Buschow K.H.J. (Ed.) Concise Encyclopedia of Magnetic and Superconducting Materials

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The temperature of the sample can be varied by

using either a furnace or a cryostat. The susceptibility

deduced for a lock-in ampliﬁer (U

¼RMS value of

the signal) is given by:

w ¼

pick

2pfkI

¼ m  1 ð5Þ

where N

pick

is the number of turns of the pick-up coil,

f is the frequency of the a.c. current, and k is the

calibration factor of the ﬁeld coil. For a ribbon

X ¼A

rib

(cross-section of the sample) because the

ribbon is much longer than the pick-up coils. For a

small sample, X takes the value of the area of the

cross-section of the pick-up coil (e.g., a hard magnetic

spherical specimen). The voltage U

is in any case

proportional to the mass of the sample—the calibra-

tion factor has to be determined experimentally.

For an absolute value of the measured susceptibil-

ity, the following points have to be considered. In the

case of thin long samples such as amorphous ribbons

one can measure on toroid and open ribbons:

Toroid ribbon: this is a magnetically closed circuit,

which means the demagnetizing factor is zero. The

disadvantage is that in the case of a soft magnetic

thin ribbon this is not a stress-free state. Also stress-

dependent measurements are not possible on a tor-

oid. It should be mentioned that the domain structure

of soft magnetic materials is strongly stress-depend-

ent (Badurek et al. 1988, Veider et al. 1986);

Straight ribbon: this is not a closed magnetic cir-

cuit. Therefore, the measurements in low ﬁelds are

very sensitive to external ﬂuctuating ﬁelds. Either a

very effective shielding or an accurate earth ﬁeld

compensation is essential for soft magnetic materials.

Additionally the demagnetizing factor N may be con-

sidered. The effective susceptibility with the internal

ﬁeld H

int

can be described as:

w ¼

int

H  NM

meas

1  Nw

meas

ð1=w

meas

ÞN

ð6Þ

For a soft magnetic material w is large which means

that the measured w is proportional to 1/N. Therefore

the samples should be very long compared with the

cross-section in order to reduce N. For hard magnetic

materials w is small and proportional to 1/K.

2. Magnetization Measurements

There are many techniques for measuring hysteresis

loops and magnetization. The majority of them is

based on the law of induction. Besides this, devices

exist where the magnetization is measured via the

force of a dipole in an inhomogeneous ﬁeld (Faraday

balance) or even magneto-optically by measuring

the Faraday rotation due to the magnetization. The

latter method is, of course, important for recording

materials. Figure 4 shows a sketch of the Faraday

method. The ﬁeld can be produced using a solenoid

which can be also superconducting or have an iron-

yoke. In the case of an electromagnet the specially

shaped pole pieces generate a ﬁeld gradient in the air

gap. The force can be measured by all kinds of force

sensors such as strain gauges or piezoelectric sensors.

Also torque magnetometers, in which a single crys-

talline sample feels a torque L ¼m H in a magnetic

ﬁeld, are based on similar principles. These systems

are very important for studying magnetocrystalline

anisotropy.

One can consider two main possibilities for induc-

tion-based methods.

The ﬁrst is that one can integrate the induction

voltage which is proportional to dB/dt and conse-

quently to dM/dt during the variation of the ﬁeld.

Generally this method is a d.c. technique because

the variation of the ﬁeld, which is produced by an

electromagnet or by a superconducting magnet, is

performed slowly. This technique needs a very stable

integrator. The sensitivity is limited principally by

the thermal drift of the system as well as by the 1/f

noise.

The second method uses a type of modulating

method either by vibrating the sample or by slowly

modulating the varying ﬁeld. This technique over-

comes the drift problem. In this case the induced

voltage u is a periodic signal which is proportional to

dM/dt sin(ot þd). Normally in this technique a lock-

in ampliﬁer is used which provides a drastically en-

hanced signal-to-noise ratio due to reduction of the

bandwidth of the system. In metallic samples the fre-

quency is chosen such that eddy current effects can be

avoided. Consequently the frequency is rather low

(o82 Hz), so drift problems can be overcome but the

1/f noise problem cannot be solved completely.

Figure 4

Sketch of a Faraday balance arranged in a solenoid.

