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IntermetallicCompunds_Vol4/4-4/22BF3A80FD58D2B095241FA1CFC835B.pdf1. Introduction

The term magnetostriction should strictly be reservedto describe the change in dimensions of a magneticmaterial as it undergoes magnetization, and often thedistinction between 'magnetostriction' and 'magneto-elasticity' is, unfortunately, not made; confusion betweenthem occurs, especially with amorphous materials inwhich the response to the application of stress produceslarge changes in the magnetization because of the highmagnetoelastic coupling, whereas dimensional changesgenerated during magnetization are usually quite small.The distinction between the two terms is that magneto-elasticity can be seen in a variety of different responses,not only as a magnetostrictive dimensional change butalso in the 'AE effect' (a change in an elastic moduluswith the state of magnetization) or stress-inducedmagnetic anisotropy, for example. Its origin can be dueto an interplay of either the magnetic exchange energyor ionic energy, which depends on the strength andsymmetry of localized crystal fields, with the elasticbonding energy of the material.

There is also a need to distinguish between magneto-elastic coupling, as described for instance in the Kittel(1949) formation of magnetostriction, and magneto-mechanical coupling, although their fundamentalorigins are the same. Whereas the former describes theinteratomic coupling of the magnetic and elastic energieswithin a single crystal structure, gauged by themagnetoelastic coupling coefficients Bx and B2 (Lee,1955), the magnetomechanical coupling is moreempirical and gauges the efficiency in the conversionof input energy density \ fxH2 to elastic energy density

j Ee2 (where E is the Young's modulus and e is theelastic strain for an isotropic body), or vice versa.Magnetomechanical coupling like magnetoelastic couplingcan be anisotropic, in keeping with the crystal structure,but it is also influenced by the frequency of excitation,metallurgical condition of the sample, the manner inwhich it is housed or even its geometry. Magnetoelasticityis a term used frequently by physicists, whereas magneto-mechanical coupling is an engineering term.

The purpose of this chapter is to review researchconducted since the early 1960s on magnetoelastic metals,alloys and compounds and to show how materials withoptimum magnetostrictive properties can be employedin devices whose extraordinary magnetomechanicalcoupling efficiencies can generate large displacementseven under substantial loads.

It is beyond the scope of this chapter to includeadvances made with amorphous materials such as iron-transition metal (TM) or nickel-transition metalcompounds and rare earth-transition metal (RE-TM)compounds. These have been reviewed by Luborsky(1980) and are now finding novel applications innoncontact torque sensors, accelerometers and pressuretransducers (i.e. passive transducers). Crystalline, highlymagnetostrictive intermetallics, such as the RE-Fecompounds, for instance, are more suited to actuationor sonar projectors (active transducers), and these are thesubject of this chapter. In crystalline materials, the aniso-tropic magnetostriction is of principal interest; volumemagnetostrictive effects are usually much smaller, isotropicand normally only observed in large applied fields.

The anisotropic magnetostrictive strains in a cubicsingle crystal are given by

Magnetic, Electrical and Optical Applications of Intermetallic Compounds. Edited by J. H. Westbrook and R. L. Fleischer 1995,2000 John Wiley & Sons Ltd

Chapter 6

Magnetostriction: Materials and Applications

Rodney D. Greenough and Michael P. SchulzeDepartment of Applied Physics, University of Hull, Hull HU6 7RX, UK

V 3/ ( i )

+ 3X1 , ^a1(X2(S1(S2 + a2a3(32@3 + Cx3Cx1^1)

where a and /3 are the direction cosines of magnetizationand strain measurement, respectively, and X100 andX1 j ! are the magnetostriction constants relating to thestrains along the and axes relative to thedemagnetized state.

The anisotropy energy is given by

E01 = KMB2 phase transformation at~ 800 K is accompanied by compression of the latticeand precipitation hardening and also by a substantialdecrease in the resistivity, as seen in Figure 4. Theobserved resistivity change at the lower temperatures(10|im)

External quantum efficiency (0Io)Commercially

available- 4

0.2- 3-15










2. Compound Semiconductors for DeviceApplications

' 2 A Group H-VI Semiconducting CompoundsGroup II-VT semiconductors are binary compounds andtheir solid solutions, consisting of Group HB elements(Zn, Cd, Hg) and Group VIB elements (O, S, Se, Te),e.g. ZnSe, ZnS and (Hg, Cd)Te, are typical.

2.1.1 ZnSe and ZnSThe forbidden band gaps Eg of Zn chalcogenides are2.26 eV for ZnTe, 2.67 eV for ZnSe and 3.7 eV for ZnS,all at 300 K. Their mutual solid solutions exist over allcompositions, and thus Eg of Zn chalcogenides canchange continuously from the yellow region to the violetregion. Attention has especially centered around ZnSeand ZnS as materials for light-emitting device appli-cations in the blue region.

ZnSe and ZnS possess a phase transition between thezinc blende type and wurtzite type. The transition tem-peratures are 1698 K and 1297 K for ZnSe and ZnS,respectively (Klakov et al., 1976).

ZnS has a high equilibrium dissociation pressure(3.7 x 102IcPa) at the melting point Tm (2103 K). TheTm (1793 K) and dissociation pressure (53 kPa) of ZnSeare not as high. However, the vapor pressures of Zn,S and Se are exceedingly high in comparison with thecompounds. Consequently, it is difficult to grow large,high-quality single crystals of ZnSe and ZnS from themelt. The vapor-growth method has been intensivelystudied to avoid these problems. High-purity ZnSe singlecrystals including few twin boundaries have been grownby means of the sublimation method (Fujita et al.,1979).

As-grown crystals are high-resistance n-type, and ananneal in the melt or under the vapor pressure of Znis essential for low resistivity. Epitaxial thin-film growthhas been achieved using liquid-phase epitaxy (LPE),vapor-phase epitaxy (VPE), MOVPE and MBE. Amongthese alternatives, the MOVPE and MBE methods havebeen intensively examined because of the nonequilibriumcharacter of film growth. Since ZnSe and ZnS wafersfor epitaxial growth are not commercially available,GaAs (mismatch of 0.27% for ZnSe) and GaP (0.77%for ZnS) are very frequently employed as substrates. Theepitaxial growth temperature is 350 to -500 0C.

Undoped films are highly resistive (specific resistivityp above 106Q cm), and the photoluminescence (PL)spectra generally show strong near-band edge emissionand a weak self-activated emission.

Formation of a p-n junction leads to realization ofhigh injection efficiency in LED applications, and henceconduction-type conversion is needed. However, controlof conduction type by means of impurity doping hasproven unachievable in Group II-VI semiconductors,except for CdTe. The reason still remains unclarified.

For n-type films, iodine is useful as a donor species,and the carrier concentration can be controlled in therange 1016-1019cm~3 (Shibata et al., 1988). The resis-tivities of films doped with 1 x 1019cm~3 iodine atomsdecreased to as little as 1.3 x 10 ~3 U cm at room tem-perature. These doped films showed a strong blue PLat 300 K. For p-type films, nitrogen (Ohki et al., 1988),phosphorus (Bhargava, 1982), lithium (Nishizawa, 1986;Yasuda et al., 1988; Cheng et al., 1988) and oxygen(Akimoto et al., 1989) have been examined as acceptorspecies. Recently, a nitrogen-doping technique using ahigh-frequency plasma cell was established. Further-more, a radio-frequency plasma gas cell was installedinto an MBE chamber and low-resistance p-type filmswere grown on a GaAs substrate by plasma-assisteddoping with nitrogen (Park et al., 1990; Ohkawa, 1990).Still further, a laser diode (LD) with a (Zn, Cd)Se/ZnSequantum-well structure has been manufactured, andlaser emission with a wavelength of 494 nm at 300 K wasobserved by electric current injection (Jeon et al., 1991;Haase^a/. , 1991).

2.1.2 (Hg, Cd)TeThe narrow band gap semiconductor (Hg, Cd)Te is asolid solution of the Group II-VI semiconductors HgTeand CdTe. The band gap changes continuously fromHgTe (-0.3eV at OK) to CdTe (1.6 eV at OK).

The terminal compound CdTe generally shows nophase transition. As-grown crystals, grown from a meltwith a Te-rich composition, proved to be p-type andto contain a high concentration of Cd vacancies(Triboulet and Marfaing, 1977). By annealing under Cdvapor pressure, the crystals become low-resistance n-type (Triboulet, 1981).

Epitaxial film growth has also been studied. CdTe,(Cd, Zn)Te and (Cd, Mn)Te have been used as sub-strates. Undoped (Hg, Cd)Te films formed by the LPEmethod contained 1016-1017cm3 Hg vacancies andshowed p-type conduction. By annealing under Hgvapor pressure, the films were converted to n-typeconduction. Both conduction types could also beproduced by impurity doping.

(Hg, Cd)Te has been developed as a material for aninfrared detector in the range 1-20 ^m. Furthermore,(Hg, Cd)Te has high electron mobility and forms an

insulating native oxide of high quality in the film form;it is therefore highly useful for electrical applications.

A 'superlattice' of modulated CdTe/HgTe andZnTe/HgTe structure is highly desirable for high-speeddevices, such as Gunn diodes and hot-electron tran-sistors. In an (Hg, Cd)Te/HgTe/(Hg, Cd)Te structurewith a double barrier type, single quantum well, anegative resistance attributed to a resonance tunneleffect was found (Reed et al., 1987). The peak to valleyratio was 1.4:1, comparable with that of AlGaAs/GaAs.

2.2 Group III- V Semiconducting CompoundsGroup IH-V semiconductors are binary compounds andtheir solid solutions consist of Group IHB elements (B,Al, Ga, In) and Group VB elements (N, P, As, Sb).Table 2 shows some fundamental characteristics ofGroup III-V semiconductors. Practical interest has beenmost intensively centered on GaAs, InP, and the solidsolutions (Al, Ga)As, (In, Ga)As, and (In, Ga) (As, P).

