloy. Figure 1 presents the results from this author’s
own study of 167 Magnetics Inc 4-79 ring cores with properties otherwise similar to
Infinetics S-1000 6-81 ring cores. These rings were made from 4-79 square loop
permalloy, with seven layers each 12 µm thick. The white histogram plots the noise
power probability density function (PDF) for the rings as received from their original
heat treatment. The black histogram plots the noise power PDF for the same rings
◦
after they had all received an extra heat treatment, 100 h at 100 C in air. Note the
general shift of the density function towards lower noise levels. Figure 3 shows a similar
probability density function for a collection of 195 “6-81” ring cores (12 µm), a function
which also exhibits an asymmetric form. These data suggest that “6-81”s requirement
for slow cooling acts as a substitute for the beneficial low temperature heat treatment.
The choice of “6-81” for fluxgate sensors thus has four distinct advantages. It
simultaneously minimizes magneto-crystalline anisotropy, magnetostriction and Bs
and, by slow cooling, also achieves a preferred statistical distribution for average noise
power. It is clearly the optimum composition in the Fe–Ni–Mo ternary system. These
properties can now provide clues to the nature of fluxgate sensor noise.
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Magnetic properties of “6-81”
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4, 319–352, 2014
The origin of noise
and magnetic
hysteresis in
permalloy fluxgate
sensors
B. B. Narod
Title Page
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Two more observations provide evidence for the last clues needed for a theory of noise.
Figure 2 plots dB/dt vs. t, an analog for DC differential permeability vs. H for five ring
cores, two of material 6-81 and three of 4-79 permalloy, each annotated with its noise
power. Lower noise power correlates with higher peak dB/dt. Similarly amorphous
alloys treated at temperatures up to 350 ◦ C will develop large magnetic domains
(Shishido et al., 1985), and treating such alloys to about 300 ◦ C will significantly reduce
their noise power (Narod et al., 1985). The 4-79 rings exhibit both higher noise power
and higher coercivity (peaks are to the right of the 6-81 curves). This is likely a grain
size effect (Pfeifer and Radaloff, 1980).
An hypothesis follows: (1) larger domain sizes lead to both lower noise and to higher
differential permeability, (2) smaller grain size leads to higher domain wall energy, noise
power and coercivity. It then follows that total domain wall area is also related to fluxgate
sensor noise.
A last clue comes from my examination of noise in a large collection of Infinetics
“6-81” ring cores. This included 195 12 µm rings with 7 layers each, and 75 3 µm rings
with 28 layers. Otherwise these rings are similar. Figure 3 plots the noise PDFs for
this collection, with 3 µm rings’ data in white, and 12 µm data in black. Counts have
been scaled to match. These data show clearly that the 3 µm rings are typically 12 dB
noisier than the 12 µm rings. Equally, the 3 µm rings have four times the free surface
area, the same ratio. With such a large number of specimens, from a large number of
independent processing batches, this statistic is robust and cannot have happened by
chance. Clearly the geometry of the fluxgate core material matters, and any theory of
fluxgate noise must be able to take this into account.
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Geometrical evidence towards a theory of noise
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The B–H curve from a fluxgate’s point of view
10
dB
.
dHd
(1)
15
∆B(Hd + ∂Hd ) − ∆B(Hd ) = ∂Hd He
d B
dHd2
.
(2)
The rate of change of flux density with time is then
2
dB
d B dHd
= He
.
dt
dH 2 dt
(3)
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d
Similarly for a general volume element C–C0 one takes into account B–H misalignment.
To calculate energy available to the sensor winding per cycle one needs a multiplication
of dB/dt with He , and integration over volume and time:
!
ZZ
2
dHd
d B
energy =
· He dV dt.
(4)
· He
dt
dHd2
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4, 319–352, 2014
The origin of noise
and magnetic
hysteresis in
permalloy fluxgate
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B. B. Narod
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This requires that one superpose the ring’s response to Hd and He (Primdahl et al.,
2002). He is not necessarily uniform. It can vary both spatially and temporally. Open
circuit sensor windings avoid temporal changes, while short circuit or capacitively
loaded sensors cause He to vary throughout the saturation cycle.