500

Magnetic Measurements: Quasistatic and ac

2.1 Vibrating Sample Magnetometer

A vibrating sample magnetometer (VSM) is a state-

of-the-art device for magnetization measurements.

There exist several commercial VSMs which operate

from very low temperatures up to about 1000 K. In a

VSM the sample is driven by a mechanically vibrating

system and the signal is measured by a pick-up sys-

tem. The ﬁrst working model of a VSM was assem-

bled by Foner (1956) using a reduced-amplitude

60 Hz line excitation of a loudspeaker. The sample

vibrates perpendicularly to the applied ﬁeld. Small

permanent magnets were used to generate a reference

signal in order to stabilize the movement of the loud-

speaker. Field modulation combined with lock-in

ampliﬁcation was used for signal processing and a

lock-in ampliﬁer was used to enhance the signal-

to-noise ratio. Later, Foner (1959) described a more

standard version of a VSM in a comprehensive paper.

In the ﬁrst system electromagnets generated the

external ﬁeld. With the development of high-ﬁeld

superconductors, superconducting solenoids were in-

troduced into the VSM, producing ﬁelds up to about

20 T. These systems are commercially available for

measurements from low temperatures up to furnace

systems. The principal setup of a VSM is shown in

Fig. 5.

2.2 SQUID Magnetometer

SQUID (superconducting quantum interference de-

vice) magnetometers are generally used to measure

the magnetization of very small samples (e.g., thin

ﬁlms) or systems that are just on the limit of magnetic

order. Therefore, SQUID magnetometers are mainly

applied to scientific investigations. Measuring mate-

rials with a high magnetization requires the use of an

attenuator, but due to their very great sensitivity

geological, biological, and medical applications are

increasing. The operation of a SQUID magnetometer

is based on two effects: ﬂux quantization and the

Josephson effect.

The classical Josephson junction consists of an in-

sulating barrier between two superconductors thin

enough to couple ‘‘weakly.’’ This allows an electron

pair tunneling via the barrier. The ﬂux F

in the su-

perconducting state is quantized in steps of F

h/2e ¼2.07 10

15

Wb, where h is Planck’s constant

and e is the electron charge. Due to the different types

of electronic readout, a SQUID consists basically of a

superconducting ring interrupted at one or two po-

sitions by a Josephson junction. SQUIDs with one

junction are RF-SQUIDs (radio-frequency) and

those with two junctions are d.c.-SQUIDs (see

SQUIDs: The Instrument).

Figure 6 shows the basic electronic circuit of a

RF-SQUID. The peak value of the voltage drop V

across the tank circuit is measured by a RF ampliﬁer

together with a detector. This signal has to be set in

relation to the RF signal at a power P

applied to the

tank circuit. The value of V

depends on the low-

frequency input signal, i.e., the ﬂux F

ext

is applied to

the SQUID ring via the inductively coupled input

coil. The advantage of a RF-SQUID is that the use of

a high-frequency a.c. signal allows the application

of a.c. techniques which improve the signal-to-noise

ratio. Unlike the inductive detector VSM, whose

output is proportional to frequency, the output of the

SQUID detector is independent of frequency.

The SQUID generally operates at 4.2 K since the

critical temperature of superconductors is low. High-

superconductors allow systems to operate at

temperatures up to 77 K. Since the SQUID is an

extremely sensitive ﬂux detector it must be carefully

Figure 6

Block diagram of a RF-SQUID.

Figure 5

Principal setup of a VSM.

501

Magnetic Measurements: Quasistatic and ac

shielded. But the high sensitivity means it can be used

to measure very small ﬁelds. There are SQUID mag-

netometers which operate in external ﬁelds up to 6 T.

They can also be used for medical applications such

as measuring brain currents. A good survey of this

ﬁeld is given by Koch (1991).