GaAs and InP melt congruently; however, they havehigh dissociation pressures. Hence, pressures aboveatmospheric pressure are required over the melt tomaintain a stoichiometric composition during the growthprocess. The pressures are 100 and 2.7xl03kPa forGaAs (Tm = 1511 K) and InP (Tm = 1335 K), respectively.

These crystals usually contain high concentrations ofcompositional defects. Thus, growth of perfect singlecrystals of large diameter has been investigated by meansof horizontal Bridgman (HB), liquid-encapsulatedCzochralski (LEC), and vertical-gradient freezing (VGF)methods. Point defects, diffusion mechanisms, andsuperlattice disordering in GaAs-based materials havebeen reviewed in depth by Tan et al. (1991). Cr, oxygenand native point defects form deep acceptor levels inGaAs (Hobgood et al., 1982; Martin et al., 1980).

In InP, transition metal elements such as Fe (Mizunoand Watanabe, 1975; Kamada et al., 1984) and Ti(Iseler, 1986; Katsui, 1988) form deep levels. The deeplevels compensate the residual carriers electrically, andas a result the crystals become semi-insulating with pabove 5 X 106 Q cm.

The semi-insulating crystals are useful as substratesfor optoelectronic integrated circuits (OEICs), capableof reducing floating capacitance and providing electricalisolation for devices.

Recent practical interest has been greatly centered onOEICs. Furthermore, marked advancement in theMOVPE and MBE methods and in the sophisticationof small-scale, high-resolution processing techniqueshave enabled producers to propose various multi-functional or high-performance devices. A voluminousand excellent secondary literature exists dealing withGroup III-V semiconductors and their applications (forexample, Willardson and Beer, 1975-1992).

2.3 Group IV-IV Semiconducting CompoundsSiC, one of the Group IV-IV semiconductors, possessescrystal polymorphism (Marshall et al.., 1974). HexagonalQf-SiC is an indirect transition type semiconductorhaving a band gap Eg of about 3.OeV at 300K. It isuseful as a material for LEDs in the blue region. Low-resistance n-type and p-type films are grown by dopingwith nitrogen and aluminium, respectively. The blueemission is probably due to a donor-acceptor (D-A)emission in the n-type layer.

High-quality a-SiC single crystals have recently beengrown by an improved Reigh method and a sublimationmethod (Tairov and Tsvetkov, 1981). A p-n junctionhas been formed in these crystals by a rotational dippingmethod (Ikeda et al., 1979). First, a p-type layer was

Table 2. Fundamental properties of Group III-V compound semiconductorsPropertiesE% (eV) at 300 KTransition typeRecombination coefficient

(Cm2S1)Hall mobility /xe (cm2 V 1 s"1)

300K77 K

Surface recombination velocity(cms"1)

Dielectric constant (C/N)Relative permittivity e/e0Thermal conductivity

(W cm"1 deg"1)Melting point (K)Hardness (kg mm 2)



7.2IxIO-1 0


2 XlO7~ 2 . 7 x l 0 1 2






45004.5 XlO4

IxIO 3






33 0001.5 XlO5






80 000IxIO 6

2.35 XlO"1218.7




14 (/*h)

5 x l 0 1 2


BN (cubic)8.0


0.2 (500 K)


7300-10 000

grown on the crystals by immersion in an Si melt dopedwith Al, and then an n-type layer was formed byintroducing nitrogen gas into the melt.

LEDs with a central wavelength of 475 nm, anexternal quantum efficiency of 2x 10 ~4, a maximumluminous intensity of 12 mcd at an operational currentof 20 mA and continuous room-temperature operationtimes above 5000h have been developed (Koga et al.,1985).

For an SiC LED, since the band gap of a-SiC is largerthan the energy of photoemission, reabsorption ofemitted light in the LED interior is negligibly small.

In cubic (3-SiC9 undoped n-type films with a carrierconcentration of 1016-1017cm~3 have been grown ona two inch diameter Si wafer by means of chemicalvapor deposition (CVD).

Just as with cx-SiC, /3-SiC is chemically and structurallystable to high temperatures and is highly thermallyconductive. Diodes and transistors operable at hightemperatures and high electric power have been studied.

* 2.4 Group IV-VI Semiconducting CompoundsTable 3 shows some fundamental properties of GroupIV-VI semiconductors. (Pb, Sn)Te compounds whichform a complete solid-solution series of the Group IV-VI semiconductors PbTe and SnTe are direct transitiontype semiconductors.

The crystal structure of (Pb, Sn)Te is f.c.c, NaCl-type Bl (cF8). The band gap of (SnxPb1 _JTe at 77 Kdecreases linearly with increasing x from 0.22 eV for theterminal compound PbTe, becomes zero at x=0.4, andagain increases to 0.27 eV for the other terminalcompound SnTe at x= 1 (Paker and Johnson, 1981).

As-grown, undoped (Pb, Sn)Te crystals are p-type,owing to high concentrations of Pb and/or Sn vacancies.

The n-type crystals are obtained by annealing as-growncrystals in a metal-rich atmosphere, so that the Pband/or Sn vacancies are compensated. Furthermore, Pband/or Sn interstitials may be generated (Paker andJohnson, 1981). The n-type conversion is also achievedby doping with Sb, Cd or In. Tl is used as a dopantfor p-type crystals (Antcliffe and Wrobel, 1970).

(Pb, Sn)Te is mechanically soft, similar to aluminummetal. Plastic deformation readily occurs and low-angleboundaries are easily formed. Efforts have been madeto grow bulk single crystals without low-angleboundaries (Kinoshita and Sugii, 1983).

Recent studies have been made of epitaxial thin-filmgrowth by means of LPE using a Te solvent and MBE.

(Pb, Sn)Te is promising as a material for variablewavelength LDs and for photodetectors for the wave-length region 5-30 /*m. Although (Pb, Sn)Te has thesame band gap as the Group H-VI semiconductor (Hg,Cd)Te, (Pb, Sn)Te has a larger effective electron massand a smaller energy loss owing to nonradiativerecombination; therefore, it is more useful for deviceapplications.

LDs with homojunctions and double hetero junctions(DH) have been studied (Tomasett and Fonstad, 1974;Horikoshi et al., 1982). In DH-type structures, the laseroscillation temperature is below about 80 K.

A 'quantum well' in semiconductors signifies aquantum-mechanical well of a certain potential energy,and within it carriers, electrons or holes, are two-dimensionally confined. A single quantum well is themost fundamental structure, and consists of anextremely thin well layer sandwiched between barrierlayers. A quantum well may also be formed such thata light beam is confined in it, as well as electrons. Then,their confinement becomes highly effective in a multiplequantum-well (MQW) structure, and therefore the upper

Table 3. Fundamental properties of Group IV-VI compound semiconductors and semimetals


Lattice constant (nm)

Density (g cm "3)Volume dilatation coefficient

(1(T6K-1)Specific heat ( J g 1 K 1 )Thermal conductivity (W m 1 K"1)Permittivity e

Vickers hardness (kg mm 2)Melting point (K)Band gap (eV) at 77 KEffective mass

















a-0.599, b = 0.434,c=l. 120















"In these semimetals a negative band gap implies an overlap of the conduction and valence bands.

temperature limit can be expected to be raised in LDsprovided with the MQW structure. Pulse oscillation at204 K was demonstrated in an MQW laser with aquantum-well layer of (Pb, Sn)Te and a barrier layerof Pb(Se, Te), which was capable of emitting at a wave-length of 6 fim (Shinohara et aL, 1985). However, evenin a simple DH-type laser of wavelength 3.27/xmstructured by (Pb, Cd) (S, Se)/PbS/(Pb, Cd) (S, Se),pulse oscillation operation at 200K was observed(Koguchi et al.9 1987).

2.5 Group V-VI Semiconducting CompoundsBi2Se3, Bi2Te3 and Sb2Te3 compounds have a layeredstructure with a rhombohedral unit cell (C33, hR5). Thephysical and mechanical properties of these compoundsare very anisotropic arising from the hexagonalsymmetry of the crystal. Solid solutions of these narrowband gap semiconductors have been utilized for thermo-electric devices (Lovett, 1977; Uemura and Nishida,1988; see also Vedernikov, Chapter 20 in this volume).

The figure of merit Z for evaluating thermoelectricmaterials is given by a2o/x, where a is the thermo-electromotive force, a is the electrical conductivity andx is the thermal conductivity. Some n-type and p-typethermoelectric materials are prepared with Bi-rich andSb-rich solid solutions, respectively. The optimum Zvalue is attained by controlling the electron and holeconcentrations with halogens and tellurium, respectively,as dopants. Commercial thermoelectric materials arecommonly made with a uniaxial solidification techniquein which the a axis is grown as the preferred directionsince it has better thermoelectric properties than the c-axis (Uemura and Nishida, 1988). The Z value of thematerial oriented along the a axis is about3.OxIO-3K"1. In order to increase the mechanicalstrength, however, the Z value of most commercialmaterials is suppressed to 2.0-2.6XlO-3K"1 byallowing some misorientation of the a axis.

Since Bi2Te3 crystals with the layered structure showpronounced mechanical cleavage on the c plane,the uniaxially solidified polycrystalline samples and thesingle crystals are readily broken along the solidificationdirection and the c plane. It was found that the mech-anical strength of Bi2Te3 can be improved by powdermetallurgy techniques without spoiling the thermoelectricproperties. Despite the fact that the thermoelectric powerand electrical resistivity of the sintered materials arevaried, optimization is again realized by controlling thecomposition of the alloying element and the dopant andby increasing the phonon thermal resistivity owing tograin boundary scattering. Recently, anistropic sintered

materials of Bi2Te3 have been fabricated by the hot-pressing technique (Ohsugi et al.9 1989). A maximumfigure of merit for n-type Bi2Te2 85Se015 of up to3.7XlO -3K1 , perpendicular to the pressing direction,has been achieved, which is much higher than that ofsingle crystals (Kaibe, 1989; Kaibe et al., 1989).

