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Magnetotransport and magnetocaloric effects in intermetallic compounds
Duijn, H.G.M.
Publication date
2000
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Final published version
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Citation for published version (APA):
Duijn, H. G. M. (2000). Magnetotransport and magnetocaloric effects in intermetallic
compounds.
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Magnetotransportt and magnetocaloric
effectss in intermetallic compounds
••••••••••••• •
» • • • • • • • • • •
Magnetotransportt and magnetocaloric
effectss in intermetallic compounds
ACADEMISCHH PROEFSCHRIFT
terr verkrijging van de graad van doctor aan de Universiteit van Amsterdam opp gezag van de Rector Magnificus prof. dr J.J.M. Franse,
tenn overstaan van een door het college voor promoties ingestelde commissie, inn het openbaar te verdedigen in de Aula der Universiteit
opp donderdag 10 februari 2000, te 12.00 uur
door r
Henricuss Gerardus Maria Duijn
Co-promotor:: Dr E. Brück
Commissie:: Prof. dr K.H.J. Buschow Prof.. dr P.F. de Chatel Prof.. dr R. Coehoorn Prof.. dr J.J.M. Franse Drr P. Schobinger-Papamantellos Prof.. dr V. Sechovsky Drr A. de Visser
Faculteitt der Wiskunde, Informatica, Natuurkunde en Sterrenkunde
Thee work described in this thesis was part of thee research program of the Dutch Technology
Foundationn 'STW' and was carried out at the Vann der Waals-Zeeman Instituut
Universiteitt van Amsterdam Valckenierstraatt 65 10188 XE Amsterdam
Thee Netherlands
wheree a limited number of copies of this thesis is available.
3 3
Contents s
Introductionn 7
1.11 General introduction 7 1.22 Outline 922 Theoretical aspects 11
2.11 Magnetism in metals 112.1.11 Local-moment versus itinerant-electron magnetism 11 2.1.22 Phase transitions in itinerant-electron systems 14
2.22 Electrical resistance in metals 17
2.2.11 Basic concepts 17 2.2.22 Band-structure effects and spin-dependent scattering 20
2.2.33 Magnetic scattering 23 2.33 Theory of group representations 27
33 Experimental 35
3.11 Sample preparation 35 3.22 Characterisation techniques 36 3.33 Electrical-resistivity measurements 373.44 Magnetic measurements 38 3.55 Measurements under hydrostatic pressure 41
3.66 Neutron-diffraction experiments 41
44 Magnetic and transport properties of (Hf,Ta)Fe
2compounds 47
4.11 Introduction 47 4.22 Sample preparation and characterisation 49
4.33 Structural properties 50 4.44 Magnetic properties 51 4.55 Powder neutron diffraction 61 4.66 Application of the theory of group representations 64
4.77 Single-crystal neutron diffraction 66
4.88 Electrical resistivity 69
5 5
55 Magnetic and transport properties of Fe
3(Ga^l)
4compounds 85
5.11 Introduction 85 5.22 Sample preparation and characterisation 86
5.33 Structural properties 87 5.44 Magnetic properties 89 5.55 Pressure dependence of the ferromagnetic to antiferromagnetic transition 94
5.66 Powder neutron diffraction 96
5.77 Specific heat 97 5.88 Electrical resistivity 99 5.99 Discussion 105
66 Electrical-transport properties of GdT
2Si2 compounds 109
6.11 Introduction 109 6.22 Experimental 111 6.33 Results and discussion 111
6.44 Conclusions 130
77 Electrical-transport properties of RMn
6Ge6 compounds 133
7.11 Introduction 133 7.22 Magnetic properties 134
7.33 Results 136 7.44 Discussion 141
88 Magnetic properties of Gd
5(Ge,Si)
4compounds 145
8.11 Introduction 145 8.22 Experimental 146 8.33 Structural properties 148 8.44 Application of the theory of group representations 151
8.55 Magnetic properties 152 8.66 Electrical resistance 160 8.77 Discussion 163 Appendixx A 169 Appendixx B 176 Summaryy 179 Samenvattingg 182 Listt of publications 185 Dankwoord d 186 6
Introduction n 7 7
Chapterr 1
Introduction n
1.11 General introduction
Afterr the discovery of the giant magnetoresistance (GMR) effect in magnetic multilayerss [1.1], subsequent research has led to a large number of materials that are suitable forr application as magnetic-field sensors. Magnetic multilayers consist of ferromagnetic layerss that are separated by non-magnetic spacer layers. By choosing the correct thickness of thee spacer layer, antiferromagnetic coupling between magnetic layers takes place. Under the actionn of a magnetic field this coupling can be overcome and the relative orientation of the magneticc moments in successive layers can be changed from antiparallel to parallel. Since electronss have a spin, according to the Mott model [1.2], the total conductivity can be thought off as a sum of one electron current with spin up and one with spin down. As the conduction-electronn scattering is spin dependent, the two spin currents have different resistances for the antiparallell and parallel alignment of the magnetic multilayers, resulting in a large change of electricall resistance: the so-called GMR effect. Since 1998, there exists a new generation of hardd disks containing GMR-based read heads, enabling a higher density of bits. Furthermore, ann increasing amount of applications of field sensors involve high-temperature operation, whichh reveals the limitations of the magnetic multilayers currently used. As multilayered systemss are metastable, at elevated temperatures (200-300 °C) diffusion of dissimilar atoms fromm the adjacent layers occurs, which generally decreases the magnetoresistance.
Itt is well known that many intermetallic compounds have crystal structures in which thee magnetic atoms are located in layers, sandwiched between layers of non-magnetic atoms. Severall of these compounds give rise to antiferromagnetic ordering, that can be forced into a ferromagneticc alignment of the magnetic moments by application of a moderate magnetic field.field. Similar to multilayered systems, in these intermetallic compounds the field-induced transitionn is accompanied by a considerable change in electrical resistivity. From a thermodynamica!! point of view, for high-temperature applications intermetallic compounds mayy be more favourable than multilayers, because in intermetallic compounds atom diffusion doess not occur until very high temperatures.