2.3 Pulsed Field Magnetometer

Generally, pulsed ﬁeld systems are used for measure-

ments in high external ﬁelds which are above those

which are available as static ﬁelds. This means that

such magnetometers are used in ﬁelds up to 100 T

(Herlach 1994) and higher. However, there are also

technical applications which are interesting because

here the easy-to-handle factor and fast measurement

becomes important. Modern high-quality permanent

magnets, such as Sm–Co- or Nd–Fe–B-based mate-

rials, need rather high magnetic ﬁelds in order to

measure the hysteresis loop. Static ﬁelds such as those

available with Fe-yokes (up to about 2 T) are not

sufﬁcient here. Additionally, static hysteresis meas-

urements take too much time for an industrial pro-

duction control. Using pulsed ﬁeld magnetometry full

hysteresis loops M(H) of a hard magnetic material

can be measured very quickly and with an accuracy

of 1% at room temperature (for details see Magnetic

Measurements: Pulsed Field).

3. Characterization of Soft Magnetic Mater ials

Soft magnetic materials (see Magnets, Soft and Hard:

Magnetic Domains), such as Fe–Si strips, amor-

phous ribbons, or nanocrystalline materials, exhibit

a special geometry. Because of its technical impor-

tance Fe–Si needs a very accurate and standardized

determination of the hysteresis properties. For meas-

uring the losses as well as the permeability in ﬁelds

that are not too high the so-called Epstein frame

(Epstein 1900) is well established (see Magnetic

Hysteresis).

New kinds of soft magnetic materials such as

amorphous ribbons based on a 3d metal (Fe, Co)–

metalloid (B, Si, C) compound (see, e.g., Luborsky

1983) are thin ribbons (typical thickness 20–50 mm,

width 1–15 mm) produced by rapid quenching

through melt spinning. Some of them can be trans-

formed after special heat treatment into the nano-

crystalline state and they also exhibit excellent soft

magnetic properties (Yoshizawa et al. 1988, Herzer

1990) (see also Amorphous and Nanocrystalline

Materials). The ribbon-like geometry requires special

equipment. Due to their very low coercive ﬁeld

(typically between 1 and 20 Am

1

) standard magne-

tometers using either an electromagnet or a super-

conducting solenoid cannot be used, since the lowest

achievable ﬁeld is determined by the remanence

of the system which is in the order of 10 kAm

1

Additionally, at very low ﬁelds, the electromagnetic

noise of the surroundings is a limiting factor. As ex-

amples, two systems suitable for scientific applica-

tions are described which allow measurements at

both low and elevated temperatures.

The magnetic properties of soft ferromagnetic rib-

bons (hysteresis loop, susceptibility) can be measured

either on toroidal samples or on single strips. The

toroidal sample (see Fig. 7) has the advantage that,

to the ﬁrst order, the demagnetizing ﬁeld can be

neglected, which is important especially for permea-

bility and loss measurements such as in an Epstein

frame. However, for determining the saturation

magnetization, this shape is no more advantageous

due to the increase in the stray ﬁeld of the system with

the increase of the applied ﬁeld, which causes an

undeﬁned error.

3.1 Low-temperature Hysteresis Loops

Low-temperature measurements are generally impor-

tant in order to test the validity of theoretical pre-

dictions such as the Bloch law (Bloch 1930) or the

l(l þ1) power law for the anisotropy and magneto-

striction (Callen and Callen 1966). For the system

that will be described here (see Fig. 8) the tempera-

ture dependence of the hysteresis loops can be meas-

ured on wires, single ribbons, or thin sheets. Due to

the special geometry of the setup, an external stress

can also be applied which is very useful for studying

the effect of the magnetoelastic energy on the hys-

teresis. This effect is especially important for sensor

applications.

Figure 8 shows a system that measures hysteresis

loops between 4.2 K and room temperature. The fa-

cilities necessary for measuring the hysteresis loop are

ﬁeld coils (cylindrical coil þHelmholtz coil), a com-

pensated pick-up coil, a current source, an ampere-

meter, and an integrator (ﬂuxmeter). The devices can

be connected and controlled by a data acquisition

and control unit forming a fully automatic hysteresis

loop system. The external stress is measured by a

strain gauge bridge. To avoid the inﬂuence of eddy

currents and magnetic relaxation effects measure-

ments are performed quasi-statically (0.05–0.03 Hz).

Figure 7

Example of a toroidal measurement facility suited for

use in a furnace: glass toroid, sample, and coils.