Moving up the B–H curve by an amount ∂Hd , the change in flux density as seen by
the sensor winding is
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∆B(Hd ) = B(Hd + He ) − B(Hd − He ) = He
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Consider a ring core lying in the plane of this page, immersed in a small external field
He (Fig. 4), with core drive field Hd , and internal flux density B, a volume element A–A0 ,
◦
comprising two mirror-symmetric elements, 180 apart on the ring, located where He is
parallel to Hd . Locally at point A, B is B(Hd + He ) and at A’ B is B(Hd − He ). As viewed
from a sensor winding parallel to He , their combined flux density is
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Thus fluxgate sensitivity results from the second derivative of the B–H curve, i.e. its
curvature. Actual power transfer goes as the product of B–H curvature and the rate of
change of the drive field. The importance of the second derivative was originally noted
by Marshall (1966) and Primdahl (1970).
A theory of noise
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The origin of noise
and magnetic
hysteresis in
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sensors
B. B. Narod
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To understand how saturation can cause both signal and noise it is necessary
to examine the processes that govern the formation and evolution of the ring’s
magnetic domains. Figure 5a–d represent for an idealized iron crystal respectively
a demagnetized ring core, a core immersed in an external field, a core partially
magnetized by an excitation field and a core saturated by an excitation field. Each
sector representation (a–c) comprises two domains jointly bounded by a common 180◦
domain wall. In Fig. 5a the core is depicted as having equal opposed domain paths,
with net magnetization zero.
In an external field H (Fig. 5b) only sectors with magnetization directions parallel to
H respond. A sensing winding develops a self-inductance higher than for a saturated
core, and can generate fluxgate signal as described above. In Fig. 5c an excitation field
has the domain path assisted by the excitation field expand, consistent with the high
effective permeability one expects in a toroidal magnetic circuit. Here the effects of the
excitation and external fields are entirely separable with the excitation field occupying
a toroidal, low-reluctance circuit and the external field response occupying a solenoidal,
high-reluctance circuit. Perfect separability also manifests as zero mutual inductance
between the two magnetic circuits. At saturation (Fig. 5d) both ferromagnetic paths
vanish, and inductances fall to air-core values (Butta and Ripka, 2008).
The picture presented above is not useful for considering domains in polycrystalline
permalloys. I present here a hypothetical hysteresis and noise model based on stripe
domains. Here domains have uniform width D, occupy the entire thickness t of the
film (or foil, or grain), and have lengths large compared to either width or thickness.
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4, 319–352, 2014
The origin of noise
and magnetic
hysteresis in
permalloy fluxgate
sensors
B. B. Narod
Title Page
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K is a parameter representing all of magnetic anisotropy, J is the exchange integral, S
2
is the magnitude of atomic spin, and a is the lattice constant (Chikazumi, 1997). Bs is
2
proportional to JS given that J is proportional to Curie temperature Tc (Weiss, 1948),
Tc is proportional to magnetic moment-squared M 2 (Chikazumi, 1997, Eq. 6.8), and Bs
correlates strongly with M (Fig. 14). The substitution with Bs requires the assumption
that FCC lattice parameters, spin magnitude and molecular field coefficient vary little
or not at all compared with variation of Bs over the range of permalloy compositions.
Magnetic anisotropy for cubic alloys has higher order terms. Reducing anisotropy to
a single parameter however does not significantly impact our aim which is to calculate
total domain wall energy and total magnetostatic energy, and minimize their sum to
determine domain width. Magnetostatic energy is required on the free surface of
a permalloy foil as all crystalline permalloys have nonzero anisotropy and an easy
axis that dips across the free surface at some angle α. Flux density across the free
surface is thus about αBs , where a small angle is assumed for α. Figure 6 indicates
the associated flux paths as curved arrows originating from one domain and returning
to an adjacent domain.