These sintered materials are quite suitable for verysmall thermomodule applications, for example, forbuilding cooling modules for optical communicationdevices and equipment for semiconductor processing, etc.

2.6 Transition Metal Silicide Semiconductors HTransition metal silicides, which have been investigatedmainly as refractory materials, have interesting physicaland chemical properties owing to the continuouschanges in the valences of the transition metal atoms.

CrSi2, MnSi2_x (0.273^ x^0.250) and /3-FeSi2 aresemiconductors developed as thermoelectric conversionmaterials for use at temperatures above 1200K (seeVedernikov, Chapter 20 in this volume). Their heat-resistant properties are attractive, even though thethermoelectric power of these materials is not as largeas that of the chalcogenide thermoelectric materials, andtheir electrical conductivities are comparatively high.

Of all the silicides, a degenerate semiconductor of p-type CrSi2 with a hole concentration of 1020cm~3 hassuperior heat resistance and is used for a high-performance thermoelectric leg (Nishida, 1972). Also,nonstoichiometric MnSi2 _x is a p-type, degeneratesemiconductor (Nishida, 1972; Kawasumi et al.9 1981).It is noted that /3-FeSi2 becomes p-type by doping withMn and/or Al (Nishida, 1973) and n-type with Coatoms. These materials have found use in high-performance, high temperature applications (Ware andMcNeill, 1964).

On the other hand, the thermoelectric performanceof the semimetal CoSi (its conduction and valence bandsoverlap by about 0.02 eV) is inferior to that of thesilicides mentioned above; it is an n-type material witha comparatively large thermoelectric power (Asanabeet al., 1964). When this material is combined with p-type CrSi2, one can obtain a CoSi-CrSi2 thermo-couple. As with the previously mentioned silicides,including FeSi2, mass production technology is estab-lished (Tokushima et al.t 1972; Uemura and Nishida,1988), and developmental research on these materialsis now in progress for thermoelectric generatorapplications. Since the electric power of these thermo-couples is sufficient for power supplies for safety devicesfor gas and petroleum combustion instruments and soon, various ideas have been proposed in this field.

Note that since these materials are useful forapplication at higher temperatures with material costsmuch lower than those of the chalcogenides, specialapplications are possible, such as high-temperaturecontrollers for combustion and chemical reactioninstruments or furnaces and for thermoelectric generatorsthat utilize waste heat.

Mesotaxially formed, buried layers of silicides havealso been investigated. Such patterned layers of CoSi2have been used as gates for permeable-base transistors,and have been proposed for application as buriedcollector contacts for high-speed bipolar transistors, asground planes for microstrip lines, and as mirrors inGaAs-Si modulators. In addition to CoSi2, Ti, Fe, Cr,Ni, and Y silicides have also been formed by mesotaxialimplantation (Celler and White, 1992).

2.7 Ternary Semiconducting CompoundsAs ternary compounds, Group H2-III-VI4 (defectchalcopyrite type), Group I3-V-VI4 (famatinite), andGroup I2-II-VII4 (stannite) are known (see Parthe,Chapter 14 in Volume 1). Chalcopyrite-type compoundshave been most intensively studied of all ternary semi-conducting compounds. They are expressed by themolecular formulae 1-HI-VI2 and H-IV-V2, and havethe same tetragonal structure with c/a 2 as chalcopyriteCuFeS2 (El1, tI16) (Shay and Wernick, 1975). High-quality single crystals of 1 cm3 have been grown by thetraveling heater method (Sugiyama et al., 1989) and bythe gradient freezing method (Ando and Katsui, 1981).

The principal features of chalcopyrite-type compoundsare birefringence and a large, nonlinear optical constant.Typical compounds are AgGaS2, AgGaSe2, ZnGeP2and CdGeAs2. These compounds with a wide band gapand large lattice distortion show a degeneracy of therefractive index at a specified wavelength (Lotspeich,1979). Recently, optical filters were prepared utilizingthis property (Horinaka et al., 1989).

Chalcopyrite-type semiconductors are direct transitiontype. Among them, CuAlS2 and CuGaS2, belonging tothe 1-HI-VI2 compounds, have Eg above 2eV andthus show potential for use in light-emitting devices inthe visible region. The conduction type of as-growncrystals is usually highly resistive p-type, the resistivityof which can be decreased by annealing under aconstituent chalcogen atmosphere.

High-quality, impurity-doped, heteroepitaxial filmshave been studied by means of MOVPE using cyclo-pentadienyl(triethylphosphine)copper(I) (CpCuTEP) asa Cu source material (Hara et al., 1987) in a closed-tubehalide VPE method with chloride sources and MBE.

Figure 3. Spinel structure Hl 1 (cF56) (unit cell of A sites).The heavy lines define a subcell which is one fourth of the unitcell of A sites

Recent interest has focused on CuInSe2 as a materialfor a high conversion efficiency, thin-film solar cellwithout optical degradation. CuInSe2 has an Eg ofabout 1.0 eV at room temperature and a high absorptioncoefficient, above 1 x 105 cm"1. The efficiency of solarcells with polycrystalline CuInSe2 films, produced mainlyby means of selenization and vacuum evaporationmethods, exceeds 14% (Michell et al., 1988; Basol andKapur, 1990). Development of CuInSe2 solid solutionsand tandem structures with amorphous Si are inprogress to realize conversion efficiencies above 20%.

2.8 Ferromagnetic Semiconducting CompoundsThere are many kinds of magnetic semiconductors; mostare binary and ternary compounds. The attention ofmany researchers has focused on a group of ferro-magnetic chromium chalcogenide-based ternary com-pounds represented by MCr2X4, where M stands for anonmagnetic metal, such as Cd or Hg, and X is S orSe. They, like ferrite, are of spinel structure (Hl1,cF56), as shown in Figure 3. The portion of the structuredefined by the thick solid lines corresponds to one fourthof the unit cell in the spinel structure. All M ions lieat A sites, each of which is a lattice point at the centerof a tetrahedron of anions. The Cr ions all lie at B sites,each of which is a lattice point at the center of anoctahedron of anions. Neighboring pairs of Cr ions areferromagnetically bonded.

Table 4 lists the properties of major chromium chalco-genide-based ternary compounds. Compounds having Cuatoms at A sites are metallic conductors and not ferro-magnetic; compounds having Zn atoms at A sites areantiferromagnetic. Although these compounds are notferromagnetic, they are listed in Table 4 because a com-parison of ferromagnetic materials with nonferromagnetic

A siteB siteAnion

Table 4. Properties of chromium chalcogenide based ternary compounds




(room temperature)(ficm)

AsymptoticCurie point


Curie pointTc(K)


(4.2 K) foe permolecule)




a (A)(lA = 0.1nm)Compound









~103~1(T4~io- 4~10~ 4


+ 152+ 204+ 137+ 192-12+10-234+14-390 40+ 390+ 465+ 400-392+ 18

+ 115

Antiferromagnetic,Neel point TN = 9 K




TN= 16 KT N = 1 8 K

TN = 23K












The u parameter gives the lattice site of the anion (ideally u = 0.375, see Figure 3).

materials, both having the same crystal structure, is ofsignificance in understanding the chromium chalco-genide-based magnetic semiconductors.

With the above exceptions, chromium chalcogenideternary compounds are both ferromagnetic and semi-conducting, and they show unique conducting behavioras general magnetic materials. These compounds havebeen the objects of numerous studies on their electrical,magnetic, and optical properties. A combination ofthese properties seems promising for new electronicdevices and is of worldwide interest from the viewpointsof physicochemical qualities and applications. Ferro-magnetic semiconductors containing Cd or Hg atomsat the A sites can be doped with impurities to exhibitp-type or n-type conduction. These materials havecharge carriers with the largest mobilities among all themagnetic materials. An appropriate chemical treatmentmakes the magnetic anistropy of these compounds lesssignificant and forces them to have a narrow microwaveabsorption width. Thus, we may be able to find newfields for their application.

CdCr2Se4 is a material that has been studied as athin film for p-n junction elements (Wen et al., 1968).HgCr2Se4 and CdCr2Se4 are promising materials fortunable semiconductor lasers for the mid-infrared region(Masumoto and Koguchi, 1976, 1982; Koguchi andMasumoto, 1978; Takahashi et al., 1983). The wave-length of their laser oscillations changes with ambienttemperature and applied magnetic field; the rate ofchange is about 10 times that of IV-VI semiconductingcompounds. For example, the optical absorption edgeof HgCr2Se4 is 0.83 eV (1.49 /on) at 300K; it sharplydecreases with decreasing temperature (a phenomenonknown as the red shift), reaching 0.28 eV (4.43 fim) at4.2 K (Harbeke et al., 1968; Lehmann and Emmenegger,1969). At the Curie temperature (about 120 K), the edgeis further decreased by 10% with the application of a10 kG magnetic field (Lehmann and Emmenegger, 1969).

3. Electrical Applications

3.1 Materials with High Electron Mobility

Si has the following characteristics: (i) its electronmobility is relatively high; (ii) high-purity and high-perfection single-crystal wafers up to eight inches indiameter are commercially available; (iii) insulatingSiO2 is quite compatible with Si and is highly useful fordiffusion masking and passivation; (iv) its strength ishigh; and (v) its Eg is moderate for integration withother materials.

These characteristics make Si highly suitable forintegrated electronic applications. Although Si at presentis certainly in the very large scale integration (VLSI) age,attempts to realize with compound semiconductorsbetter integrated circuits (ICs) than those with Si havebeen increasingly made.

In materials for high-speed device applications, highelectron mobility and high saturation carrier velocityare required. Table 5 shows a comparison of character-istics among high-mobility Group IH-V semiconductors.The narrower the band gap and the smaller the effectivemass, the higher is the electron mobility. Saturationvelocity has the same trend as the mobility. The mobilityof carriers is affected by various scattering mechanisms,and generally scattering by optical phonons is pre-dominant in compound semiconductors.