502

Magnetic Measurements: Quasistatic and ac

In this case the measurement of one hysteresis loop

takes 20–30 s. The magnetic ﬁeld is deduced from the

current measured by the amperemeter and the induc-

tion from the magnetic ﬂux is measured by the inte-

grator. Since the change of the induction voltage

is slow, a very stable industrial integrator (e.g., Flux-

meter Steingroever EF4) is required. Effectively it is a

d.c. method where the drift is a limiting factor.

When a time-dependent magnetic ﬁeld is applied,

the induction signal in the pick-up coil is:

pick-up

¼N

pick

pick-up

¼N

pick

rib

þ A

pick



ð7Þ

and the induction signal in the compensation coil is:

comp

¼N

comp

¼N

comp

ð8Þ

Then in the compensated system the signal is:

ind

¼ u

pick-up

 u

comp

¼N

pick

rib

ð9Þ

For this measurement system there are three impor-

tant points.

The resistance of the pick-up system is tempe-

rature-dependent for low-temperature measurements.

The resistance of a pick-up system with many

windings can be several hundreds of Ohms at room

temperature, which can change by a factor of 7 to 10

when it is cooled down to 4.2 K. Generally, the input

resistance of a ﬂuxmeter is not very high (typically

10 kO); therefore, this has to be taken into consider-

ation in the calibration procedure.

A zero-signal must be subtracted. This is necessary

because the compensation changes with temperature.

Equation (9) contains also the cross-section A

rib

the ribbon. Therefore, it is very difﬁcult to obtain an

accurate value of A

rib

of amorphous or nanocrystal-

line ribbons due to ﬂuctuations in thickness as well as

in the width. This often limits the accuracy of the

determined magnetization values.

3.2 High-temperature Hysteresis Loops

For the study of magnetic properties at high temper-

ature, generally three aspects are considered: (i) the

temperature range up to 100 1C is technically relevant

for many applications; (ii) investigations up to T

are important to study the break-down of the order-

ing process or phase transition; and (iii) higher

temperature ranges are necessary to determine the

susceptibility above T

from which the effective

paramagnetic moment can be deduced. They are

also used for the study of metastable materials. As

examples, investigations of relaxation effects in the

amorphous state (Gro

ssinger et al. 1996) or of the

Figure 8

Hysteresigraph for quasi-static hysteresis measurements and device for force measurements (Polak 1992, Holzer

1998).

503

Magnetic Measurements: Quasistatic and ac

onset of crystallization in situ to obtain nanocrystal-

line materials (Gro

ssinger et al. 1995, Schwetz et al .

1998) can be cited. Additionally, the thermal stability

investigation in metastable hard magnetic materials

such as Sm

N (Kou et al. 1995b) should be men-

tioned here. However, a complete experimental curve

of M

(T) is of great importance because, by analyzing

this curve, important magnetic parameters such as

the exchange constant or critical exponents can be

obtained.

The setup for high-temperature hysteresis loop

measurements is similar to that of Fig. 8 when the

cryostat is replaced by a furnace. One of the technical

difﬁculties of the construction of high-temperature

systems is to obtain a sensitive and well compensated

pick-up system that survives higher temperatures.

One other very useful high-temperature hysteresi-

graph is the system described in Fig. 9, which uses the

technically relevant line frequency of 50 Hz (Schwetz

et al. 1998). The basic idea of this system is the use of

one or two periods of the main line to generate a

reasonable high ﬁeld in an air coil. The apparatus

consists of:

A ﬁeld-generating coil with a high-power trans-

former which is connected on the primary side to the

line as power supply;

A pick-up system which is placed inside the inner

diameter of the ﬁeld generating coil;

Figure 9

High-temperature hysteresis measurement system.

504

Magnetic Measurements: Quasistatic and ac

A switching unit with a solid-state, high-current

switch;

Electronic devices to compensate, amplify, and in-

tegrate the signal induced in the pick-up system;

A storage oscilloscope and a personal computer

(PC);

A heating system employing a biﬁlar wound heater

in order to avoid a ﬁeld due to the heating current.

For producing the ﬁeld a current (several cycles of

50 Hz) is applied to the ﬁeld-generating coil which

will induce a signal in the pick-up system. This signal

is compensated, ampliﬁed, integrated, and then saved

in the memory of a storage oscilloscope. The ﬁeld-

producing current is measured by shunts and stored

also in the oscilloscope. Afterwards the measured

data are transferred to the PC. The switching and

heating procedures, the oscilloscope, and the elec-

tronic devices are controlled by a PC.