Thee aim of our research project at the outset, was two-fold. From the applicational pointt of view, we aimed at investigating intermetallic compounds in bulk and thin-film form, inn order to establish the appropriateness for sensor applications. From the fundamental point
Chapterr 1 off view, we attempted to gain a deeper insight in the mechanism responsible for the magnetoresistancee effects in bulk intermetallic compounds. This insight may guide the search forr compounds suitable for applications. In the course of the investigations, the field-induced transitionn in intermetallic compounds was found to be temperature dependent [1.3]. Furthermore,, at temperatures far below the critical temperature, very large magnetic fields are requiredd to induce the magnetic transition, up to five orders of magnitude larger than in multilayeredd systems. This makes intermetallic compounds less suitable for application in magnetic-fieldd sensors. Additionally, by using different constituents, the operating range of multilayeredd systems is being extended to higher temperatures. For these reasons, the emphasiss of the study presented in this thesis has shifted to the fundamental part.
Settingg up a quantitative theory of the GMR effect in multilayers is hampered by the factt that there are several types of scattering processes which determine the GMR effect: (i) scatteringg at the interfaces, (ii) scattering in the bulk of the layers (electron-phonon and electron-magnonn scattering) and (iii) scattering at grain boundaries and other defects. Therefore,, it is useful to study the magnetoresistance effects in structurally more perfect and welll characterised systems, like (single-crystalline) intermetallic compounds, in which the electron-phononn and electron-magnon scattering mainly contribute to the electrical resistivity. Still,, a thorough comparison of the magnetic and electrical-transport properties is difficult, becausee a limited number of theoretical models is available. To cite Ashcroft and Mermin [1.4]:: "The development of a tractable model of a magnetic metal, capable of describing both thee characteristic electron spin correlations as well as the electronic transport properties predictedd by simple band theory, remains one of the major unsolved problems of modern solidd state theory".
InIn this thesis, we present the magnetic and electrical-transport properties of several systemssystems of intermetallic compounds: (Hf,Ta)Fe2, Fe3(Ga,Al)4, GdT2Si2 (T= transition
element),, and RMn6Ge6 (R = rare-earth element). The common property of these systems is
ann antiferromagnetic(-like) phase that can be modified into a ferromagnetic phase by changingg the temperature and/or by application of a magnetic field. We connect the magnetic andd electrical-transport properties of the above-mentioned systems using the theoretical modelss available.
Additionally,, we have investigated some physical properties of the system Gd5(Ge,Si)4.. Recently, an extraordinarily large magnetocaloric effect has been discovered in
Gd5(Geo.5Sio.5)44 [1.5]. The magnetocaloric effect, which is the heating or cooling of magnetic materialss due to a varying magnetic field, has potential application in magnetic refrigeration inn the room-temperature range. The origin of the large magnetocaloric effect in Gd5(Gei-xSi^)4
liess in a simultaneous magnetic and structural phase transition, that can be evoked as a functionn of both temperature and magnetic field.
Introduction n 9 9
1.22 Outline
Thee outline of this thesis is as follows. In chapter 2, we briefly introduce some models describingg the properties we have been investigating. The first section is devoted to the magneticc properties of metals. Furthermore, phase transitions between ferromagnetism and antiferromagnetismm are discussed. The second section deals with the electrical-transport propertiess of metals, and the concomitant (magneto)resistance effects. In the third section, the theoryy of group representations is introduced. This theory supplies us with a solid basis of the magnetic-structuree analysis of crystalline solids from symmetry considerations.
Inn chapter 3, we describe the experimental techniques and equipment employed in the investigationn of the physical properties of the materials studied in this thesis. Furthermore, the samplee preparation and characterisation are included in this chapter.
Inn chapter 4, we present and discuss the magnetic and transport properties of the systemm Hfi^TaJ^. For 0.1 <, x <, 0.25, this system displays a phase transition from an antiferromagneticc to a ferromagnetic state as a function of temperature and magnetic field. Wee have investigated the detailed magnetic structures in the antiferromagnetic and ferromagneticc state by means of magnetisation and neutron-diffraction experiments on poly-andd single-crystalline samples. Furthermore, the magnetic phase diagram and the (magneto)resistancee effects at the magnetic phase transition are discussed.
Inn chapter 5, we report on the magnetic and transport properties of the system Fe3(Gai-^AU)4.. Similar to Hfi.^TaJ^, the compound Fe3Ga4 has a temperature- and field-inducedd antiferromagnetic to ferromagnetic phase transition, which is accompanied by a reductionn of the electrical resistivity. Furthermore, the magnetic phase diagram of Fe3Ga4 is
studiedd upon Al substitution and as a function of hydrostatic pressure.
Chapterr 6 is devoted to GdT2Si2 (T = transition element) compounds. The Gd sublatticee orders antiferromagnetically at low temperatures, and can be aligned ferromagneticallyy by application of a magnetic field of the order of 10 T. We have carried out resistivityy measurements as a function of temperature and magnetic field. The results are comparedd with the results of magnetisation measurements.
Inn chapter 7, we investigate the RMnöGeö (R = rare-earth element) compounds, that possesss complex magnetic structures due to the existence of magnetic moments on both the R andd the Mn sublattices. Furthermore, most of the RMnóGeg compounds undergo spin-reorientationn transitions as a function of temperature. In this chapter, we investigate the relationn between the magnetic structure and the observed resistance effects in the RMnöGeó compounds. .
Forr 0.25 <x< 0.50, the system Gd5(Gei^Si^)4 has a simultaneous magnetic/structural
phasee transition, that is accompanied by an extraordinarily large magnetocaloric effect. In chapterr 8, we report on the magnetic and transport properties of a Gd5Ge2.4Sii.6 single crystal.
Thee entropy change associated with the magnetic/structural phase transition is calculated from thee magnetisation data using a thermodynamical Maxwell relation. Furthermore, the
anisotropyy observed in the magnetisation is discussed in relation to the results of symmetry considerationss with the theory of group representations.