For soft magnetic materials the calculation is slightly more complicated, with three
energies to equilibrate: domain wall energy Ed , magnetostatic energy Es and magnetic
anisotropy energy Ec . Assuming uniform spin rotation and neglecting free surface
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Typically D and t are of the same magnitude with D < t except for very thin films.
Figure 6 extends the concepts of Fig. 5 to examine a small portion of a core, within
a single crystal, one that occupies the entire thickness of the core foil material.
Areal domain wall energy density can be approximated as
s
s
2
Bs2 K
JS K
Ed = 2π
∝ 2π
.
(5)
a
a
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(6)
Ec = (α − β)2 K
β=α
(7)
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4, 319–352, 2014
The origin of noise
and magnetic
hysteresis in
permalloy fluxgate
sensors
B. B. Narod
Title Page
Discussion Paper
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For the thick foils used in typical ringcores the two angles become equal, and D is
1/2
proportional to t . In this case Ec vanishes and the equilibrium approaches that
derived by Kittel (1949). In Amos et al. (2008) for films below 1 µm, a square-root
relation between D and t held, implying a value for K such that the angle β = α.
The above calculation is for uniformly spaced, flat domain walls. When an external
field is applied in a direction along the length of the domains the initial response is to
move the domain walls such that net magnetization adapts to the applied field. However
the magnetostatic energy constraint at the free surface requires that the domains’
surface expression remains essentially unchanged at a uniform distribution, i.e., 50–
50 % magnetic flux in and out. (Any deviation from 50–50 % would require a very large
energy expense. Magnetostatic energy Es could no longer be localized near the free
surface.) It follows that for a small field the domain walls must adapt as shown in Fig. 7.
Es increases with negligible surface change while domain wall energy increases due
to the longer arc length of the domain wall across the foil thickness.
Increasing the applied field further expands the domain wall (Fig. 8a). Eventually
a field value is reached where adjacent domain walls make contact (Fig. 8b) at which
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B2
1 + Dt K µs
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where β is the actual magnetization dip angle with respect to the free surface. Similar
to the approach taken in Butta and Ripka (2008), the total energy is minimized when
Ed equals Es and when the dip angle
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depolarization these energies (per unit volume), are
q
Bs2 K/a
D
; Es = (βBs )2
Ed ∝ 2π
tµo
D
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4, 319–352, 2014
The origin of noise
and magnetic
hysteresis in
permalloy fluxgate
sensors
B. B. Narod
Title Page
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point a domain wall reconnection occurs (Fig. 8c), forming a new configuration in
which a pair of arched domains is left to satisfy the magnetostatic energy boundary
condition. Domain wall area is reduced by this reconnection. The event is irreversible
and exothermic, with excess energy dissipated thermally as eddy currents. Such an
irreversible event must have random activation energy and thus must be a source of
noise. One can identify such a reconnection with a Barkhausen jump. Further field
increases serve to reversibly compress these new domains (Fig. 8d), which I shall call
“channel domains” to distinguish them from the original stripe domains. However the
free surface expression of the channel domains remains, and the domains themselves
cannot denucleate.
In Coïsson et al. (2009) their Fig. 4 (here as Fig. 9) provides evidence for the
formation of channel domains. MFM images of the domains’ surface expressions for
the two polarizations of a saturated specimen are practically indistinguishable, and
substantiate the strength of the magnetostatic energy boundary condition. (To be
clear, in crystalline alloys the stripe domains would form due to magneto-crystalline
anisotropy and not spin reorientation (Coïsson et al., 2009; Sharma et al., 2006).
Their test material was an iron-metalloid sputtered film, but arguably the magnetostatic
energy constraints should be similar.)
The saturated state thus is one of channel domains, which being shallow and
bounded by a free surface are relatively difficult to compress (low permeability). The
demagnetized state is dominated by stripe domains. Walls are relatively easy to move,
and reconnections present as large values of differential permeability (Barkhausen
jumps). In general the distribution of individual crystals’ easy axes will be random.