This a.c. method system is very easy to realize.

However, very close to application, due to possible

50 Hz noise, the accuracy is rather limited. With this

system many interesting in situ high-temperature

studies can be performed (Schwetz et al. 1998). Due

to the high intensity of the applied ﬁeld (maximum

ﬁeld of 300 kAm

1

), this system can be used to in-

vestigate the semihard magnetic materials.

These are only two examples of high-temperature

systems. There are many other possibilities to realize

a high-temperature magnetometer that works in the

gap of an electromagnet or in the temperature-

isolated hole of a superconducting solenoid. Here,

various methods such as the integration of the in-

duction voltage in a pick-up system or the detection

of an a.c. voltage (of a VSM) can be used.

4. Characterization of Hard Magnetic Materials

The magnetic methods characterizing hard magnetic

materials (see Hard Magnetic Materials, Basic Prin-

ciples of ) differ slightly from those of soft magnetic

materials. In hard magnetic materials the shape of the

initial curve, minor loops as well as recoil curves, is

important for studying the magnetization process.

Hard magnetic materials have a high coercive

ﬁeld (typically between 240kAm

1

and 1600 kAm

1

)

where the origin is a high magnetocrystalline aniso-

tropy. Therefore, high external ﬁelds are necessary

for studying these types of materials. A standard tool

for studying permanent magnet materials is an elec-

tromagnet as shown in Fig. 10. The ﬁeld is generated

by the electromagnet and the sample is in the air gap

of pole pieces. In the case of room-temperature

measurements the sample should have polished and

plane surfaces in order to avoid undeﬁned air gaps

which cause stray ﬁeld errors. The magnetization can

be measured by pick-up coils and the actual ﬁeld by a

magnetic potentiometer or by Hall probes. The signal

is then integrated using a ﬂuxmeter. Such a system is

well suited for measuring minor loops and recoil

curves; however, for determining the full loop or the

saturation magnetization, the achievable external

ﬁeld, which is about 800 kAm

1

, is not sufﬁcient.

Additionally, a determination of H

for high coercive

ﬁeld materials is also limited by the insufﬁciently low

external ﬁeld. Because an electromagnet has a high

induction the change of ﬁeld with time is slow.

Therefore, the measurement usually takes seconds

and the sensitivity of the system is limited by drift

problems as well as by noise.

If measurements at variable temperatures are to be

performed this system loses its advantage (magneti-

cally closed system) because the use of a furnace or a

cryostat causes an air gap, which means that the de-

magnetizing ﬁeld of the sample has to be considered.

Generally, the available ﬁeld in the air gap of an

electromagnet is too low (800–1200 kAm

1

) for mod-

ern hard magnetic materials. This situation can be

improved using a superconducting solenoid which

allows the ﬁeld range to be extended such that ﬁelds

up to about 20 T become available.

5. Conclusions

The main techniques to characterize hard magnetic

materials besides the a.c. susceptibility measurements

are magnetometers as described in Sect. 2. In short,

the advantage of a VSM is that it gives good accuracy

in reasonablly high ﬁelds covering a wide ﬁeld and

temperature range. A SQUID is an extremely sensi-

tive device where a magnetic ﬁeld resolution in the

order of femtoteslas can be achieved.

See also: Magnetic Units; Magnetism: High Field

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Magnetic Micro-actuators (MAGMAS)

Magnetic micro-actuators and systems (MAGMAS;

also called ‘‘MAG-MEMS,’’ (micro-electromechanical

systems)) present outstanding performance for pow-

erful integrated conversion from electrical to mechan-

ical energy. MAGMAS, due to available forces and

energies, belong to the recently appearing ‘‘Power-

MEMS’’ family. Besides, the diversity of structures

and the range of possible applications, including

distance and remote-controlled actuation, explains

the many recent publications and the large scope of

research in this ﬁeld. The topics addressed range from

power energy microsources, to micromotors, micro-

actuators, microswitches, levitation and remote-

controlled magnetic micro-objects. The corresponding

devices comprise optical, electrical, radiofrequency

(RF), biomedical, and ﬂuid dynamics applications.