GaAs, InP, InSb and InAs show high electronmobilities and high saturation velocities relative to Si.They therefore are suitable for high-speed deviceapplications. In GaAs, semi-insulating crystals arecommercially available, and thus low power con-sumption is expected, as well as high-speed operation.Both InSb and InAs have higher electron mobility thanGaAs. However, conduction in both of these semi-conductors, owing to their band gaps at 300 K, is intrinsicand, hence, low temperatures are needed for operationwith a high margin of reliability.

There are two conceptions in the further developmentof high-speed integrated circuits using compound semi-conductors. One direction aims at speeds above thoseof Si ICs, and the other direction is toward Si/VLSIdevices resistant to radiation damage and with lowpower consumption for space development applications.In the latter case speed is sacrificed. In the former, aswell as in GaAs metal Schottky field effect transistors(MESFETs), the IC being at an LSI level, superhigh-speed devices such as the high electron mobilitytransistor (HEMT), the heterojunction bipolar transistor(HBT), the permeable-base transistor (PBT), the staticinductance transistor (SIT), and the metal insulatorsemiconductor field effect transistor (MISFET) areobjectives.

3.2 GaAs MESFET ICsThe driving force for GaAs FET IC developments isthe exploitation of electron mobilities several timeslarger than that of Si by direct circuit integrationon a semi-insulating substrate using ion implan-tation and extension of recently developed, ultrahigh-resolution processing techniques to Group IH-Vsemiconductors.

Currently under development are MESFETs, metaloxide semiconductor (MOS) FETs, junction-type FETsand so on, the FET being a fundamental device forGaAs ICs. Because of the underdeveloped planar tech-niques for p-n junctions, the leading device type is theMESFET.

Studies of GaAs ICs started in 1970, at which timethe Si LSI was developed. By 1974, the normally-onGaAs IC was proposed. In 1977 and 1978, integratedcircuits and normally-off ICs were produced,respectively. Since then, remarkable progress in thedegrees of integration and speed achieved has been madeyearly.

A MESFET is a transistor with a structure which isformed from a Schottky gate electrode and source anddrain ohmic electrodes on an n-type semiconductorlayer. A metal-semiconductor ohmic contact is used asa gate electrode in the MESFET, instead of a p-njunction. The operational function is such that the gateinput signal changes the spread of the Schottky contactdepletion layer, and thus changes the thickness of the

channel layer, which provides the conductance betweenthe source and drain. Figure 4 shows the two typicalstructures of MESFETs. There are normally-onelements (depletion type) and normally-off elements(enhancement type) in fundamental circuit constructionsof MESFETs. In the normally-on type, the current iscut off by applying a negative electric field. On the otherhand, in the normally-off type, the source and drainpinch off at the built-in voltage and are turned 'on' bya positive voltage.

Although normally-on types operate at very highspeed, the power consumption is large, the circuit iscomplex, the occupied area is large, and hence they areunsuitable for integration. Normally-off types havesimple circuit construction and small power con-sumption, and thus are suitable for high integration.However, normally-off types possess several problemsstill to be solved: the manufacturing conditions aresevere, the logic amplitude is small, the limitation onthe pinch-off voltage is strict and the switching speed israther low. Developments of GaAs FET ICs are making

Table 5. Property comparison


_ (eV) 300 K77 K

Lattice constant (nm)Electron affinity (eV)Electron mass0 m/m0


of high electron mobility semiconductors




Hall mobility at 300 K (cm2 V-1 s"1)


Zi6(Cm2V"1 s"1) at 77 KSaturation drift velocity vm

( c m s 1 )Saturation field Em (V c m 1 )Intrinsic carrier concentration

nx ( c m 3 ) 300K77 K

80 000450

IxIO 6

5.2 XlO75.6 XlO2

2 XlO163.8 XlO9




33 000450

1.5 XlO5

3.2 XlO74 XlO3

1 x 10153.5 XlO3





< 6 x l O 6

2.7 XlO12





4.5 XlO4

2.3 XlO71.2X104

1.4 XlO7





1.7-2.OxIO75.2 XlO3

1.2 XlO7





3.6 XlO4

6 XlO64 XlO3






2.4 XlO4

IXlO72 XlO4

1.2 XlO10

Figure 4. Fundamental structures of the two MESFET types


Source Gate


Source Gate Drain


(b) Normally off type

Layer ofn-type


(a) Normally on typeDepletion


steady progress through a reiterative process of experi-mental production and evaluation.

A serious problem on the performance side of theGaAs MESFET LSI is that, compared with the SiLSI, the speed is not as high as expected. Furthermore,there is a problem of low yield. Uniformity of character-istics is important for LSI, and hence reliable achieve-ment of a high-quality, semi-insulating GaAs substrateis necessary.

3.3 New High-speed DevicesSuperhigh-speed devices based on new operationalprinciples have been proposed as a result of markedprogress with heteroj unction techniques and high-resolution processing techniques. Here, HEMTs, HBTs,PBTs, SITs and MISFETs are introduced.

3.3.1 HEMTsA modulated, doped GaAs/AlGaAs heteroj unction isa junction formed from undoped GaAs and AlGaAsheavily doped with an n-type impurity. In single hetero-junctions and in a 'superlattice' using modulated,doped GaAs/AlGaAs, the mobility, especially at lowtemperatures, is greatly increased compared to GaAs


of a nearly equal carrier concentration (Dingle et al.,1978).

Since in such a junction as GaAs/Al0 3Ga0 7As theelectron affinity of GaAs (iig = 4.07 eV) is larger thanthat of Al03Ga07As ("g = 3.75 eV), electrons suppliedfrom the donor in n-type AlGaAs flow to the GaAs sideand are accumulated in quite a narrow region of theheteroj unction, which then forms a two-dimensionalelectron gas (Dingle et al., 1978). Conduction electronsare spatially isolated from the donor and are less scatteredby ionized and neutral impurities; therefore, the mobilityis exceedingly high (Hiyamizu and Mimura, 1981).

The HEMT is a superhigh-speed device utilizing asingle, modulation-doped heterojunction (Mimura et al.y1980). Fundamental to the HEMT is a heterojunctiongate. Figure 5 shows the structure and schematic banddiagram near the gate of the MESFET and the hetero-junction gate FET. In the MESFET, the channel layermust have impurity concentrations of about 1016cm~3in order to switch the channel and to obtain the requiredchannel current. Consequently, subsequent impurityscattering caps the electron mobility at the channel. Onthe other hand, in heterojunction types impurityconcentrations in the channel are below 1015 cm~3. Aheterojunction gate with high impurity concentrationsis constructed by forming a layer with a larger band gap

(b) Heterojunction gate FETHeteroepitaxial film (large n)

Source Gate Drain

Semi-insulating GaAs


Source Gate Drain

Semi-insulating GaAs


Channel layer


Channel layer

Figure 5. Comparison of MESFET and heterojunction gate FET (upper, structures; lower, schematic band diagrams near the gates)

and smaller electron affinity than the channel. Asufficient supply of the required electrons is obtainedfrom the high-concentration layer.

In the HEMT structure, the low-field electronmobility is 8 x 103 cm2 V"] s"l at 300 K, twice that ofthe GaAs MESFET, and the saturation velocity is1.5-1.9XlO7CmS"1; hence the current gain cutofffrequency/T in switching is high. For example,/T was80 GHz at a gate length of 0.25 /mi, twice that of a GaAsMESFET with the same gate length (Abe et al., 1987).

As materials for HEMTs, AlGaAs/InGaAs/GaAs(Chao et al., 1987) and InAlAs/InGaAs/InP (Henderson,1987) have also been studied.

3.3.2 HBTsThe concept of the HBT is defined in Shockley's patent(Shockley, 1948). With the advancement of hetero-junction technology and ion implantation technology,devices have since been experimentally produced andtheir performance confirmed.

The features of the HBT are a high degree of freedomin structural design and high speed, because movementsof electrons and holes are separately controlled(Kroemer, 1982).

Injection of minority carriers from base to emitteris so restricted, compared to homojunction bipolartransistors, as to allow doping with high concentrationsof carriers. Thus, base resistance, base thickness andconductance between base and collector decrease, andthe design of transistors with high speed and excellenthigh-frequency characteristics can be realized.

Recently, n-p-n-type (Beneking and Su, 1981) andp-n-p-type (Su et a/., 1981) bipolar transistors of homo-junction or heterojunction types using GaAs withAlGaAs have been examined. Also, an HBT with an/T of 105 GHz and a maximum oscillation frequency/MAX of 55 GHz was developed through improvementsin device structures and processing (Ishibashi, 1987).

A review article dealing with recent HBT studies hasbeen published (Sugeta and Ishibashi, 1990).

3.3.3 PBTs and SITsIn both PBT (Bozler and Alley, 1980) and SIT(Nishizawa et al., 1975) devices there is a vertical FET,as is expected for superhigh-frequency operation.

The PBT has the same structure as a base-embeddedSIT (Alley et al., 1980; Vojak et al., 1983), and consistsof a grating made from a thin film of tungsten whichis embedded in an epitaxial GaAs layer. The electronsflow from emitter to collector through slits in the metal

film. The tungsten grating is used as a gate, and a gatelength of about 0.1 /mi is easily attainable. In atheoretical comparison of PBT, MESFET and HEMTdevices with very short gate lengths, an /MAX above500 GHz was projected in PBTs for an effective gatelength of 0.15^m (Das, 1987).

The structure of an SIT is fundamentally the sameas that for a PBT but the dimensions and the impurityconcentrations of the active layer differ from those ofthe PBT. In an SIT, the distance between source andgate is extremely short, and the structure has asufficiently small resistance Rs between source and gatethat the product of Rs and mutual conductance Gm is< < 1 (Sugeta, 1986). Furthermore, the impurity con-centration is so low that the whole active layer isdepleted at the gate voltage.

