Bushow K.H.J., de Boer F.R. Physics of Magnetism and Magnetic Materials
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162
CHAPTER 14.
SOFT-MAGNETIC MATERIALS
thin films are currently indicated as materials giving rise to giant magnetoresistance (GMR)
effects.
In most applications of vertical magnetoresistive heads, one has in the quiescent
the anisotropy field
state, owing to the magnetic anisotropy. In Fig. 14.5.2.1, this anisotropy is represented by
During operation of the head, the magnetization vector will
rotate over an angle
given by
where is the sum of the field emanating from the recording medium and a bias
field
applied to linearize the response of the head to the field of the medium. The field
in Eq. (14.5.2.2) accounts for local demagnetization effects of the magnetoresistive
element, the latter effects being more pronounced at the edges of the magnetoresistive
element. Assuming that the bias and demagnetizing fields are constant, and that only the
field from the medium
depends on
y
and
z
one finds for the output voltage
The signal levels of magnetoresistive heads are much higher than that of conventional
inductive heads. Furthermore, the output signal of the magnetoresistive head depends only
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SECTION 14.5.
HEAD MATERIALS AND THEIR APPLICATIONS
on the instantaneous fields of the media, and hence is independent of the media velocity
or the time rate of change of the fields. This offers a significant advantage for reading
low-velocity media. Here, we recall that the sensitivity of inductive reading is strongly
dependent on the relative velocity between head and medium, since this type of recording
is based on Faraday‘s law (see Section 14.5.1).
References
Alben, R. J., Becker, J. J., and Chi, M. C. (1978) J. Appl. Phys., 49,1653.
Bernards, J. P. C. and Schrauwen, C. P. G. (1990) Thesis, University of Twente, The Netherlands.
Boll, R. (1994) in R. W. Cahn et al. (Eds) Materials science and technology, Weinheim: VCH Verlag, Vol. 3B,
p.
401.
Bozorth, R. M. (1951) Ferromagnetism, New York: Van Nostrand.
Brabers, V. A. M. (1995) in K. H. J. Buschow (Ed.) Magnetic materials, Amsterdam: North Holland Publ. Co.,
Vol.
8
, p.
189.
Chikazumi, S., Suzuki, K., and Iwata, H. (1957) J. Phys. Soc. Japan, 12,1259.
Chin, G. Y. and Wernick, J. H. (1980) in E. P. Wohlfarth (Ed.) Ferromagnetic materials, Amsterdam: North
Holland Publ. Co., Vol. 2, p. 55.
Fedeli, J. M. (1993) in K. H. J. Buschow, G. J. Long and F. Grandjean (Eds) High density digital recording,
Dordrecht: Kluwer Academic Publ., NATO ASI Series E, Vol. 229, p. 251.
Ferguson, E. T. (1958) J. Appl. Phys., 29, 252.
Fujimori, H., Yoshimoto, H., and Matsumoto, T. (1981) J. Appl. Phys., 52,1893.
Goss, N. P. (1935) Trans. Am. Soc. Met., 23, 511.
Herzer, G. (1989) IEEE Trans. Magn., 25, 3327.
Herzer, G. (1996) in K. H. J. Buschow (Ed.) Magnetic materials, Amsterdam: North Holland Publ. Co., Vol. 10,
p.
415.
Hibst, H. and Schwab, E. (1994) in R. W. Cahn et al. (Eds) Materials science and technology, Weinheim: VCH
Verlag, Vol. 3B, p. 211.
Mee, C. D. and Daniel, E. D. (1990) Magnetic recording handbook. New York: McGraw-Hill.
Slick, P. I. (1980) in E. P. Wohlfarth (Ed.) Ferromagnetic materials, Amsterdam: North Holland Publ. Co., Vol. 2,
p.
189.
Soinski, M. and Moses, A. J. (1995) in K. H. J. Buschow (Ed.) Magnetic materials, Amsterdam: North Holland
Publ. Co., Vol. 8, p. 189.
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15
Invar Alloys
The origin of thermal expansion is the presence of anharmonic terms in the potential energy
expression describing the mutual separation of a pair of atoms at a temperature T. If x
represents the displacement of the atoms from their equilibrium position, the potential
energy may be written as
The term in
is a measure of the asymmetry of the mutual repulsion of the atoms, and
the term in
can be regarded as describing the general softening of the
vibrations at large
amplitudes.