References References
[1.1]] M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G.
Creuzet,Creuzet, A. Friederich and J. Chazelas, Phys. Rev. Lett. 61 (1988) 2472; G. Binasch, P.. Griinberg, F. Saurenbach and W. Zinn, Phys. Rev. B 39 (1989) 4828
[1.2]] N.F. Mott, Proc. Roy. Soc. 156 (1936) 368
[1.3]] J. van Driel, Thermal stability of magnetoresistive materials, Thesis, University of Amsterdam,, 1999
[1.4]] N.W. Ashcroft and N.D. Mermin, Solid state physics, page 673 (Saunders College Publishing,, Forth Worth, 1976)
Theoreticall aspects 11 1
Chapterr 2
Theoreticall aspects
Inn the study of solids, metals have continued to play a dominant role, since they exhibit aa variety of fascinating properties. During the last century, countless models of the characteristicc metallic properties have been constructed. Still, the theoretical description of metalss is far from complete, and a variety of experimentally observed phenomena does not yet havee a solid theoretical explanation. In this chapter, we will briefly introduce some models describingg the properties we have been investigating.
Thee first section is devoted to the magnetic properties of metals. Two approaches of thee description of magnetism, namely the local-moment model and the itinerant-electron model,, are introduced. Furthermore, the concept of spin fluctuations and their role in the vicinityy of phase transitions between ferromagnetism and antiferromagnetism, are discussed.
Thee second section deals with the electrical-transport properties of metals. There are threee basic approaches to describe transport phenomena: the phenomenological description basedd on thermodynamics, the semiclassical model based on the Boltzmann transport theory, andd the many-body quantum theory. We will briefly discuss the former two models. Besides, wee will describe several contributions to the (magneto)resistivity arising from band-structure effectss and arising in the case of certain types of magnetic interactions.
Finally,, in the third section, the theory of group representations is introduced. This theoryy supplies us with a solid basis of the magnetic-structure analysis of crystalline solids fromm symmetry considerations.
2.11 Magnetism in metals
2.1.11 Local-moment versus itinerant-electron magnetism
Thee magnetic properties of solids are usually described in the framework of two approaches,, namely the local-moment model and the itinerant-electron model. The former startss with electronic states localised in real space, while the latter starts with those localised inn reciprocal space. Both models only describe limiting cases of the actual magnetism occurringg in solids. Still, it is instructive to introduce these approaches, since they provide importantt insight in the concept of magnetic ordering and since they serve as starting points of
12 2
moree sophisticated models.
Inn the local-moment approach, the magnetic moments are assumed to be localised at thee atoms of a solid [2.1], and are described in the framework of atomic theory. Here, the magneticc moment of an atom originates from the spins st and orbital angular momenta /, of the
nn electrons (i = 1,..., n) associated with this atom. The total angular momentum J of an atom is thee resultant of the spin angular momentum S = £ i s{ andd the orbital angular momentum
LL = Y, th Since the moments of completely filled electron shells are equal to zero, only the
electronss of incomplete shells contribute to the magnetic moment. As a consequence, in solids,, a magnetic moment arises only at an atom with a partially filled shell. The s and p electronss usually take a principal part in the bonding of solids, and their states are quite differentt from those of free atoms. In contrast, the wave functions of the ƒ electrons are spatiallyy more confined to the atom, and hence are usually less important in bonding. In particular,, the 4/-electron wave functions of rare-earth atoms usually keep nearly their free-atomm form. Due to the localisation of their wave functions, the 4 / electrons have a fully developedd orbital angular momentum and, consequently, also the associated orbital magnetic moment.. A clear manifestation of the localised character of 4/electrons in metals is that the observedd magnetic moments are close to the values expected from Hund's rules, which predict thee values of L, S and J for the free atom in its ground state.
Sincee in the localised-moment model the electron wave functions of the different atomss are well separated, the direct exchange coupling will be small [2.2]. Such a system wouldd not exhibit long-range magnetic order. However, there exist several indirect exchange mechanisms,, like the RKKY interaction [2.3] and interaction via polarisation of 5d spins [2.4],, which may result in long-range magnetic order. For example, in the RKKY model the magneticc moments partly polarise the conduction electrons around them, leading to an indirect interactionn that is long-range and is oscillatory. The standard approach to describe magnetic-orderingg phenomena in the local-moment model, is by the isotropic Heisenberg exchange Hamiltoniann [2.5]
H „ = - 2 V
i- 5
jj(2.1)
i.j jwheree the summation extends over all spin pairs in the crystal lattice, and Jy is the exchange-couplingg constant describing the interactions between the spins 5; and S}. By generalising the
signn and range of Jij, this phenomenological model can describe ferro-, antiferro-, ferri- and helimagneticc ground states. Furthermore, the model can account for low-lying excited states, likee spin waves.
Alreadyy in an early stage of the development of the localised-moment model, it becamee clear that this model is not appropriate for the description of ferromagnetism in transitionn metals, where the d electrons contribute to both the bonding and the magnetism. In thiss case, the unpaired electrons are no longer localised and accommodated in energy levels
Theoreticall aspects 13 3 associatedd with particular atoms. Instead, the unpaired electrons are described by (itinerant) Blochh wave functions, the energy levels having broadened into energy bands. As a good approximation,, only the spins of the d electrons contribute to the magnetic moment, because thee orbital moments are quenched. Nonetheless, a small contribution due to spin-orbit couplingg may be present.