The SIT is also useful for high electric powerapplications (Nishizawa, 1986).

3.3.4 MISFETsThe electron saturation drift velocity of InP is 1.5 timesas large as that of GaAs. Theoretical (Frey and Wada,1979) and experimental studies of high-speed InP FETshave been made.

To realize an MESFET in InP is difficult, owing tothe small potential barrier of the Schottky junctions inthe n-type layer. In an InP MIS-type structure, anelectron accumulation layer and an electron inversionlayer can be easily formed at the insulating film-InPinterface (LiIe et al., 1978), and hence InP is suitablefor MISFET applications. This situation is quite incontrast to the case of GaAs (Meiners, 1978).

The largest factor in the realization of high-speed InPMISFETs is the formation of a high-quality gate-insulating film. Enhancement-type MISFETs usingSiO2, Si3N4, P3N5 or anodic Al2O3 as the gate-insulatingfilm have been investigated. The channel layer wasdirectly formed on a semi-insulating substratesurface in InP, and an effective mobility of about3000Cm2V-1S"1 was attained for an Al2O3/nativeoxide double-gate structure.

A problem common to InP MISFETs is the drift ofthe drain current. Recently, drain current drift below 10% of the initial value was attained (Hasegawa,1984). From this result, it was speculated that the driftis probably owing to interface levels being distributedenergetically and spatially below a conduction band.

Furthermore, enhancement-type MISFETs usingInGaAs have been studied and an effective mobility of1500cm2 V -1S"1 was obtained using plasma oxidationof Al2O3 as the gate film.

Next Page

Front MatterTable of Contents4. Semiconductor Applications4.1 Introduction4.2 Compound Semiconductors for Device Applications4.3 Electrical Applications4.4 Optical Applications4.5 Other Applications4.6 Conclusions4.7 ReferencesAddendum


IntermetallicCompunds_Vol4/4-4/6CAD44E0508A83C2455412F419D07356.pdf1. Introduction

Intermetallic compounds containing one or more ofthe iron-group, rare-earth, or actinide metals continueto provide an arena for enormous advances inthe understanding and applications of many aspectsof magnetism. The years since the publication ofthe original edition of Intermetallic Compounds(Westbrook, 1967) have seen many importantunanticipated developments. Foremost examples include:the compounds of rare earths with transition metals thatare extraordinarily strong permanent magnets; theso-called heavy-fermion compounds of cerium oruranium; the magnetic intermetallics with partialchemical disorder that exhibit spin-glass properties; and(in a related area) the superconducting intermetalliccompounds of niobium, whose critical temperatureswere a record high until the more recent discovery ofthe superconducting oxides.

These recent developments simply extend the long anddistinguished history of intermetallic compounds asmagnetic materials. It is now almost a century ago thatHeusler (1903, 1904) discovered that certain ternaryalloys containing none of the common ferromagneticmetals (iron, cobalt, or nickel) were neverthelessferromagnetic, and it was later found by X-raydiffraction that many of these alloys were chemicallyordered at specific stoichiometries. However, theprototypical Heusler compounds, Cu2MnSn andCu2MnAl, do contain manganese, which is in the sametransition-metal series as iron and is therefore similarlycapable of deriving an atomic magnetic moment fromits unfilled 3d-electron shell. Analogous examples,

discovered more recently, are the weakly ferromagneticcompounds Au4V, Sc3In, and ZrZn2, where vanadiumand scandium are also in the iron-group series whilezirconium is in the heavier transition-metal seriescharacterized by a partially filled 4d electron shell.Indeed, as the magnetic implications of the electronicbasis of the periodic table of elements came to beunderstood with the advent of quantum mechanics inthe 1920s, it became increasingly puzzling as to why sofew of the intermetallic (or ionic) compounds of thetransition-group elements (even of iron) showed anyevidence of ferromagnetism.

This puzzle was largely resolved in the early 1930sthrough the intuition of Neel (1932), who conceived ofthe antiferromagnetic state, in which the magneticmoments of neighboring atoms are aligned antiparallelto each other and thus produce no magnetization in thematerial as a whole, in contrast to the parallel momentsin the case of ferromagnetism. The existence of anti-ferromagnetism has subsequently been confirmed innumerous intermetallic compounds of transition metals,e.g. MnAu (Bacon and Street, 1958; Bacon, 1962), andof rare earths, e.g. TbAg (Cable et al,, 1964), by meansof neutron diffraction measurements, the experimentaltechnique of prime importance in determining atomicmagnetic configurations (see Chapter 19 by Roth in thisvolume). In various other intermetallics, e.g. Mn2Sb(Wilkinson et al., 1957), as well as in the ionic ferritesand garnets, this technique has also established theexistence of the ferrimagnetic state, where theantiparallel atomic moments are unequal in size, whichNeel (1948) had postulated as an extension to theantiferromagnetic case. In such a case, there is a net

Chapter 1Magnetic Principles

James S. KouvelDepartment of Physics, University of Illinois at Chicago, Chicago, IL 60680, USA

Magnetic, Electrical and Optical Applications of Intermetallic Compounds. Edited by J. H. Westbrook and R L Fleischer 19952000 John Wiley & Sons Ltd

bulk magnetization, though smaller, of course, than ifthe atomic moments were aligned parallel.

Even in a ferromagnetic material, the bulk magnet-ization in zero field is normally smaller than thesaturation magnetization in large fields. Demagnetizationeffects associated with the shape of the sample causethe material to subdivide into macroscopic regions calledferromagnetic domains, which have magnetizations ofthe same magnitude (per unit volume) but differentorientations. When the material is subjected to amagnetic field H, the domains with magnetizationsoriented closest to the direction of H will grow in sizeat the expense of the others by means of domain-wallmotion. Moreover, the domain magnetizations will allturn by various amounts toward the direction of H. Theease with which these processes can occur separates the'soft* from the 'hard' magnetic materials, the extremesof the latter being the strong permanent magnets.

2. Basic Magnetic Concepts

Certain general features of the ordered magnetic statesfound in intermetallic compounds have had profoundtheoretical implications pertinent to all magneticmaterials. Their significance can best be appreciatedwith reference to various interpretations of the basicmagnetic parameters: (i) the magnitude of the atomicmoments and (ii) the strength and sign of the couplingbetween these moments. (Conventionally, a positive(ferromagnetic) or negative (antiferromagnetic) couplingfavors a parallel or antiparallel alignment of magneticmoments, respectively.) All modern theories of magnetismagree that the origin of both (i) and (ii) lies in thequantum-mechanical part of the electron Coulombenergy known as the exchange energy, but they differin the approximations made in their application todifferent types of materials.

At one extreme, in the atomistic theories, theviewpoint is taken that (i) and (ii) can be consideredquite separately. This approach is especially valid forionic crystals, in which all the electrons are tightly boundto the ion cores. The energy levels of the unfilled d- orf-electron shell of a transition-metal or rare-earth ionin such a crystal, though modified in their spacing byinteractions with the electrostatic fields of neighboringions, are nevertheless fairly discrete and are thereforeoccupied, if at all, by an integral number of electrons(per ion). Consequently, the magnetic moment per ionequals a certain number of individual electron moments(or Bohr magnetons). In most cases, in which theelectrostatic interactions between Adjacent ions are

rather weak, the ionic moment in a crystal is nearly thesame as that of the free ion. Consistent with thisapproximation, the exchange coupling between themoments of magnetic ions in such a crystal is a minorperturbation and does not appreciably affect the sizeof the ionic moments. These considerations continueto apply to rare-earth atoms in a metallic system, wherethe f electrons are still tightly bound and do notparticipate significantly in the structural binding.However, the same cannot be said for the d delectronsin transition-metal systems.

At the other extreme is the collective electron theory.As applied to the transition metals, this approach hasstrong interactions among the d and s (conduction)electrons, giving rise to a broadening of their energylevels into bands. Furthermore, because the d and sbands overlap in energy and are both occupied up tothe same maximum (Fermi) energy level, the numberof electrons in each band will not be integral. If, inaddition, a sufficiently strong exchange coupling favorsa parallel alignment of the electron moments, theenergies of those with moments parallel or antiparallelto a reference direction will be lowered or raised,respectively, relative to the Fermi energy, and theirnumbers will adjust further. As a result, there will bea net bulk magnetization corresponding in general toa non-integral number of Bohr magnetons per atom,which agrees qualitatively with the situation forelemental iron, cobalt, and nickel. However, since thistheory handles all the d and s electrons collectively, itmakes no distinction between the constituent elementsof disordered transition-metal alloys and has thereforeenjoyed only a limited success in explaining their basicmagnetic properties.

It is with respect to the magnetic properties of orderedalloys or intermetallic compounds that the limitationsof these extreme theoretical viewpoints are particularlyevident. As neutron diffraction experiments haverevealed, the magnetic moment of a transition-metalatom such as iron varies appreciably from one compoundto another and even among crystallographicallydifferent sites in the same compound. Such a variabilityin magnetic moment testifies against the validity of thecollective electron model in the simple form describedabove. On the other hand, the fact that the magneticmoments of transition-metal atoms in intermetalliccompounds are usually not integral in Bohr magnetonstells us that the simple atomistic model also is not valid.Indeed, as theoretical work has shown (White, 1970),the appropriate magnetic model is a complicated (butsolvable) intermediate between these two extremes.Moreover, in intermetallic systems, the exchange

interactions between the atoms with magnetic momentsdo not derive entirely from a direct process involvingthe d or f electrons that are responsible for the atomicmoments. There is also an indirect exchange process inwhich the conduction electrons act as intermediaryagents, and this process predominates in cases wherethe moment-bearing atoms are separated by largedistances, as they often are in compounds with largeunit cells.