This results in a range of H values for which reconnections will occur in a bulk
specimen, and any attempt to estimate differential permeability will need to take this
into account. The return from saturation starts by expanding the channel domains,
similar to inflating a bubble (Fig. 10). Channel domains self-align across the thickness
of the foil, due to the transverse magnetic moments of the channels having opposing
magnetic charges, attracting each other by a mechanism broadly similar to one found
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(8)
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Two of the four parameters given in Eq. (8), Bs and anisotropy, are well known to be
involved in producing fluxgate noise. With Marc Lessard and others at University of New
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Putting it to the test
4, 319–352, 2014
The origin of noise
and magnetic
hysteresis in
permalloy fluxgate
sensors
B. B. Narod
Title Page
Discussion Paper
A hypothesis that fluxgate noise power is proportional to Ed can now predict all of the
principal observations. Noise power increases as foil thickness decreases, or as Bs or
anisotropy increase. Equation (8) also predicts that easy axis dip angle is an important
element in creating noise, with small dip angles to be desired, and may explain some of
the results of Fig. 2, whereby lower dip angles lead to reduced magnetostatic energy.
Stripe domain reconnection energy should not vary with the rate dHd /dt. Thus energy
released per cycle, and thus noise energy per cycle is independent of a fluxgate’s drive
frequency, consistent with this author’s experience.
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× K 1/4 .
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Ed ∝ t −1/2 × α × Bs
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by O’Brien et al. (2009) for transverse domains in permalloy nanowires. Initially the
domain walls extend slowly, requiring a lot of variation of H for relatively little gain
in net B, thus the slowly changing but accelerating tops of the hysteresis curve. The
channel domains begin to expand adding domain wall area and requiring energy to
do that – the energy of hysteresis. Eventually the channels themselves reconnect via
a second, smaller Barkhausen jump to again form stripe domains and the process
repeats, now in the opposing polarity. For stripe domains reconnecting into channels
a lot of domain wall energy is released. For channel domains converting to stripes the
amount of energy available is much less clear. The field value at which this occurs is
likely closely related to what we can observe as coercivity Hc .
Using Eqs. (6) and (7), solving for and eliminating D, one finds that for
a demagnetized stripe domain system, domain wall energy per unit volume is
proportional to
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Infinetics thin foils develop high differential permeability and more square B–H curves.
The 100 µm rings exhibit much “rounder” curves and lower remanence, yet have
excellent noise performance. In fact the new seven-layer rings at 5.2 pT are amongst
the best performing permalloy ring cores ever produced. How round B–H curves
improve noise performance requires a further development of the noise model.
Referring now to Fig. 11c, from point A to point B there is little curvature, so little
or no power is transferred to the sensor winding. Changes in B are dominated by
large Barkhausen jumps which are identified in the noise model with domain wall
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The origin of noise
and magnetic
hysteresis in
permalloy fluxgate
sensors
B. B. Narod
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Another look at B–H loop curvature
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Hampshire I have put to the test foil thickness as a controlling parameter, by making
6-81 ring cores differing from the Infinetics design only in foil thickness. We produced
two groups of rings, respectively one-layer and seven layers, as follows: cast an ingot
of 6.0 % Mo, 81.3 % Ni, Fe balance. Fabricate flat strips approximately 3 mm × 25 mm ×
◦
125 mm. Homogenize at 1100 C in 95 % Ar/5 % H, for seven days. Cold roll to 100 µm,
with no in-process anneal. Slit to width. Coat with a thin layer of MgO. Spot weld to an
Inconel X750 bobbin. Heat treat in 95 % Ar/5 % H four hours at 1100 ◦ C, cool to 600 ◦ C,
◦
−1
cool at 35 C h to room temperature. At no point through this development was any
attempt made to optimize a process step. This was simply our first attempt to fabricate
a 6-81 polycrystalline permalloy ring core.