Thee possibility of magnetic ordering in itinerant systems is described by the Stoner-Wohlfarthh model of itinerant-electron magnetism [2.6; 2.7]. In this model, the electron states aree divided into spin-up and spin-down subbands. The energy levels of the subbands are filled accordingg to the Pauli exclusion principle. Depending on the balance between the kinetic energyy and the exchange energy lex of the electrons, the ground state of the system may be eitherr ferromagnetic or paramagnetic. The occurrence of ferromagnetism is given by the well-knownn Stoner criterion
II = Ie xn ( EF) > l (2.2)
wheree I is the effective Stoner interaction parameter, and n(Ep) is the density of states per spin subbandd at the Fermi energy. When equation 2.2 is fulfilled, the net magnetisation is a direct consequencee of the unequally occupied spin-up and spin-down subbands. Although ferromagnetismm is not realised for I < 1, the magnetic susceptibility % is enhanced with respect too the Pauli susceptibility %pauu according to
XX = SStonerJCpauü =
T^yXpauU (2-3)
Here,, Sstonw = (1 - 1 )_1 is called the Stoner enhancement factor. A generalised Stoner criterion alsoo describes the occurrence of itinerant-electron antiferromagnetism. In this case, the Pauli susceptibilityy is replaced by a wave-vector-dependent susceptibility Xq- The periodicity of the antiferromagneticc structure is characterised by a wave vector q = Q at which the enhancement off the susceptibility is strongest. This mechanism has been discussed extensively by Herring [2.8].. At finite temperatures, the Stoner theory based on the Hartree-Fock approximation [2.9] iss insufficient. For example, the often observed Curie-Weiss behaviour above the ordering temperaturee is hard to explain within the Hartree-Fock approximation. Furthermore, the calculatedd Curie or Néel temperature is generally too high compared with experiment.
Inn the Stoner theory, thermal excitations exist as spin-flip excitations of electrons acrosss the Fermi surface, leaving holes behind. Due to interactions between the excited electronss and holes, another excited quasi-stationary state is formed. These collective excitationss are referred to as spin fluctuations. If the life time of spin fluctuations is long enough,, they can have a significant effect on the physical properties like the magnetic susceptibilityy [2.10], the electrical resistivity [2.11] and the specific heat [2.12].
Thee concept of spin fluctuations is often taken as a starting point to improve the descriptionn of the magnetic properties of metals [2.13]. Izuyama et al. [2.14] have developed a
generall theory of spin fluctuations based on the so-called dynamical Hartree-Fock approximationn or random-phase approximation (RPA). At elevated temperatures, however, thee results of RPA theory do not agree with experiment, as it assumes the Stoner equilibrium statee in calculating the spin fluctuations and neglects the effect of the spin fluctuations on the thermall equilibrium state. As a further improvement, renormalisation of the thermodynamical quantitiess or the effect of the spin fluctuations on the equilibrium state has been considered [2.13;; 2.15]. The thus developed theory successfully describes almost all the interesting propertiess of weakly ferro- and antiferromagnetic metals. Furthermore, it became apparent thatt spin fluctuations show various properties ranging between the two opposite extremes: the local-momentt limit (spin fluctuations localised in real space) and the itinerant limit (spin fluctuationss localised in reciprocal space, i.e.: having a well-defined wave vector).
Inn the next section, we discuss the coexistence of ferro- and antiferromagnetism and phasee transitions between ferromagnetic, antiferromagnetic and coexistent phases. Furthermore,, some effects of localised and itinerant spin fluctuations on the electrical resistivityy are discussed in section 2.2.
2.1.22 Phase transitions in itinerant-electron systems
Usually,, the magnetic interactions in a metal lead to either ferromagnetism or antiferromagnetismm below the ordering temperature. Only under special conditions, a temperature-inducedd transition from one magnetic configuration to another occurs. To explain suchh a transition in localised systems, Kittel [2.16] has proposed a model in which the Heisenbergg exchange constant changes sign with lattice expansion. This approach has been used,, for example, to explain the temperature-induced transitions observed in many Mn-based compounds,, like RMn6Ge6 and RMn2Ge2 (R = rare-earth atom) [2.17]. Furthermore, in these
compounds,, the temperature dependencies of the sublattice interactions and anisotropics play ann important role.
Inn localised magnetic systems, the coexistence of ferromagnetic and antiferromagnetic structuress is practically excluded. In itinerant-electron systems, however, this coexistence can reallyy take place as is shown by Moriya and Usami [2.13; 2.18]. They consider a strongly-interacting-electronn system without magnetic anisotropy and express its free energy <I» in termss of the uniform (Mo) and the staggered (MQ) magnetisation up to the fourth power
*(M
0,M
Q,B,T)) = M
+^L
+' X + k
M«
zzXoXo ZXQ l L (2.4)
+^YÜ SMJMJ+|YUS(MO-MQ)2-BM0 0
wheree B is the uniform external field, Xo and XQ are m e uniform and staggered susceptibilities, respectively.. The temperature dependencies of Xo and XQ are dominated by the effect of spin
Theoreticall aspects 15 5 p p u u P A A B B
a a
--^^. --^^. AF F 11 "" T T 0 0 \ B B AF+FF ' ƒƒ \ ! !V V
F F T T P Pb b
:: n i i i B B T T f f - ,, B ^ NN k S S s s \ \ AF F // 1 -i-i 1 a'' w ii / i i i F F T T P P ff c ' V V Too Tc T B B F Fd d
— ^ s . . T f f B B kk x ' s s / / / / 1 1 1 1 :: AF \ T T _^^ B > .. c1 1
\ \V V
T T P P T* * TT TFiguree 2.1. Temperature dependence of p = %Q I Xo and possible phase diagrams. The dashed and full curvess represent first- and second-order phase transitions, respectively.
fluctuationss and are quite strong in a system where magnetic ordering takes place. The coefficientss of the fourth power of the magnetisation yu, Ys> Yus and Y us are assumed to be temperaturee independent. The first four terms on the right-hand-side of equation 2.4 are the usuall terms of the Taylor expansion of the free-energy functional. The fifth and sixth term are cross-termss describing the interaction between the uniform and the staggered magnetisation. Ass a result of this, also non-collinear magnetism is covered by this theory.