Another type of variability of magnetic moment isseen in certain intermetallic compounds containingprototypically cerium or uranium, the lightest of thelanthanide (rare-earth) and actinide elements withelectrons in the 4f or 5f shell. In these compounds (e.g.CeCu2Si2 and UPt3, whose antiferromagnetism isaccompanied by superconductivity at very lowtemperatures), the number of electrons in these shells,which defines the effective valence, is not a stablequantity but fluctuates rapidly with time, resulting ina fluctuating magnetic moment. These so-called heavy-fermion systems are of considerable current interest (seeChapter 10 by Aronson and Coles in this volume).

More generally, an appropriate model must alsoexplain the origin of magnetocrystalline anisotropy, thephenomenon defined by the orientational preference ofatomic moments in a crystal for certain crystallo-graphically equivalent directions. It is broadly understoodthat the anisotropy associated with any magnetic atomderives from the coupling between the spin and orbitalmoments of its d or f electrons. The orbital electronicmotion interacts, in turn, with the electrostatic fieldsproduced by the local crystalline environment. For rare-earth atoms, the interactions of the nearly localized felectrons with the crystal fields are not very strong andcan be readily calculated. In the case of transition-metalatoms, whose d electrons are much more extendedspatially, the crystal-field interactions are correspondinglystronger and they restrict the degrees of freedomassociated with the orbital states. This 'orbitalquenching' results in an atomic moment that reflectsprimarily the electronic spin state and which,paradoxically as it may seem, has a much weakeranisotropy than in the case of the rare earths. In allcases, the magnetocrystalline anisotropy also dependson the crystalline symmetry and is weakest when thesymmetry is highest (i.e. cubic).

3. Categories of Magnetic Behavior

3.1 Intrinsic Properties

The intrinsic magnetic properties of an intermetalliccompound (or any other system) are most easily

Figure 1. Typical variation with temperature of the magnetization(M) or susceptibility (x) below the Curie point (Tc) and of x~'above Tc for (a) a ferromagnet, (b) an antiferromagnet, and(c) a ferrimagnet. In (c), M versus Tis shown for cases A andB, as described in the text

described within the framework of the molecular-fieldtheory, in which the net exchange coupling between agiven atom and all its neighbors is approximated by theinteraction of its magnetic moment with an effective'molecular' field. The situation is particularly simplefor a ferromagnetic system when all the exchangeinteractions are positive, in which case the spontaneousmagnetization corresponds at 0 0K to a parallel alignmentof all the atomic moments and diminishes monotonicallywith increasing temperature as the moments fluctuatein direction. The manner in which the magnetizationdecreases from its value M0 at 0 K and finally vanishesat the Curie temperature (Tc) is represented in Figurel(a). Above Tc, where the system is magneticallydisordered (i.e. paramagnetic), the magnetic susceptibility(x) decreases with rising temperature according to thefollowing relation known as the Curie-Weiss law:

x = CZ(T-O) (1)

where C is the Curie constant. Thus, as shown in thefigure, 1/x versus temperature T describes a straight line

intersecting the temperature axis at T= 6. When all theatomic moments of the system are the same, the theorypredicts that 6 and Tc will be identical. However, inferromagnetic systems whose magnetic atoms are ofdifferent elements and/or are not crystallographicallyequivalent, 6 and Tc will not be equal and equation (1)will not be strictly valid except at very high temperatures.Deviations from equation (1) can also arise fromshort-range magnetic ordering, which is ignored inthe molecular-field theory.

In the simplest antiferromagnetic system, all themagnetic atoms are chemically and crystallographicallyalike and, as a result of negative exchange interactionsbetween them, subdivide in equal numbers into twogroups (or sublattices) having moments of oppositedirection. Hence, the spontaneous magnetization ofthe system is zero at all temperatures. Instead, asshown in Figure l(b), there is a susceptibility that(for a randomly oriented polycrystal) increases asthe temperature rises from OK, reaching a peakvalue at the magnetic disordering temperature Tc(which for an antiferromagnet is usually referred to asthe Neel temperature). The susceptibility decrease aboveTc again follows equation (1), except that 6 will benegative, roughly equal to - Tc. An^ inequalitybetween 6 and - Tc is accounted for by the theory asevidence of interatomic interactions within (in additionto those between) the magnetic sublattices.

The simplest ferrimagnetic system is alsocharacterized by two magnetic sublattices withantiparallel magnetizations. Here, however, thesublattices are not equivalent and the magnitudes oftheir magnetizations are not the same. Consequently,the system will have a net spontaneous magnetization,whose variation with temperature will often resemblethat of a ferromagnet, as illustrated by curve A in Figurel(c). More unusual forms for the magnetization-temperature behavior of a ferromagnet are theoreticallypossible and indeed have been encounteredexperimentally. The most exotic example perhaps is thatdemonstrated by curve B in Figure l(c), which is readilyunderstood in terms of the magnetization of onesublattice being larger at low temperatures butdecreasing more rapidly with temperature than that ofthe other sublattice. Since the two sublatticemagnetizations are opposite in direction, the net bulkmagnetization (when measured in a small field) mayactually reverse in polarity, as shown by the dottedcurve, and the initial temperature at which it goesthrough zero is called the compensation point.Regardless of their magnetization behavior below TCiall ferrimagnetic systems are expected to show a similar

temperature dependence of susceptibility above Tc. Asindicated in Figure l(c), the theoretical \/\ versus Tcurve follows a complicated hyperbolic variation beforeit ultimately approaches a straight line that obeysequation (1) at high T, usually with a negative value of6. Again, as in the antiferromagnetic case, 6 will deviatefrom - TQ owing to any magnetic interactions betweenatoms of the same sublattice.

A more general statement regarding 6 in equation (1)is that it represents an algebraic sum of all the positive(ferromagnetic) and negative (antiferromagnetic)interactions in the system, in contrast to Tc, which isincreased or decreased depending on whether a giveninteraction helps or hinders the atomic-momentalignment that describes the ordered magnetic state.Hence, in an intermetallic compound of any complexity,a deviation of B from Tc is to be expected and, infact, can be used in determining the magnitudes andsigns of the different interactions. Another importantdistinction is that between the ferromagnetic moment/iF and the 'effective' paramagnetic moment /AP of amagnetic atom in a simple ferromagnetic system. Thelatter is derived from the following expression for theCurie constant C in equation (1):

C = Nfi2P/3k (2)

where N is the number of magnetic atoms per unitvolume and k is the Boltzmann constant. For transition-metal atoms, it is defined by

ZiP = S[S(S+I)]1^B (3)where g is the spectroscopic splitting factor (with a valueclose to 2), S is the spin quantum number (the orbitalmomentum being quenched), and /iB is the Bohrmagneton. On the other hand, fi *S simply the low-temperature spontaneous magnetization M0 divided byN9 and it is defined by

fiF = gSfiB (4)

In the case of rare-earth atoms, the distinction betweenequation (3) and (4) persists, but S is replaced by J, thetotal angular momentum (spin plus orbital) quantumnumber, and g has various values of order unity.

An interesting phenomenon arises when the exchangeinteractions between magnetic atoms are conflicting withregard to the orientations of the atomic moments. Sucha situation obviously cannot exist when all theinteractions are ferromagnetic, but it is a distinctpossibility when some or all of the interactions are

(a) (b)Figure 2. Magnetization vectors of antiferromagneticallyinteracting sublattices A, B, and C in an apparent dilemma(a) and in a stable but frustrated configuration (b)

antiferromagnetic. In a simple example of the latter,cleverly provided by Nature in the hexagonal compoundMn3Sn, (DO19, Ni3Sn type (hP8)) the magnetic (Mn)atoms lie in equal numbers on three different but relatedcrystallographic sites (A, B, C), where each atominteracts antiferromagnetically with its magnetic nearestneighbors, which are of equal number on the other twosites. Schematically, in Figure 2(a), we see that if themagnetic moments of atoms A and B are alignedantiparallel to each other, thus completely satisfying theantiferromagnetic A-B interaction, then the magneticmoment of atom C is in a dilemma as to how to satisfyboth the antiferromagnetic A-C and B-C interactionssimultaneously. Energetically, the resolution of thisproblem is a compromise, whereby the magneticmoments of atoms A, B, and C are oriented at 120relative to each other, such that their vector sum is zero,as shown in Figure 2(b). This triangular magneticconfiguration has in fact been found in Mn3Sn, andalso in the related Mn3Ge and Mn3Rh compounds, bymeans of neutron diffraction (Kouvel and Kasper,1964). This phenomenon, in which the exchangeinteractions are not totally satisfied, is aptly calledfrustration. It is known to occur in many stoichiometricintermetallic compounds, where it often producescomplicated but periodic magnetic configurations thatmay or may not be commensurate with the spatialperiodicity of the atomic structure.

In non-stoichiometric compounds, the atomicdisorder on one or more of the crystallographic sitescan combine with any magnetic frustration to give riseto magnetic properties characteristic of a spin glass (SG).These properties include pronounced irreversibilitiessuch as the differences in the magnetization-fieldbehavior after cooling to below the SG orderingtemperature with and without a magnetic field. Suchproperties, for example, have been observed in the cubicpseudobinary compounds Tbj^Y^Ag as cleardepartures from the simple antiferromagnetism of

stoichiometric TbAg (Said et al., 1988, 1990). In the SGstate, the combination of magnetic frustration andatomic disorder produces a nearly random orientationalarrangement of atomic moments, and its occurrence inpartially disordered intermetallic compounds has furtherexpanded the study of this intriguing subject. (Magneticstructures are discussed in Roth, Chapter 19 in thisvolume.)