Noise PSD at 1 Hz for the two types of 100 µm rings are respectively 14 and
5.2 pT rtHz−1 , the difference consistent with rms stacking. Figure 11 presents the B–H
curves for these and Infinetics 3 µm and 12 µm rings, measured at 25 Hz. All four rings
were assembled with 25.4 mm diameter bobbins. The magnetic material was 6-81 in
all cases, and heat treatments were nominally identical. The 100 µm ring has a single
layer and the 400 µm ring has a single circular wire of diameter 400 µm. The magnetic
masses of the four rings match within 20 %, allowing the B–H curves to be recorded
without any changes to the measurement settings. Between measurements only the
−1
ring was changed. Coercivities for the four rings are in the range 3–7 A m .
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The origin of noise
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reconnections from stripes to channels. All the energy released by these events
couples back into the Hd drive circuit and no noise energy gets into the sensor winding.
From B to D curvature is larger, reaching a maximum at point C. Over this
interval energy is transferred to the sensor winding. Since in this range changes
in B are still dominated by stripe to channel reconnections, it is here that fluxgate
noise power is generated, ending at D. From D to E changes in B can result from
domain wall movement in the form of channel domain compression. Our new ring
cores exhibit significant curvature in this range, resulting in its round B–H curve.
This curvature generates additional sensor energy, and since the process of channel
domain compression is nominally a reversible process this additional sensor energy
comes with little noise energy when compared with interval B–D.
The inset to Fig. 11c is a detail showing in the 100 µm data a well-defined transition
from irreversible behaviour to reversible behaviour, a transition not noticible in the return
path. Such a sharp transition requires a magnetization process that can support such
transitions and that provides for an entirely reversible process above the transition
point. Conventional thinking calls for reversible rotation to explain much of the approach
to saturation from D to E (Chikazumi, 1997, Fig. 18.26), the realignment en masse of
individual electronic spins. Being a body force such realignment should not depend
on geometry, particularly material thickness. Comparing the 12 and 100 µm curves in
Fig. 11 dispels this possibility. The hysteresis loop closure points marked “+” indicate
where reversibility begins. Given that all other variables have been constrained one
would expect the two B–H curves to the right of these points to coincide, and clearly
this is not the case. Reversible rotation on its own would require average easy axis
misalignment of 45 % to explain the 30 % reversible approach to saturation in Fig. 11c.
This is unlikely, implying a need for reversible domain wall motion.
From E to D and part way to point F, an interval of channel domain expansion,
energy may be recaptured from the sensor winding. From D to F the B–H curve is
dominated by small Barkhausen jumps which can be identified with channel-to-stripe
reconnections (Chikazumi, 1997, Fig. 18.43). From point F to point G the B–H curve
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Grain sizes of our new material are about 20 µm, thus there exists a large intergrain
boundary area within the foil. Mismatch of easy axes at these interior surfaces
also serves to localize magnetostatic energy, and if channel domains form at free
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is again of low curvature, dominated by large Barkhausen jumps (stripe-to-channel
reconnections).
The channel domain concept can also explain remanence variability in permalloy
foils, and in particular why thin foils exhibit greater remanence. Remanence exists
because channel domain walls, while satisfying internal boundary conditions, do not
lead to a symmetry such that net magnetization is zero.
Figure 12 visualizes the proposed channel domains for very thin foils (a) (or high
anisotropy), or very thick foils (b) (or low anisotropy). For thin foils (Fig. 12a), at point
0
P on channel domain centerline section A–A , the internal boundary condition requires
at P that H = 0. The very wide aspect of the channel domain means that the field at
P is mainly influenced by a neighborhood dominated by relatively planar domain walls.
This neighborhood is depicted as the dashed square. A simple calculation using oval
channels leads to a remanence ratio of 59 %, in rough agreement with the B–H curves
in Fig. 11.
For thick foils the domains are larger, but their aspects are very different, appearing
as thin laminations. A point at the tip of a channel domain is influenced by a much
smaller neighborhood. This is depicted in Fig. 12b as the small dashed square.
Destructive interference causes material outside this square to have little influence.
In the limit as channels come close to touching, and using ovals to approximate the
domain shape one obtains a remanence ratio of 21 %, a value found in many natural
materials, and also by this author for 400 µm material. Thicker foils thus reduce fluxgate
noise in two ways, both of which reduce total power to the Barkhausen jumps. They
reduce total domain wall energy (Pfeifer and Boll, 1969), and deep channels reduce
the range in B occupied by the hysteresis loop.