Lett us now consider the case that the wave-vector-dependent susceptibility %<, has two peakss at q = 0 and q = Q. The equilibrium state is obtained by minimising the free energy with respectt to M0 and MQ. Considering the sixth term in equation 2.4, we may take Mo i . MQ (i.e.:
Figuree 2.2. Typical forms of the magnetisation curve showing the transition from a coexistent or antiferromagneticc phase to a ferromagnetic phase.
cantedd state) when y us > 0, while for Yus<0 we have Mo || M Q (i.e.: ferrimagnetic state). Therefore,, we may regard Mo, M Q and B as scalar quantities, taking B parallel to Mo. By furtherr investigation of the free energy, Moriya and Usami have constructed four kinds of magneticc phase diagrams. The results are depicted in figure 2.1. The relative values of the coefficientss determine which phase diagram is relevant. These may be expressed as
PP = X Q / X O , PA = YUS/YS, PF = YU/YUS, PO = (YU/YS)'/ 2
, p* = 3 pF- 2 pA (2.5)
Thee characters F, AF and P in figure 2.1, indicate the ferro-, antiferro- and paramagnetic phases,, respectively. F + AF stands for the coexistent phase at B = 0. For example, phase diagramm d applies when XQ < 0 and YuYs<Yus2; i.e.: the coupling between the uniform and staggeredd modes is stronger than the coupling within the uniform and staggered modes themselves. .
Furthermore,, Moriya and Usami have described the possible magnetisation processes fromm the zero-field state to the field-induced ferromagnetic state. The magnetisation curves for thee various phase diagrams are shown in figure 2.2. Curves (' and ii are expected to be observedd in the cases described in figures 2.1a and c. Curves Hi and iv belong to the phase diagramss depicted in figures 2.1b and d.
Isodaa [2.19] has extended the theory developed by Moriya and Usami by including a termm in the free-energy relation 2.4 that takes into account the effect of uniaxial magnetic anisotropy.. Then, additional canted and ferrimagnetic states are possible. He thus obtained 18 differentt kinds of phase diagrams.
Theoreticall aspects 17 7 Summarising,, ferromagnetic and anti ferromagnetic spin fluctuations due to the strong interactionss between them and their temperature dependencies may give rise to the coexistencee of ferromagnetic and antiferromagnetic phases, and may lead to temperature- and
field-inducedfield-induced magnetic phase transitions. In the following chapters, we will discuss the measuredd magnetic phase diagrams of (Hf,Ta)Fe2 and Fe3(Ga,Al)4 in the above-described
terms. .
2.22 Electrical resistance in metals 2.2.11 Basic concepts
Whenn a solid is exposed to an electric field E, the internal equilibrium is distorted, (possibly)) inducing transport of charge. The electrical conductivity a (or the electrical resistivityy p - 1/a) is defined to be the proportionality constant in the phenomenological transportt equation that relates the electric field and the induced current density j
jj = o- e (2.6)
Historically,, the modern treatment of the electrical conductivity originates from the Drude theory.. The conduction electrons in a metal are assumed to behave like a gas of non-interactingg particles, and experience a force due to the applied electric field. The accelerated electronss are subjected to collisions with the fixed ions of the crystal, with an average time T betweenn the collisions. If the motion of a conduction electron is completely disrupted at each collision,, the conductivity is given by Drude's formula [2.20]
wheree n is the number of conduction electrons per unit volume, and e and me are the electron
chargee and mass, respectively. Although this expression has been obtained on classical arguments,, it can also be derived in a quantum-mechanical treatment of the conduction electronss in the relaxation-time approximation.
Inn the quantum approach, the conduction electrons are treated as a quantum gas, obeyingg a Fermi-Dirac distribution /o originating from the Pauli exclusion principle. Furthermore,, the Bloch theorem applies, since the conduction electrons are subjected to a periodicc potential arising from the ions in the crystal. Hence, the conduction-electron states shouldd merely be denoted as Bloch eigenstates, having momentum k and energy E(k). A consequencee of the theorem is that in a perfect crystal, electrons can travel freely without beingg scattered by the ions, resulting in a vanishing resistance.
Chapterr 2 Inn thermal equilibrium, the change in the distribution function due to the electric field iss balanced by the change resulting from scattering of the electrons on various imperfections inn the crystal, giving the Boltzmann equation
dtt dt dt t == 0 (2.8) )
Whenn an electric field zx is applied in the x direction, all electrons are displaced uniformly in
kk space by an amount 8kx = ezjh 5t. Thus, the new distribution function ƒ is the same as ƒ<>
butt displaced by bkx: f=fo(kxSkx, ky, kz). As the deviation from equilibrium usually is small
inn the steady state, we can write for the new distribution function
ff
--
ff°-dk-7T*°-dk-7T*
xx--
ff°-m^°-m^
eeee**
(2.9) ) bt btwheree v ^ = ^ ' 9E(k)/9k is the group velocity. The change in the distribution function due to thee electric field is then given by
dt t 3E E v,ce, , (2.10) )
Forr elastic, isotropic scattering and a spherical Fermi surface, the scattering can be describedd in the relaxation-time approximation
v. v.
dt tf~fo f~fo (2.11) )
Combinationn of equations 2.8, 2.10 and 2.11 yields for the distribution function in the steady state e
f(k)f(k) = f0(k)-^vxeExx (2.12) )
Usingg equation 2.12, we can write for the current density
L = ^ J / ( k ) v , ( k ) d k = ^ j f /
0( k ) - | | v ^ E
j rT | v
j cd k k
47t3 3 ; ; 3E E dk== -4it3 3 f fTl I ^ - d S d E E MM BE
Theoreticall aspects 19 9 Thee integral over/0(k) vanishes by symmetry. The first integral on the right-hand side is over a
surfacee of constant energy, and the second one is over all energies. As the derivative of the distributionn function appears in equation 2.13, only electrons at the Fermi surface contribute too the conduction. Furthermore, by taking the free-electron-energy expression
E(k)) = h2k2/lme, the Drude result for the conductivity is recoverable in the Boltzmann
approach. .