In order to calculate the average displacement, we will follow Kittel (1953) and use
the Boltzmann distribution function (analogous to Eqs. 3.1.3 and 3.1.4), which weights the
possible values of x with a factor representing their thermodynamic probability.
For small displacements, the anharmonic contribution to the potential energy is relatively
small. In this case, the integrands may be expanded as
and
so that
This result shows that the temperature coefficient of the thermal expansion is a constant.
In classical mechanics, the mean value of the energy E of an oscillator in the harmonic
165
166
CHAPTER 15.
INVAR ALLOYS
approximation is equal to For this reason, one may also write Eq. (15.3) in the form
which suggests that the approximate quantum-mechanical result would be obtained by
substituting for the corresponding quantum-mechanical expression for the energy of
a harmonic oscillator with frequency This then leads to
It is useful to bear in mind that the specific heat as well as the thermal-expansion coefficient
are temperature derivatives of E, which means that the thermal-expansion coefficient is
proportional to the specific heat.
On the basis of Eq. (15.5), one would furthermore expect the thermal-expansion
coefficient to decrease rather abruptly when the temperature falls below the characteris-
tic temperature of the oscillator and to go to zero if the temperature goes to zero kelvin.
This is what is commonly observed. The third law of thermodynamics requires that the
thermal-expansion coefficient vanishes if the temperature goes to zero.
A schematic representation of the thermal-expansion behavior expected, if only the
lattice anharmonicity contributes, is shown in Fig. 15.1 (dashed-dotted curve). In many
magnetic materials, the thermal expansion takes quite a different form, as shown for instance
by the full curve in the same figure. The total thermal expansion can be subdivided into a
lattice contribution
and a contribution due to magnetic effects. The latter contribution
is called the spontaneous volume magnetostriction and is indicated by the broken curve
in
Fig.
15.1.
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CHAPTER 15. INVAR ALLOYS
The most conventional treatment of magnetovolume effects is based on a model in
which the magnetic moments are assumed to be localized. The magnetic contribution to the
volume change can be represented by the two-spin correlation function via
where the summation is taken over all magnetic sites, and where is the compressibility.
The quantity
is the magnetovolume coupling constant. It originates from the volume
dependence of the exchange constant responsible for the magnetic coupling between
two magnetic moments
i
and
j
(see, for instance, Eq. 4.4.2). This means that is
proportional to
In the case of alloys or intermetallic compounds based on 3d metals, one has to realize
that the 3d electrons occupy a narrow energy band having a width of a few electron volts,
as has been discussed in Section 7.1. In order to describe magnetovolume effects in these
materials, it is therefore necessary to take the band character of these electrons into con-
sideration. The reason for this is that there is an intimate connection between interatomic
distances, bandwidth and magnetic properties, as will be further discussed below.
It was outlined in Section 7.2 that the spin polarization of the 3d band that causes the
formation of magnetic moments is a trade-off between exchange energy (which is gained)
and the kinetic energy (which is lost). However, the increase in kinetic energy required
for the band polarization can be kept low if this band polarization is accompanied by
volume expansion. This may be seen as follows. If volume expansion occurs, one expects
a concomitant decrease of the bandwidth W on the basis of Eq. (7.1.1). It can be easily
verified by means of Fig. 7.1.1 that the expenditure in kinetic energy required to realize
a given 3d-band polarization (i.e., to realize a given amount of electron transfer from the
minority band to the minority band) will be lower, the smaller the bandwidth (i.e., the higher
the density of states).
To a first approximation, the increase in kinetic energy is proportional to the square of
the magnetization. The volume change due to band polarization can therefore be written as
where represents the magnetovolume coupling constant associated with the band
character of the 3d electrons.
in Eq. (15.6) may
At low temperatures the spin-correlation function
be approximated by so that the total volume magnetostriction can now be written as
Well-known materials with Invar properties are alloys of iron and nickel in a concen-
tration range close to the composition
It is interesting and instructive to compare
the Invar properties of these alloys with results of calculations of their electronic band
structure. The volume dependence of the total energies of non-magnetic and ferromag-
netic states derived from these calculations (Williams et al., 1983) is shown in Fig. 15.2.