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Equation (8) can be used to guide choices in magnetic materials and in ring
core geometry. Regarding the choice of material, clearly “6-81” has very desirable
properties. To improve further we need to look for lower Bs materials while
simultaneously maintaining low anisotropy. Within the quaternary system Fe–Ni–Mo–
Cu it is possible to estimate Bs by calculating an alloy’s average atomic moment:
Bm = at% Fe · 2.8 + at% Ni · 0.6 − at% Mo · 4.6 − at% Cu · 0.4, and correlating that with Bs .
Figure 14 summarizes such data. The open square marked “3” locates “6-81”. The
circle marked “6” locates Neumann’s “1040” alloy, which is a possible contender for an
improved fluxgate core material due to its lower Bs .
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surfaces so too should they form at intergrain boundaries. This source of magnetostatic
energy must give rise to additional domain walls. Energy needed to reconnect the
intergrain channel domains back to stripes will manifest as an increase in hysteresis
loss, and relates to the known inverse relation between grain size and coercivity
(Couderchon et al., 1989; Herzer, 1990; Pfeifer and Radaloff, 1980). Speculating
further, if larger grains lead to longer magnetic domains then considered as a solenoid
each domain’s local field near its surface would decrease with increasing grain size.
This implies a given amount of domain wall displacement requires less drive field to
achieve equilibrium, and by extension that domain wall reconnections require less field
(Fig. 13). These are respectively interpretable as higher initial permeability and lower
coercivity, both known to correlate with lower fluxgate noise (Nielsen et al., 2010).
Our 6-81 material, with over twice the coercivity of the Infinetics material, must have
significantly more magnetostatic energy. An electron micrograph (Kuyucak, 1996) of
an Infinetics 3 µm ring core showed grains with 10 µm average size – much larger than
the foil thickness. For the 3 µm rings free surface magnetostatic energy must dominate
its energy budget.
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I have presented evidence by both data and theory that noise power in the parallelgated fluxgate has its origin in domain wall energy that transforms irreversibly and
exothermically to eddy current energy when domain walls reconnect, identifiable with
Barkhausen jumps. Domain wall energy available for noise generation is maintained in
a balance with magnetostatic energy, which in thin foils is dominated by the anisotropydriven surface magnetostatic energies. Reducing surface area thus reduces noise
energy.
Ed , and thus fluxgate noise is reduced by treating the permalloy for low anisotropy.
However the biggest influence on Ed is the saturation total magnetic flux density
Bs . Reducing Bs simultaneously reduces all energies. Its only limitation is a need to
concurrently reduce Curie temperature.
These argue for the existence of a previously unidentified magnetic domain state
I have called “channel domains.” A channel domain hypothesis can be expressed as
follows: in the presence of no external field an individual crystal of a soft magnetic
metal can assume one of three magnetized states. One state, in the form of stripe
domains, has zero net magnetization. The second and third states, in the form of
channel domains of different polarizations, have non-zero net magnetization. A stripe
domain or a channel domain state can convert to the other by domain wall expansion
and irreversible reconnection, respectively a large or small Barkhausen jump. The
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Reducing magnetostatic energy by reducing free surface area is an obvious method
to reduce domain wall energy. The use of thicker foils or even wires are two such
means, both relatively easily achieved. Reducing intergrain surface area is also
a means to reduce magnetostatic energy, to be achieved by the application of specific
heat treatments tailored to growing large crystals. Ideally a typical grain would be the
full thickness of the foil or wire and its diameter much larger. Cold forming to control for
easy axis direction might also be an approach to further reduce domain wall energy.
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Acknowledgements. I thank J. R. Bennest, G. Cross, and S. Allegretto, who assisted on this
manuscript. The Geological Survey of Canada made available the collection of “6-81” ringcores. I thank S. Kuyucak of CANMET, P. Dosanjh at UBC, M. Lessard and the others at UNH,
and L. Jewitt, Laura K Jewitt Design Gallery, for their efforts in creating the new 6-81 ring cores.