MEMS are presently diversifying towards diffe-

rent principles of actuation, either electrostatic, mag-

netic, piezoelectric, or thermal, or actuation based on

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Magnetic Micro-actuators (MAGMAS)

shape memory alloys. Each principle presents specific

properties so as to cover a wider range of possibilities

and applications. MAGMAS’ development is recent,

rapid, and promising as most specific and limiting

technological problems are progressively being sol-

ved. It combines magnetic materials now available

on silicon: microcoils, soft magnetic materials, micro-

magnets, magnetostrictive thick layers, etc., with

either an integrated or remote-control ﬁeld resulting

in numerous possibilities. MAGMAS deserve more

credence and attention from MEMS designers, and

materials scientists should continue to develop better

thick-ﬁlm patterned permanent magnets compatible

with microsystem technologies.

Stress will be put on downscaling laws. Due to

both new available materials and the impressive in-

crease of current density in microcoils, it is demon-

strated that downscaling actually results in more

effective electromagnetic converters. That magnetic

interactions actually improve on downscaling is quite

new and revolutionary.

1. A Brief History of MEMS and Magnetism

Developments in microelectronics led to MEMS,

where the ﬁrst actuations of thin beams employed

were electrostatic, aluminum conductors and elec-

trodes being standard. Thermal bimorph actuation

came later. Development of other actuation principles

(electromagnetism, piezoelectricity, magnetostriction,

shape memory, etc.) has been slower because basic

components and specific materials are not standard.

Early on, it was felt strongly that electromagnetism

would not be very relevant for MEMS. Based on the

simpliﬁed interactions between conductors in air or

between a conductor in air and a piece of iron, but no

effective electromagnetic power conversion, down-

scaling is catastrophic! This is even more so when

drastic changes of essential parameters such as gra-

dients or current densities were not considered when

downscaling! Obviously, many recently published

‘‘downscaling laws’’ still favor electrostatics consid-

ering electromagnetism in its less appropriate and less

downscalable conﬁguration. These theories did not

consider direct and homothetic scale reduction of the

familiar electrical machines in daily use.

Generally, the perspectives for downscaling of

different types of actuation for MEMS were neutral

or pessimistic for magnetism, whereas a rigorous

electrical engineering approach led to very positive

conclusions (Jufer 1994). In the late 1980s, the ﬁrst

teams designed and promoted magnetic micro-actu-

ators (Wagner and Benecke 1990, Guckel et al . 1991,

Benecke 1994). They wisely remarked that perma-

nent magnets are vital to magnetic actuation but

that unfortunately their integration needs to be mas-

tered (Guckel 1996). This is still a bottleneck. Many

other works deserve to be read (Nami et al. 1996,

Busch-Vishniac 1992). More details on MAGMAS,

principles, materials, and applications have been col-

lected in the book of Cugat (2002). The reader is also

referred to the Feynman (1992, 1993) articles for

perspectives in the early 1990s.

2. Magnetic Interactions and Downscaling at

Constant Current Density d

For simple energy considerations, electrical engineer-

ing clearly indicates that macroscopic devices (elec-

trical engines and actuators that dominate the

macroscopic world) are mainly composed of ferro-

magnetic materials (either soft magnetic materials or

permanent magnets). Hence, the focus is on laws of

scale reduction applied to such structures.

2.1 Magnetic Field of Magnets

A magnet of volume v

and magnetic polarization J

generates a scalar potential V at any point P located

at a distance r from the magnet. The magnetic ﬁeld H

is the local gradient of the scalar potential V:

VðPÞ¼

4pm

J r

; H ¼rV ð1Þ

Homothetic miniaturization (all dimensions re-

duced by the same factor k ¼10, 100, 1000, y)ofa

given magnet (homogeneous material and constant

polarization J

) generates a constant magnetic ﬁeld

H, whereas the scalar potential V(P) is divided by k.

The relative geometry and magnitude of the ﬁeld map

around a magnet remain unchanged after scale re-

duction (Fig. 1). A direct and remarkable conse-

quence is that the ﬁeld gradients are multiplied by k.