3.2 Soft and Hard FerromagnetsIn order that a strong magnetocrystalline anisotropy ina ferromagnet may be useful as a permanent magneticproperty, it is necessary in response to a reversingmagnetic field that the magnetization be constrained torotate against the anisotropy forces. If this constraintis absent, the magnetization may reverse at much lowerreverse fields by means of the motion of the domainwalls that separate adjacent ferromagnetic domains. Theresult, as shown schematically in Figure 3(a), will be athin hysteresis loop with a very small intrinsic coercive

Figure 3. Typical hysteresis loops of magnetization (M) ormagnetic induction (B) versus applied field (H) for a softmagnetic material (a) and a hard magnetic material (b). MRand BR are remanent values at H= 0; Hd and Hc are coercivefields at M= 0 and B = 0> respectively

field //ci, which characterizes many homogeneous bulkmaterials that are soft ferromagnets, especially thosewith weak anisotropy. However, if a ferromagneticmaterial with strong anisotropy is subdivided into smallparticles of dimensions comparable to the thickness ofa domain wall (typically of the order of 1000 A), itbecomes energetically unfavorable for any domain wallsto exist in the particles. Each particle is then a singledomain, whose magnetization can reverse only by arotation process requiring a reverse external field equalto the so-called anisotropy field. Since the anisotropyfields of known materials range up to several thousandoersteds, the hysteresis loop of a compact of single-domain particles can be extremely wide, as shown inFigure 3(b). It is not this hysteresis loop of M versus//, however, but the loop of the magnetic flux densityB versus H (shown as broken curves in the figure) thatsets the quality of a material as a practical permanentmagnet. Because B = H+4TTM, the coercive field Hcdefined at B = 0 is generally smaller than Hd defined

at M=O and, in fact, is limited to a maximum valueequal to BRf the remanent flux density, regardless ofhow large Hd may be. Thus, the magnetization as wellas the anisotropy of a material must be fairly large forit to qualify as a hard ferromagnet capable of use asa permanent magnet.

Many intermetallic systems have atomically orderedphases that are strongly ferromagnetic with a largeanisotropy, where the atomic ordering process isinhomogeneous and can be easily controlled. A suitablethermal treatment of such a system produces very smallregions of the ordered phase in a matrix of a weaklymagnetic disordered phase, which is magneticallyanalogous to a compact of single-domain particles withdesirable permanent-magnet properties. Moreover, there

are now several known ferromagnetic intermetallicsnotably, the compounds R2T14B, where R is a rareearth and T is iron and/or cobalt whose magneto-crystalline anisotropies are so extraordinarily large that,when a bulk homogeneous sample is driven to itssaturation magnetization, it stays in this condition (i.e.without subdividing into domains) even when verysizable reverse fields are applied. The discovery ofthese compounds has had an enormous impact on thepermanent-magnet industry. (Applications of magneticIMCs are reviewed in Stadelmaier and Rausch, Chapter14 in Volume 2, and by McGahan, Chapter 19 inVolume 2.)

4. References

Bacon, G. E. (1962). Proc. Phys. Soc, 79, 938.Bacon, G. E., and Street, R. (1958). Proc. Phys. Soc, 72, 470.Cable, J. W., Koehler, W. C , and Wollan, E. O. (1964). Phys.

Rev., 136, A240.Heusler, F. (1903). Verh. Dtsch. Phys. Ges., 5, 219.Heusler, F. (1904). Z. Angew. Chem., 17, 260.Kouvel, J. S., and Kasper, J. S. (1964). Proc. Intl. Con/, on

Magnetism, Nottingham. Institute of Physics, London,p. 169.

Neel, L. (1932). Ann. Physique, 17, 5.Neel, L. (1948). Ann. Physique, 3, 137.Said, M. R., Kouvel, J. S., and Brun, T. O. (1988). J. Appl.

Phys., 63, 4340.Said, M. R., Kouvel, J. S., and Brun, T. O. (1990). J. Appl.

Phys., 67, 5961.Westbrook, J. H. (ed.) (1967). Intermetallic Compounds.

Wiley, New York.White, R. M. (1970). Quantum Theory of Magnetism.

McGraw-Hill, New York.Wilkinson, M. K., Gingrich, N. S., and Shull, C. G. (1957).

J. Phys. Chem. Solids, 2, 289.

This chapter was originally published in 1995 as Chapter 40 in Intermetallic Compounds,Vol. 1: Principles, edited by J. H. Westbrook and R. L. Fleischer.

Front MatterTable of Contents1. Magnetic Principles1.1 Introduction1.2 Basic Magnetic Concepts1.3 Categories of Magnetic Behavior1.4 References


IntermetallicCompunds_Vol4/4-4/7B6C25FEB97BBC5C3816FA3496255C.pdf1. Introduction

Magneto-optical (MO) effects were first discovered byFaraday nearly 150 years ago in experiments on glasssamples placed between the pole faces of a magnet(Mattis, 1988), yet only relatively recently have theseeffects found practical applications. The first proposalfor MO-based data storage by Mayer (1958) launcheda period of intense study of MO effects in variousmaterials, with the promising goal of high-densityoptical storage based on thermomagnetic writing andreadout via the polar Kerr effect. MO data storage offerstwo distinct advantages over other techniques. First, theareal density of stored data can be very large, as it islimited primarily by the wavelength of the laser usedto read and record the bits. The compact disk is a goodexample of the type of storage densities achievable withoptical data storage systems. The second advantage isthat the read/write head in the MO disk drive neverphysically contacts the disk. Thus the fatal 'head crash*associated with conventional magnetic disk storagenever occurs in the MO system. Also, the MO storagetechnique provides removable disks capable of storingmore data than a typical conventional hard drive. Thegreatest disadvantage associated with MO recording isvery slow random access times. As a result, MO systemsare most commonly used for archival data storage, withtypical applications being the storage of satellite images,CAT scans, etc. In this chapter, we review thephenomenology and physical origins of the Kerr andFaraday effects, which are the most important first-order MO effects for data storage and other deviceapplications. We also outline the principal materials

requirements for MO storage media, and discuss theproperties of various intermetallic compounds (IMCs)with respect to these requirements. We would point outthat the discovery by Chaudhari et -al. (1973) ofamorphous rare-earth/transition metal alloys all butended research on IMCs for direct use as storage media,and the amorphous class of materials dominates the MOmedia market today. However, the properties of theIMCs are still at least of academic interest, and the factthat several IMCs (MnBi, PtMnSb, for example) exhibitvery large Kerr effects at room temperature has led tocontinuing research into this interesting class of materials.

2. Phenomenological Description of Magneto-Optical Effects

Michael Faraday discovered in 1845 that linearlypolarized light undergoes a rotation of the polarizationplane when transmitted through a magnetized mediumif the light propagates along the direction of themagnetization (Mattis, 1988). Towards the end of thatcentury, Kerr found that a similar rotation of thepolarization vector occurs when linearly polarized lightis reflected at normal incidence from the surface of amagnetized material. The Faraday and Kerr effects arethe most important MO effects in terms of storage anddevice applications, and we review the phenomenologyof these effects in this section. The following is by nomeans a complete review, and the interested reader isreferred to the many reviews in the literature (Bloombergand Connell, 1988; Meiklejohn, 1986; Parker, 1987;McGahan et al., 1992a, b) for more details.

Chapter 8Magneto-Optical Applications

William A. McGahanCenter for Microelectronic and Optical Materials Research, Department of Electrical Engineering,

University of Nebraska at Lincoln, NE 68508-0511 USA

Magnetic, Electrical and Optical Applications of Intermetallic Compounds. Edited by J. H. Westbrook and R. L. Fleischer 1995,2000 John Wiley & Sons Ltd

We consider the geometry shown in Figure 1. Thesample is assumed to be infinitely thick, and to extendinfinitely in the x and y directions. A linearly polarizedbeam of light propagating along the z direction strikesthe surface of the sample, which reflects part of thebeam and transmits the rest into the bulk of the sample.When the material is isotropic and unmagnetized, thepolarization of the reflected and transmitted beams isidentical to that of the incident beam, and only theamplitudes of the reflected and transmitted componentsare altered by the material. The presence of magnetizationin the material leads to an alteration of the polarizationstate of the reflected and transmitted components aswell. We will discuss the physical reasons for this in thenext section, and present the phenomenology fortreating these effects in this section.

In order to describe the amplitude and polarizationstate of the incident, reflected, and transmitted beamswe will employ the Jones vector formalism. Thisformalism is valid for all angles of incidence, but wewill only consider the special case of normal incidence.Since the electric field of the incident and reflectedbeams is perpendicular to the direction of propagationof the beam, two unit vectors in the plane perpendicularto the wave vector of the light are required to describeany possible polarization of the beam. For the case ofnormal incidence, we can choose any two orthogonalvectors which lie in the plane perpendicular to thewave vector of the light. We will choose these vectorsto lie along the x and y axes, and give them the labelsp and s, corresponding to the plane of incidence (/?) andthe direction perpendicular to the plane of incidence (5).

These labels are chosen in accordance with the obliqueincidence case, where the incident and reflected wavevectors define the plane of incidence (containing the xaxis in this case). Any arbitrary polarization can bewritten as a two-component vector, where the com-ponents are the projections of the electric field of thelight beam along the p and s directions. These com-ponents are in general complex, as the relative phaseas well as the magnitude of the two componentsdetermines the polarization state of the beam.

Within this framework, we will now consider thereflection of light from the surface of a semi-infinitematerial, as depicted in Figure 1. It can be easily shownthat for a perfectly smooth surface, the reflected andtransmitted electric fields are given in terms of theincident fields by the following equations (Azzam andBashara, 1989)

( p ) = ( fp )( p )\ Es / reflected \ ?s I \ ^s I incident

/ Ep \ = / tp 0 W Ep \

^ Es

'transmitted^ 0 *s ' \ Es I i n c i d e n t

where Ep and Es represent the (complex) componentsof the electric fields along the p and s directions,respectively, fp and r~s are Fresnel reflection coefficientsfor the surface, and fp and fs are Fresnel transmissioncoefficients for the interface. The Fresnel coefficientsare given (for normal incidence) in terms of the complexindex of refraction n of the medium by the followingequations (Azzam and Bashara, 1989)

r~p=-j^T /*>TT^r = ~/> (3)

W'-TTT (4)

Note that for the present case, in which there is nosample magnetization, the matrices which relate thereflected and transmitted electric fields to the incidentfield are diagonal. Furthermore, it can be shown thatregardless of the choice of basis vectors (p and sdirections, in this case), these matrices will always bediagonal at normal incidence. As a direct consequenceof the diagonal nature of these matrices, an incidentbeam which is linearly polarized will always yieldreflected and transmitted beams which have the samelinear polarization as the incident beam, albeit withdifferent amplitudes.