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three states, stripe domains or channel domains, can satisfy nearly identical surface
boundary conditions. The combination of crystals in a stripe domain state and those
in a channel domain state is arbitrary and will depend on the history of external fields.
DC hysteresis results from the cycling between stripe and channel domain states, via
Barkhausen jumps.
Much work remains. Specimens need to be created which have much larger grain
sizes than in existing materials. This will require heat treatments at higher temperatures
and of longer durations than normally used for permalloys. Domain imaging of large
grain, low anistropy crystalline permalloys throughout a magnetization cycle should
provide further testing of the the hypothesis. Combining domain imaging with the
manipulation of final heat treatments could determine the relations between anisotropy,
domain sizes and fluxgate noise. Fabric analysis could provide data regarding easy
axes alignments. Micromagnetic numerical simulations of stripe domain systems with
pinned free surface expressions could lead to understanding the energetics of domain
wall reconnections.
In the future careful examinations of reconnection processes might be able to draw
connections between micromagnetic mechanisms and large scale, phenomenological
models (Bertotti, 2008). The existence of channel domain states may go a long
way to understanding many phenomena in magnetically soft polycrystalline metals
including: remanence, DC coercivity, asymmetry in the B–H loop; solving the “domain
nucleation problem”; an irreversible, exothermic process that produces both fluxgate
and Barkhausen noise.
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Figure 1. Histograms of noise power for 4-79 permalloy ring-cores.
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120!
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80!
21 pT
16 pT
60!
23 pT
B. B. Narod
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40!
INF-Q!
M331-70!
M331-49!
0!
0.75!
1.00!
1.25!
1.50!
1.75!
2.00!
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INF-2!
M331-104!
Time (ms)!
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this insert and all other data in this paper.
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Figure 2. dB/dt plots for 6-81 and 4-79 ring-cores. Noise values are in pT rtHz−1 at 1 Hz, for
2
Figure. 2. dB/dt plots for 6-81 and 4-79 ring-cores. Noise values are in pT/rtHz
this insert and all other data in this paper.
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20!
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dB/dt (arbitrary
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100!
4-79
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Figure 3. Histograms of noise power for 6-81 permalloy ring-cores.
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He
B
B
∆B(Hd+∂Hd)
C'
C
θ
B
Hd+∂Hd±He
A'
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Hd±He d
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Ring core schematic and B-H loop detail.
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Ring4.core
schematic and B–H loop detail.
3
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a
b
c
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Figure.domains:
5. Ring-coreresponses
domains: responses
to external
or excitationfields.
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Figure 5. Ring-core
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or excitation
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Figure 6. Detail of stripe domains in a demagnetized core. Thin arrows depict magnetic flux, the
2
Figure. 6. Detail of stripe domains in a demagnetized core. Thin arrows depict magnetic flux,J
larger, vertical arrows indicating the dominant flux internal to the core, and the looped arrows
the but
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verticalflux
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indicating
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dominant flux internal to the core, and the looped
depicting the3smaller
significant
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the free
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arrows depicting the smaller but significant flux crossing the free surface.
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Figure. 7. Domain wall motion for initial permeability.
Figure 7. Domain wall
motion
for initial permeability.
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Figure. 8. Stripe and channel domain configurations.
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Figure 8. Stripe
and channel domain configurations.
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Figure 9. “Hysteresis loop of the 150 nm-thick film annealed at 250 C for 60 min. Selected
2 are
Figure.
“Hysteresis
loop of
150-nm-thick
filmpoint
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at 250C for 60min. Sele
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the hysteresis loop is indicated”. From Coïsson et al. (2009). (Non-SI units are in the original
3 1 AMFM
images are shown, taken at different applied field values. The Interactive
corresponding
Discussion poin
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m−1 ∼
= 0.0126 Oe.) The specimen was a sputtered FeSiB amorphous film.
the hysteresis loop is indicated.”