This has a tremendous impact on the interaction

forces.

Figure 1

Top: Magnet—magnetic ﬁeld unchanged with scale.

Bottom: Interactions—magnet–current and magnet–

magnet.

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Magnetic Micro -actuators (MAGMAS)

2.2 Interactions between Magnets and

Current-Carrying Conductors

Intuitively, one considers the forces of magnets on

current-carrying conductors. Homothetical downscal-

ing of the magnetic ﬁeld maps due to the magnets

leaves the Lorentz forces on conductors unchanged.

Obviously it is equivalent to consider the forces

exerted by the conductors on the magnets. The Biot–

Savart law states that the magnetic ﬁeld H created in

point P by a conductor of length dl and cross-sec-

tional area S, carrying a current density d (current

I ¼d S), is

HðPÞ¼

d  S  dl r

ð2Þ

Hence, scale reduction by a factor of 1/k decreases

the ﬁelds H due to the conductors by a factor of k.

The magnetic force F on the magnet derives from

their mutual magnetic interaction energy W

¼J  v  H; F ¼rW

ð3Þ

Scale reduction by a factor k reduces the energy W

by a factor 1/k

, F by a factor 1/k

, and the force-

to-weight (or force-to-volume) ratio does not change.

2.3 Torque on a Magnet Due to External Field

The torque experienced by a magnet of polarization J

and volume v immersed in a homogeneous ﬁeld H

is proportional to M and H (where the magnetic

moment equals M ¼vJ). During the homothetic 1/k

reduction, both H and J remain constant, thus the

torque-to-volume ratio also remains constant.

2.4 Magnet–Magnet and Magnet–Iron Interactions

Once again, the magnetic ﬁeld of magnets remains

unchanged upon downscaling. Hence, for magnet–

magnet or magnet–iron interactions, Eqn. (3) shows a

decrease of W

by a factor 1/k

, F by a factor 1/k

and

an increase by k of the force-to-volume ratio.

Not only are macroscopic magnetic systems the

most performing electric-to-mechanical converters

but their scale reduction by 1/k can increase the

force-to-weight ratio by k, allowing MAGMAS to

provide large forces exactly as an ant can carry much

heavier charges relative to its own weight than can an

elephant.

2.5 Scale-reduction Laws on Magnetic Interactions

Similar calculations can be done for the magnetic in-

teractions involving soft ferromagnetic materials

combined to currents, and time-variation-induced

currents (Cugat 2002, see chapter 2); see Fig. 2.

For a constant current density d, the following

conclusions can be drawn:

The main magnetic interactions beneﬁt from

scale reduction.

The most efﬁcient magnetic interactions in

devices of small dimensions involve permanent

magnets.

The forces between two magnets or between a

magnet and a soft iron core increase by a factor k

upon scale reduction by a factor 1/k.

The forces between a magnet and current-car-

rying conductors, when effective on a macroscopic

scale, remain as effective once miniaturized.

Inductive effects are to be avoided.

3. Permanent Magnets for MAGMAS

The application of MAGMAS requires the integra-

tion of small but high-quality magnets. Material

researchers are addressing this bottleneck and new

results are presently being reported. Large availabil-

ity of thick high-quality magnets in microtechnology

will boost the performance of electrical-to-mechani-

cal conversion and, more generally, of MEMS.

3.1 Micromagnet Fabrication Techniques

The most common and most powerful micromagnets

currently used in MAGMAS are those individually

micromachined from bulk rare-earth permanent

magnets (REPMs) such as Nd–Fe–B or Sm–Co mag-

nets (see Magnets: Sintered), generally using machin-

ing by spark cutting. However, this method is hardly

compatible with full integration or batch fabrication.

Many other ways to obtain micromagnets for

MAGMAS are being developed (see Magnetic Films:

Hard). These methods all provide good results but

each suffers from a moderate or major drawback

(Cugat 2002, chapter 5). Some techniques are well

adapted to microfabrication (electroplating of Co–Pt,

screen printing of bonded powders) but the resulting

magnetic properties are relatively poor compared to

Figure 2

Effect of scale reduction 1/k on volume force for basic

magnetic interactions, for a constant current density.