Incident Reflected




Figure 1. Geometry for consideration of optical reflection andthe Kerr and Faraday effects

When the sample is magnetized along the z direction,the Jones matrices for reflection and transmission at thesurface are no longer diagonal (for reasons which willbe discussed in the next section) and have the form

\k Fj \tt tj

where k and u are small (complex) off-diagonal com-ponents, and [J]1. and [J]1 are the Jones matrices forreflection and transmission at the surface of the sample,respectively. For the case of linearly polarized incidentlight, these off-diagonal components lead to a smallalteration of the reflected and transmitted polarizationsfrom the incident linear polarization. The reflected andtransmitted polarizations in this case become slightlyelliptical, with the major axis of the ellipse rotated awayfrom the original linear polarization by a small angle0, as shown in Figure 2. This small angle is the polarKerr rotation, and the ratio of the minor to the majoraxis of the resultant polarization ellipse is termed thepolar Kerr ellipticity. The label * polar' refers to the factthat the sample magnetization is assumed to lie along thesample normal. There are two other orientations forwhich the Kerr effect can be measured, in which thesample magnetization lies along the x or y axis, referredto as the longitudinal and equatorial Kerr effects. Theseeffects are generally much smaller than the polar Kerreffect, and. are much less commonly measured. Therotation angle and ellipticity of the reflected beam areoften combined into a single complex quantity \=0 +177,which is given by (in the small-angle approximation)

*-F At normal incidence, the off-diagonal Fresnel co-

efficients are given by (Sokolov, 1967)

- inQk=

( T W (7)where Q is the magneto-optical Voigt parameter, definedfundamentally in the next section. We can combineequations (7) and (3) to yield a useful expression forthe Kerr effect in reflectance from the surface of a bulksample magnetized perpendicular to the sample surface

-inQ ixvt= T- = - F (8)

( l - 2 ) V?(l-Ca)

Figure 2. The incident and reflected polarization states for thenonzero polar Kerr effect

where we have shown the formula in terms of thecomplex index of refraction h and the Voigt parameterQ as well as in terms of the diagonal (Ix^ and off-diagonal (exy) terms of the dielectric tensor. We willclarify the relationship between these quantities in thenext section.

There are several interesting consequences of theabove functional form for the complex Kerr effect.First, the Voigt parameter Q will be shown in the nextsection to be linear in the magnetization, such that areversal in the direction of the magnetization (from + zto - z) yields a change in the sign of the Voigt parameterand hence a change in sign of both the Kerr ellipticityand rotation. Hence, by measuring the position of themajor axis of the polarization ellipse of the reflectedbeam with respect to the original linear polarization,we can determine whether the sample magnetization isup or down with respect to the sample surface normal.This is the principle by which bits (small domains) ofup and down magnetization are read optically inmagneto-optical memories, as the transitions betweenregions of up and down magnetization can be detectedby the associated change in sign of the polar Kerr effect.Second, the Kerr effect is a linear function of the Voigtparameter (and hence the magnetization), but dependson the complex index of refraction of the material aswell, and in a somewhat more complicated manner.As a result, fundamental studies of MO effects inmaterials require both optical (ellipsometry, reflectance,transmittance) and MO (Kerr rotation and ellipticity)measurements (McGahan et al.91991,1992a, b; Hansenand Krumme, 1984).

Reflected Incident

A second important magneto-optical effect, theFaraday effect, occurs when a beam of linearly polarizedlight propagates through a material which has a non-zerocomponent of the magnetization along the direction ofpropagation of the beam. If the material is isotropic andunmagnetized, the beam will retain the same linearpolarization at all times as it propagates through thematerial. When magnetized, the material induces arotation of the polarization axis as well as a smallellipticity, and these effects are cumulative as the beampropagates through the material. If the material istransparent, i.e. there is no absorption ( = 0), there willbe no induced ellipticity and the polarization axis of thebeam will only be rotated as the beam propagatesthrough the material. The total rotation is proportionalto the length of material through which the beam haspropagated, and is also proportional to the componentof the magnetization along the wave vector of the beam.As a result, reversal of the magnetization will inducea change in sign of the rotation of the propagatingbeam. The Faraday effect has been used with nearlytransparent films for optical data storage, but is morecommonly used in integrated optical devices andpolarization modulators. Specifically, the Faradayrotation and ellipticity of a linearly polarized beam oflight due to passage through a sample of thickness /,neglecting surface reflection effects, is given by(Sokolov, 1967)

where X0 is the wavelength of the light in vacuo. Wewill not consider the Faraday effect further, as IMCsare by nature strongly optically absorbing, and appli-cations of these compounds involve the Kerr effect.

The above discussions are valid for the cases of thenormal incidence polar Kerr effect and the Faradayeffect for beams propagating along the direction of themagnetization. However, real magneto-optical storagestructures, for example, usually consist of multilayerstaqks in which only one or two of the layers arema^gneto-optically active. The other layers are used foroptical enhancement of the Kerr rotation and opti-mization of the thermal properties of the stack, and canserve to protect the MO layer from oxidation. For thecase of reflection from multilayer stacks at normalincidence, equations (5) are still valid, but equations (8)are not, and it becomes somewhat involved to evaluatethe dependence of the Jones matrix elements on theindices of refraction, Voigt parameters, and thicknessesof the individual layers and the substrate. The reader is

referred to the literature for details on the calculationof the optical and magneto-optical response of multi-layered systems and for systems illuminated at obliqueangles of incidence (Smith, 1965; Parker, 1987; Gambleand Lissberger, 1988; McGahan and Woollam, 1989;Visnovsky, 1991).

3. Physical Origins of Magneto-Optical Effects

The Faraday effect occurs when light propagates in amagnetized material, and the Kerr effect occurs whenlight is reflected from the surface of a magnetizedmaterial, yet both effects have the same physical origin.Consider a specimen having cubic crystalline symmetry.In this case Neumann's principle indicates that due tothe invariance of the crystal under all symmetryoperations in the cubic group, all physical propertiesof the crystal must be equivalent along any direction,i.e. the material is isotropic. If the material is magnetized(for simplicity the magnetization is taken to be uniformand along the z direction), the cubic symmetry of thecrystal is broken and at best we can expect rotationalsymmetry about the axis of the magnetic field (z axis).In this case we may invoke Neumann's principle to showthat invariance under rotations about the z axis requiresthe following general form for the dielectric tensor (aswell as any second-rank tensor associated with a physicalproperty of the material) (see Burns, 1985, for athorough discussion)

7= exy exx 0 (10). 0 0 izz .

The dielectric tensor *e relates the displacement fieldD in the material to the applied electric field E via therelation

l U ? E = E + 4irP (11)

where P is the electrical polarization vector in thematerial. We note that the off-diagonal elements areequal in value and opposite in sign, which is requiredfor symmetry under reversal of the magnetizationdirection. Since the diagonal elements are unchangedby reversal of the magnetization and the off-diagonalelements change sign, we can develop these quantitiesas power series in the magnetization M. The off-diagonal series is usually truncated after the first term,such that exy is nominally linear in the magnetization.

The diagonal elements are assumed to be independent ofthe magnetization, although the second-order term leadsto the (very small) Voigt effect. The elements of the di-electric tensor are related to the complex index of refrac-tion n and Voigt parameter Q in the following manner

n = K Q = ^ (12)6XX

The cubic material under discussion can be shownfrom symmetry considerations to exhibit an isotropicdielectric tensor in the absence of a magnetic field, suchthat the off-diagonal elements are all zero and thediagonal elements are equal. In this case the dis-placement, polarization, and applied electric fieldvectors are collinear, and linearly polarized light willretain its polarization state when reflected at normalincidence from a planar surface of the material or whentransmitted through the material in any direction.

When the material is magnetized and the dielectrictensor has the form shown in equation (10), thedisplacement, polarization, and applied electrical fieldvectors are no longer collinear, as the off-diagonalcomponents of *e mix the x and y components of thefields. This can be understood in a classical sense asbeing due to the Lorentz force on the atomic electronsas they move in the presence of the field M. In theabsence of magnetization, the electrons experience onlythe force due to the electric field eE, which is alwaysin the direction of the applied field E. The response ofthe electrons to the applied field, which determines thepolarization P (and hence the displacement field D), isalways in the direction of E and all the fields arecollinear. When a magnetic field is present, the movingelectrons are deflected by the Lorentz force e\ x M inaddition to the electrical force_eE, and do notjpnerallv^move along the direction of E such that P, E, and Dare no longer collinear.

In order to understand the consequences of this wemust recall that the wave equation for E yields threeequations (one for each spatial dimension) which arecoupled if the dielectric tensor is not diagonal. If weperform the coordinate transformation to the principalaxis system in which the dielectric tensor is diagonal,the equations are uncoupled, and we find a separatesolution for the component of E along each of theprincipal axes. Each component propagates with anindex of refraction equal to the square root of thecorresponding principal (diagonal) element of the(diagonalized) dielectric tensor. For a cubic materialwith no magnetization, the dielectric tensor is alreadydiagonal, the three principal solutions are along the x,

y, and z axes, and all three waves propagate with thesame index of refraction. This is the isotropic case,as a wave of arbitrary polarization state retains that statewhen propagating in any direction in the material.

The material becomes anisotropic when magnetized,as diagonalizing equation (10) yields three solutionscorresponding to right-circularly and left-circularlypolarized light and z-polarized light. The right-circularlyand left-circularly polarized waves propagate withdifferent indices of refraction