From Coïsson et al., (2009). [Non-SI units are in the orig
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Figure 10. Expanding 2channel
domains.
Figure.
10. Expanding channel domains.
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Figure
Infinetics
and 3 µm 12µm
B–H curves
(top row).
100 µm
400 µm
(bottom
J
2 11.
Figure.
11. 12
Infinetics
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no4Barkhausen
jump related
lossesfor
above
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and that
the largedetail
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B distinct transition from
behaviour.
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irreversible behaviour to reversible behaviour. The argument presented here is that there are
compression.
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no Barkhausen jump related losses above the transition point, and that the large variation
in B
7
above the transition point requires reversible domain wall motion in the form of channel
8
domain compression.
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2 12.
FIG.Channel
12. Channel
domain
sections:
(a) thin
(b) thick
Figure
domain
sections:
(a) thin
foils,foils,
(b) thick
foils.foils.
3
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and magnetic
hysteresis in
permalloy fluxgate
sensors
B. B. Narod
Title Page
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11
directly to higher permeability and lower coercivity.
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351
Discussion Paper
wall displacement for reconnection implies lower coercivity. Thus increasing grain size leads
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Discussion Paper
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Figure 13. 3-dimensional representation of a single grain and stripe-to-channel reconnection.
3
Figure.
13. the
3-dimensional
representation
of a single
grain andwithin
stripe-to-channel
reconnection.
The left panel
depicts
unmagnetized
stripe
domains
a grain.
A larger grain would
4
The leftdomains,
panel depicts all
the unmagnetized
stripe domains
grain
would
contain additional
of the same
width.within
Thea grain.
rightA larger
panel
depicts
the start of
reconnection
the additional
domaindomains,
wall midpoint.
fixed
of domain
wall displacement is
5 at
contain
all of the sameA
width.
The amount
right panel depicts
the start of
required, regardless
of atthe
size of
grain.
A larger
also have
longer stripe
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reconnection
the domain
wallthe
midpoint.
A fixed
amount ofgrain
domainwould
wall displacement
is
domains and
require
less
external
field
for
a
given
amount
of
wall
displacement,
that is,
7
required, regardless of the size of the grain. A larger grain would also have longer stripe
higher initial permeability. Higher initial permeability in combination with a fixed amount of wall
8
domains and require less external field for a given amount of wall displacement, that is,
displacement for reconnection implies lower coercivity. Thus increasing grain size leads directly
9
higher initial permeability. Higher initial permeability in combination with a fixed amount of
to higher permeability and lower coercivity.
Abstract
Introduction
Conclusions
References
Tables
Figures
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Bohr moment (magnetons)!
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Discussion Paper
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The origin of noise
and magnetic
hysteresis in
permalloy fluxgate
sensors
B. B. Narod
Title Page
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4, 319–352, 2014
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density (T)
flux(T)!
Saturation
induction
total magnetic
Saturation
1!
GID
Abstract
Introduction
Conclusions
References
Tables
Figures
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Discussion Paper
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Figure 14. Saturation total magnetic flux density vs. average atomic moment. Data are
from 2Neumann
(1934),
Auwers and total
Neumann
(1935), Bozorth
(1951) or
calculated
Figure.
14.vonSaturation
magnetic
flux density
vsare
average
atomic moment.
Data/ Esc
are from
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estimates. Large, open symbols plot alloys: (1) 14 % Cu, (2) mumetal, (3) “6-81”, (4) 4-79, (5)
supermalloy,
(6) “1040”, (7)
28 % Cu.von
Small Auwers
symbols plot
Bs data
for a range(1935),
of ternary Bozorth
alloys.
3
Neumann
(1934),
and
Neumann
(1951) or are calculated
Printer-friendly Version
estimates. Large, open symbols plot alloys: (1) 14%Cu, (2) mumetal, (3) “6-81”,
4-79, (5)
Interactive (4)
Discussion
5
supermalloy, (6) “1040”, (7)
35228%Cu. Small symbols plot Bs data for a range of ternary alloys.
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