Thee use of the Boltzmann equation may be avoided by employing the more general Kuboo formalism [2.21]. Here, the external field is taken as a perturbation of the Hamiltonian, andd one considers the linear response of the observable under this perturbation as a function of time.. The conductivity derived in linear response theory is given exactly in terms of a time correlationn between the current component j(0) at time 0 and j(t) at time t [2.15; 2.22]
00 =
7^J'0
( t )J
( 0 ))
c l t<
2-
14)
KQKQ 1
AA more detailed account of this approach falls beyond the scope of this thesis, although later onn we will use some results based on the Kubo formalism.
AA treatment of the temperature dependence of the electrical conductivity requires a detailedd consideration of the scattering mechanisms involved. If the scattering processes act independently,, the various different scattering mechanisms may be characterised by a distinct relaxationn time Tit and the total relaxation time x is given by Matthiesen's rule [2.23]:
1/xx = X i VTi Thus, the resistivities arising from different independent scattering mechanismss are simply additive. Generally, deviations from Matthiesen's rule are not too large.. Under the assumption that Matthiesen's rule applies, the resistivity of a simple non-magneticc metal is usually expressed as
ptot(T)) = p0+pp h(T) (2.15)
wheree po is the residual resistivity and pPh is the resistivity due to scattering by phonons.
Thesee two contributions to the resistivity will be discussed below.
Withinn the framework of the Bloch model, scattering occurs when deviations from the perfectt periodicity of the crystal are present. There are always impurities, vacancies, or other imperfectionss that can scatter electrons. At very low temperatures, solely these imperfections limitt the electrical conduction, yielding a finite residual resistivity p0. The scattering at these
latticee imperfections is temperature independent. As the temperature is raised, thermal vibrationss of the ions about their equilibrium position give rise to scattering of the conduction electronss by phonons. Using a Debye model for the vibrational response of the lattice, the phononn contribution to the resistivity is given by the Bloch-Griineisen relation [2.24]
Chapterr 2
PPh(T)) = 4
e e
TT f e»/T x5dx
«e- > !! ( . - - i ) ( i - , ~ ) ( 2 1 6 )
wheree 0 D is the Debye temperature. Expression 2.16 yields a T5 behaviour at low temperaturess (T < 0D) and a linear temperature dependence at high temperatures (T > @D).
Deviationss from this behaviour may occur due to e.g. gaps in the phonon spectrum and Umklappp processes.
2.2.22 Band-structure effects and spin-dependent scattering
Drude'ss theory and its simplest generalisation to particles obeying Fermi-Dirac statisticss rely on the free-electron approximation. This implies a spherical Fermi surface, whichh may be a reasonable approximation for the so-called 'simple metals', in which the conductionn electrons originate from atomic s and (to a lesser extent) p states. In many transition-metall systems, however, the band structure is much more complicated, due to the bandss originating from d states. The description of such systems can be simplified by ignoring thee mixing (hybridisation) of conduction bands and d bands. This approach was first put forwardd by Mott [2.25], who considered a two-band model: the s (and p) electrons are in a broadd band with a low density of states at the Fermi energy, while the d electrons are in a relativelyy narrow band with a high density of states. He assumed that the current is largely carriedd by the s electrons, since they have an effective mass comparable to that of free electrons,, while the d electrons have a much higher effective mass and therefore have a much lowerr mobility. Hence, the electrical resistivity is determined by the scattering of s electrons. Thiss scattering can either be from one s state into another (characterised by xss), or from an
ss state into a d state (x^). The total resistivity is then given by
mme e
PP = —7 ne ne (2.17) )
Sincee the scattering probability depends upon the density of states into which the electrons are scattered,, s-d scattering occurs much more frequently than s-s scattering. In this way, Mott couldd explain the relatively high electrical resistivity of Pd and Pt at high temperatures comparedd to that of the 'simple metal' Cu.
Becausee electrons have a spin, the electron states are divided into a up and a spin-downn subband. If the two d subbands are not filled equally, the system has a net magnetisation.. For a strong ferromagnet, the corresponding band picture is given in figure 2.3. Ass was suggested by Mott [2.26], at least at low temperatures the spin direction of the conductionn electrons is conserved in the vast majority of scattering processes. Then, the currentt can be thought of as a sum of one electron current with spin up and one with spin
Theoreticall aspects 21 1
n(E)) - n(E)
Figuree 2.3. Schematic band picture of a metal containing both sip and d electrons with opposite spin directions. .
down.. Moreover, the density of states may be quite different for the spin-up and spin-down bands,, resulting in a spin polarisation of the total current, i.e.: for one type of spin the current iss larger than for the other type. In this picture, a rapid change in the density of states of the dsubbandss with increasing energy can lead to a significant change in the resistivity as a functionn of e.g. temperature, magnetic field and pressure. Although the band-structure descriptionn is very crude, it can at least qualitatively account for a variety of observed resistivityy effects. Quantitative use of this model is limited, as the proper application of the Boltzmannn equation requires a detailed knowledge of the band structure, and a realistic band structuree seldom allows for a clear-cut distinction between conduction-electron and rf-band states. .
Above,, we have discussed the spin-dependent-scattering mechanism as being a consequencee of a band-structure effect. Another mechanism resulting in spin-dependent scatteringg arises from scattering of electrons by a magnetic ion. The perturbation Hamiltonian off an interacting electron at position r is given by
HH = V ( r ) - 2 J ( r ) a - 5 (2.18)
Here,, V(r) represents the non-magnetic scattering potential due to Coulomb interactions, and J(r)) represents the exchange interaction between the conduction electron in spin state o and a magneticc moment with spin S. Supposing only elastic collisions (without spin-flip scattering), thee scattering probability is given by the squares of the matrix elements of the total potential.
Chapterr 2
Ferromagnet t Antiferromagnet t
Figuree 2.4. The two-current model for parallel (left) and antiparallel (right) alignment of the magnetic moments. .
Forr spin-up electrons, the scattering probability is given by (V2 + msJ2 - 2/nJV), where J and
VV are the matrix elements of J(r) and V(r), respectively, and ms is the magnetic quantum
numberr of the magnetic ion. Similarly, for spin-down electrons one obtains (V2 + m j2 +
2m2mssJV).JV). The asymmetry of the scattering arises from the interference term.