In fcc FeNi (top part), the ferromagnetic state is the ground state, having an energy lower
than the paramagnetic state. The situation for fcc Fe is shown in the bottom part of the
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CHAPTER 15.
INVAR ALLOYS
figure. It is seen that here the non-magnetic state has the lower energy. Computational
results for the Invar alloy are shown in the middle part of the figure. There is not
much difference in energy between the non-magnetic state and the ferromagnetic state.
At low temperatures, only the ferromagnetic state will be populated, having its minimum
energy at a comparatively high volume. Williams and co-workers ascribe the Invar prop-
erties to thermal excitations into the non-magnetic state for which the energy minimum
is seen to occur at a significantly lower volume. Increasing temperature, therefore, leads
to a gradual loss of the spontaneous volume expansion associated with the ferromagnetic
state.
Invar alloys are employed in many devices for which a low thermal expansion is desir-
able. A detailed description of the physics and application of Invar alloys is presented in the
169
CHAPTER 15.
INVAR ALLOYS
surveys of Kaya (1978), Wasserman (1991), and Shiga (1994). The properties of a number
of Invar alloys based on stainless steel are shown in Fig. 15.3. Invar properties are also
found in many intermetallic compounds. For example, compounds of the type,
discussed extensively in Section 12.5, also display such properties (Fig. 15.4).
170
CHAPTER 15.
INVAR ALLOYS
References
Kaya, S. (1978) Physics and application of Invar alloys, Tokyo: Maruzen Co.
Kittel, C. (1953) Introduction to solid state physics, New York: John Wiley.
Shiga, M. (1994) Invar alloys, in R. W. Cahn et al. (Eds) Materials science and technology, Weinheim: VCH
Verlag, Vol. 3B, p. 159.
Wasserman, E. F. (1991) Moment-volume instability in transition metal alloys, in K. H. J. Buschow (Ed.)
Ferromagnetic materials, Amsterdam: North Holland, Vol. 5, p. 237.
Williams, A. R., Moruzzi, V. L., Gelatt Jr., C. D., and Kübler, J. (1983) J. Magn. Magn. Mater., 10, 120.
16
Magnetostrictive Materials
Magnetostriction can be defined as the change in dimension of a piece of magnetic material
induced by a change in its magnetic state. Generally, a magnetostrictive material changes
its dimension when subjected to a change of the applied magnetic field. Alternatively,
it undergoes a change in its magnetic state under the influence of an externally applied
mechanical stress. By far the most common type of magnetostriction is the Joule mag-
netostriction where the dimensional change is associated with a distribution of distorted
magnetic domains present in the magnetically ordered material. It is well known that fer-
romagnetic and ferrimagnetic materials adopt a magnetic domain structure with zero net
magnetization in the demagnetized state in order to reduce the magnetostatic energy. In
a material showing Joule magnetostriction, each of the magnetic domains is distorted by
interatomic forces in a way so as to minimize the total energy.
Concentrating on a single of these domains, for materials with positive (negative)
magnetostriction, the dimension along the magnetization direction is increased (decreased)
while simultaneously the dimension in the direction perpendicular to the magnetization
direction is decreased (increased), keeping the volume constant. This means that for a piece
of magnetostrictive material, consisting of an assembly of many magnetostrictively distorted
domains, one expects dimensional changes when an external field causes a rotation of the
magnetization direction within a domain, and/or when the external field causes a growth of
domains, for which the magnetization direction is close to the field direction, at the cost of
domains for which the magnetization direction differs more from the field direction. We will
return to this point later.
The magnetostrictive properties will reflect the symmetry of the crystal lattice when the
piece of material is a single crystal. In this case, the length changes observed at magnetic
saturation depend on the measurement direction as well as on the initial and final direction
of the magnetization of the single crystal. As shown in more detail in several reviews
(Cullen et al., 1994; Gignoux, 1992; Andreev, 1995), frequently only two magnetostrictive
constants are required to describe the fractional length change associated with the saturation
magnetostriction
in cubic materials:
171