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Magnetic Micro-actuators (MAGMAS)

bulk REPMs (Zangari et al. 1996). Other techniques

giving excellent magnetic properties are sputtering

(Linetsky and Kornilov 1995), pulsed laser deposition

(Cadieu et al. 1998), low-pressure plasma spraying

(Rieger et al. 2000), or direct sintering (Yamashita

et al. 2002). But either the thickness of the deposited

layer is too thin, or it is difﬁcult to adapt the process

to microtechnology and batch fabrication (high dep-

osition temperature, chemical pollution, slow depo-

sition rate, small deposition surface). An important

aspect currently emerging is the patterning of thick

magnet ﬁlms (Budde and Gatzen 2002). Optimal

magnetic orientation of micromagnets is also studied

(Kruusing 2002). The perspectives of achievements

with low-temperature electroplated high-quality

REPMs are discussed by Schwartz et al. (1998).

The large impact of the expected results justiﬁes the

increasing number of research efforts made in this

ﬁeld. However, perfect candidates for cheap, com-

patible, and fast integration of thick patterned layers

of good-quality micromagnets remain to be found.

Recently, technologies have improved rapidly and

breakthroughs in this critical ﬁeld are likely.

3.2 Equivalence between a Permanent Magnet and a

Current-carrying Coil

A magnet is equivalent to a current-carrying coil

(current density i, radius r) whose circumferential

current density (i/2pr) remains constant (Amperian

current model). This means that the equivalent vol-

ume current density increases by k when the magnet

(or the equivalent coil) is downscaled by 1/k.

4. Current Density in Microcoils

Magnetic sensors use microcoils and MAGMAS re-

quire them for variable ﬁelds and forces. Microcoils

of various shapes and sizes can stand very high cur-

rent densities. In macro-applications, the limitation

to 3–5 A mm

2

is mainly due to thermal overheating.

In micro-applications, thermal downscaling laws al-

low densities of 10

–10

Amm

2

. While Joule con-

duction losses are volume related, the heat ﬂow (for

cooling) is proportional to the cross-section and in-

versely proportional to the thickness the heat ﬂows

through. Hence, heating scales as k

, whereas heat

ﬂow dissipated by conduction scales as k. For the

same temperature, Joule losses can increase by k

k ¼k

. These Joule losses being proportional to the

square of the current density, the current density can

increase as k. Hence microcoil downscaling by 1/k

can be done with the current density increasing by a

factor of k. This is equivalent to a constant surface

current density or, in other words, magnets and ac-

tual microcoils downscale in the same way. As forces

and energies are proportional either to that current

density or to the square of that density, this deeply

boosts scale reduction laws.

The small volume-to-surface ratio of microcoils

(planar geometry) and the direct contact with good

heat-conducting substrates (e.g., Si), allows even

higher DC currents, not to speak of pulsed currents

(Cugat 2002, chapters 3 and 4, MacKay et al. 2000).

However, it cannot be avoided that such increases

in current densities and the concomitant increases of

thermal losses in conductors can reduce both the

efﬁciency and the autonomy (if used on batteries),

and needs adapted heat sinks. These side-effects

are practical limiting parameters for real design of

MAGMAS.

5. Optimal Downscaling Laws (dBk)

With good magnets and current densities increasing

as k, the downscaling laws can become significantly

modiﬁed (Fig. 3).

One can draw the following outstanding conclu-

sions (see Table 1):

All magnetic interactions beneﬁt from scale

reduction.

Magnet–magnet or magnet–iron core forces in-

crease as k.

Current volume forces increase the same way

and possibly faster (pulses). It depends on the specific

characteristics of the application and the associated

side-effects stated earlier.

current

frequency

× k × k × 1/k X

Magnet

Induction

Iron

or current Magnet

Reduction

factor

1/k

Figure 3

Effect of scale reduction 1/k on basic magnetic

interactions, with current density increasing as k

(constant temperature).

Table 1

Optimal downscaling laws.

Downscaling 1/k

Volume, mass, or gravity force 1/k

Heat ﬂow dissipated by conduction 1/k

Electrostatic forces 1/k

Current density (constant temperature) k

Most magnetic interactions (1/k

)1/k

Force/volume for most magnetic interactions k

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Magnetic Micro -actuators (MAGMAS)