Thus,, spin-polarised transport may arise from band-structure effects, as well as from magneticc interactions. However, in the former case the spin-dependent-scattering probability iss due to a different density of possible final states for spin-up and spin-down electrons, while inn the latter case it is due to a difference in the scattering amplitudes.
AA standard way to describe the magnetoresistivity arising from spin-polarised transport iss by means of the two-current model. Under the assumption that the total current is carried by twoo independent parallel currents, one for each spin direction, and that the majority of scatteringg events is elastic, the resistivity in the ferromagnetic state is given by l/pF== l/pT + l/px, where pT ( p j denotes the resistivity of the spin-up (spin-down) channel.
Thee resistivity in the antiferromagnetic state is given by pAF = (pT + Pi) / 4 (see figure 2.4).
Thee normalised difference in resistivity between the two magnetic states yields a magnetoresistance e
(Pf-P|)2 2
( P T+P I )2 2
(2.19) )
Fromm this expression, it is clear that an asymmetry in the spin-polarised current yields a lower electricall resistivity in the ferromagnetic state compared to the antiferromagnetic state.
Theoreticall aspects 23 3
2.2.33 Magnetic scattering
Ass we have seen above, when a material exhibits magnetic interactions, an additional contributionn to the electrical resistivity must be taken into consideration. This contribution pm
describess scattering processes of conduction electrons due to disorder in the arrangement of thee magnetic moments. A treatise of the resistivity arising from magnetic interactions is a complicatedd task, as it depends critically on the details of the magnetic interactions. One can discriminatee between localised and itinerant, ferro- and antiferromagnetic structures, which mayy have long-range as well as short-range spin correlations. Furthermore, there are three mainn temperature regions to be considered: T < Tc, T ~ Tc and T > Tc, where Tc is the
magneticc ordering temperature. During the last five decades, numerous models have been constructedd that take into account different aspects of the magnetic interactions. In addition to thee temperature dependence, often the magnetic-field dependence of the electrical resistivity is consideredd as well. Here, we will consider some selected models, that will be used later on in thiss work. For more extensive discussions on the electrical resistivity in bulk materials arising fromm magnetic interactions, we refer to 2.22, 2.27, 2.28 and 2.29.
Temperaturee dependence of magnetic scattering
Thee exchange scattering of conduction electrons in a magnetic metal with localised spinss has been treated by several authors [2.30; 2.31; 2.32]. Following the approach of Van Peski-Tinbergenn et al. [2.30], well above the ordering temperature the Hamiltonian 2.18 gives riserise to a temperature-independent resistivity resulting from scattering of the conduction electronss by isolated magnetic ions
p =mg2kk N. [V2+ J25(5 + 1)] ( 2 2 0 )
nnnneeee h
wheree kF is the Fermi wave vector, N, is the number of isolated scattering centres per unit volumee and ne is the density of electrons. The second term in the brackets of equation 2.20
reflectss the contribution due to magnetic interactions and is denoted as the spin-disorder resistivityy pspd.
Nearr the ordering temperature, spin fluctuations prevail, and the range of the spin-spin correlationss is enhanced. Since the resistivity is proportional to the magnitude of such correlationss [2.32], the detailed behaviour of the resistivity at the critical point is determined byy these fluctuations. Nevertheless, the behaviour of the spin correlations is not very system dependent,, provided that the various systems have the same dimensionality and symmetry in thee ordered phase. Then, the variation of the resistivity around the ordering temperature can be
= 3 3 co o T
c c
1 ^ - ^ ^
// // // / // / // // s // s // s // s __„-**** _ „ - " p p rspd d p p total l P P mag g " " "" P phonon nTT (a.u.)
Figuree 2.5. Schematic temperature dependence of the electrical resistivity of a ferromagnetic
material.. The dashed lines represent different contributions to the total resistivity (solid line).
describedd with a universal relation in terms of critical exponents. Fisher and Langer [2.33] havee found for a localised ferromagnet taking into account fluctuations in the short-range part off the spin-spin correlation function
dT T
T - Tr r
(2.21) )
wheree A is a positive constant and a is the critical exponent of the specific heat. An extensive treatisee of critical behaviour of the electrical resistivity in magnetic systems has been given by Rossiterr [2.22] and Alexander et al. [2.34].
Beloww Tc, the localised spins begin to order so that spin-flip scattering becomes impossiblee and the conduction-electron-relaxation time sharply increases, causing a pronouncedd kink in the resistivity at the ordering temperature. In the simplest case, the temperaturee dependence of the magnetic scattering is given by [2.31]
Pm(T)) = P spd d
{sy {sy
S(S+l) S(S+l) (2.22) )
wheree (S) is the average ion spin. The behaviour of pm together with the non-magnetic contributionss to the resistivity is depicted schematically in figure 2.5.
Accordingg to equation 2.22, the reduction of the spin-disorder scattering in the magneticallyy ordered state is governed by the magnetic properties via (S)2. In the
low-Theoreticall aspects 25 5 temperaturee regime, low-lying excited states like spin waves dominate the magnetic properties,, giving rise to scattering of conduction electrons. Assuming that the spin-wave spectrumm (u(q) °= q2, several authors [2.35; 2.36] derived a T2 dependence for the spin-wave contributionn to the electrical resistivity.
Inn antiferromagnetic or spiral spin structures, the periodicity of the magnetic structure mayy be different from that of the crystallographic structure. Hence, the conduction electrons experiencee an exchange interaction with a periodicity different from the periodicity of the lattice.. This introduces so-called superzone boundaries in the Brillouin zone and distorts the Fermii surface, leading to a distinct effect in the resistivity [2.37; 2.38]. In a simple model, the temperaturee dependence of the electrical resistivity reads as
P„(T)) = « f ( 2 , 3 ,
11 - g m(T)
wheree m(T) is the normalised sublattice magnetisation, and g (< 1) is a constant which should bee deduced from the geometry of the Fermi-surface, but is usually left as an adjustable parameter,, g characterises the reduction of the number of conduction electrons due to the Fermi-surfacee gapping. Thus, according to equation 2.23 the Fermi-surface gapping may lead too an upturn in the resistivity below the antiferromagnetic ordering temperature TN. In this case,, the power-law relation 2.21 describing the ordering phenomenon has to be extended to
^Pm__ = A - ' " - B ^^£L (2.24) d T T
wheree B is a positive constant and X is an exponent arising from the energy gap [2.39; 2.40]. Mannarii [2.41] has pointed out that in the low-temperature regime the electrical resistivityy due to spin waves has a T4 dependence. He obtained this result merely by replacing thee spin-wave spectrum co(q) <* q for ferromagnetic systems by oo(q) <* q for antiferromagnets.. Furthermore, a gap in the spin-wave dispersion relation can lead to an exponentiall temperature dependence of the magnetic contribution to the electrical resistivity at loww temperatures [2.42].
Magnetic-fieldd dependence of magnetic scattering
Welll above the ordering temperature, the magnetic moments are completely disordered yieldingg a scattering contribution given by equation 2.20. Application of a magnetic field to a localisedd paramagnet leads to alignment of the magnetic moments and hence to a reduction of thee resistivity. According to De Gennes and Friedel [2.32], the magnetoresistance is related to thee correlation function (Si Sj) between the spins (cf. equation 2.22), or more simply to the
T - TN N
TN N
-a a
- B B T - TN N TN N
2 .. 1
O) ) 03 3 E E CD D D l l 03 3 E E ivv \ ii x \ ii \ \ \\ \ \\ \ \\ \ \\ \ \\ \ \\ \ xx \ NN \ ^ SS \B B
BB (a.u.)
Figuree 2.6. Magnetoresistance of an antiferromagnetic metal as obtained by Yamada and Takada
[2.46].. Dashed line: molecular-field approximation. Solid line: random-phase approximation. squaree of the magnetisation in the paramagnetic range
Pm( B , T ) - pm( 0 , T ) ) Pm(0,T) )
/ M ( B , T ) )
\\ Msa,
(2.25) )
wheree a is a positive proportionality constant. The reduced magnetisation M / Ms a t can be describedd with the Brillouin function, yielding a B ' behaviour of the magnetoresistance for smalll fields (UBB « k^T).
Forr localised magnetic moments, a quantitative link between the magnetoresistivity andd the magnetisation has been worked out by several authors. Using a theoretical approach of Vann Peski-Tinbergen et al. [2.30] which is based on the Hamiltonian 2.18, the field-induced changee of the resistivity is found [2.43; 2.44; 2.45] to behave as
p ( 0 , T ) - p ( B , T )) = Ac J2( S2) t a n h ( a / 2 )
--4 V2J2( SZ)2 2
V22 +J2[S0S + l ) - ( S;) t a n h ( a / 2 ) ] (2.26) )
wheree c indicates the atomic fraction of magnetic scattering sites, A = 3izm"Q./2he2Er
incorporatess details of the band structure (m is the effective mass of the conduction electrons withh Fermi energy Ep, and £1 is the atomic volume). The dimensionless parameter a equals
gpiBBj/A:BTT , where g is the Lande g-factor, and Bj is the internal field.
Theoreticall aspects 27 7
correlations,, or in other words, suppresses the spin fluctuations for small q, leading to a significantt negative magnetoresistance. In the case of antiferromagnets, the magnetoresistance iss generally expected to be less significant than in ferromagnets, since the external field does nott couple directly with the staggered component of the magnetisation (cf. equation 2.4). The effectt of spin fluctuations on the magnetoresistance of antiferromagnetic metals has been investigatedd by Yamada and Takada [2.46]. Their results are shown in figure 2.6. In the molecular-fieldd approximation, the resistivity is constant in the antiferromagnetic state and decreasess exponentially well above the antiferromagnetic to ferromagnetic transition field (Bc).. Taking into account the effect of spin waves in the random-phase approximation (RPA)
yieldss an enhancement of the magnetoresistance near Bc. The peak at Bc becomes sharper with
decreasingg temperature: pm(Bc,T)/pm(0,T)«>/TN/T . Usami [2.47] has investigated the
magnetoresistancee of a system with both ferromagnetic (q = 0) and antiferromagnetic spin fluctuationss (q = Q) (cf. section 2.1.2). He finds, based on numerical calculations in the renormalisedd RPA, a qualitatively similar behaviour as Yamada and Takada [2.46].
Abovee the ordering temperature, Usami predicts for large magnetic fields a square-root behaviourr of the magnetoresistance
pm(B,T)) - pm(0,T) = - a VB-BC (2.27)
wheree a is a positive proportionality constant. The dominant contribution to the magnetoresistancee arises from the antiferromagnetic spin fluctuations, as scattering of conductionn electrons at spin fluctuations around q - 0 leads to small-angle scattering, which hass only a moderate effect.
Inn summary, the temperature and field dependence of the resistivity due to magnetic interactionss pm(B,T) depends critically on the details of these interactions and the way they are
treatedd in the different models. Thus, for a full description of the various observed (magneto)resistancee effects, detailed knowledge of the magnetic structure is a prerequisite. Therefore,, in the next section the theory of group representations is introduced, which suppliess us with a solid basis of the magnetic-structure analysis of magnetically ordered crystallinee solids from symmetry considerations.
2.33 Theory of group representations
Inn a crystalline solid, the magnetic moments are coupled to a periodic lattice. In itinerant-electronn magnets, the magnetic-moment density is generated by conduction electrons,, which are subjected to the periodic electrostatic potential of the ions forming the crystal.. In localised systems, the magnetic moments are confined to the ions. One thus can imaginee that the symmetry of the crystal structure imposes confinements on the possible magnetic-momentt configurations in magnetically-ordered crystalline solids.
Chapterr 2 Similarr to a crystal structure, a magnetic structure may be described by a unit cell that iss repeated in space. The periodicity of a magnetic structure will be related to the periodicity off the crystal structure if the components of the propagation vector q of the magnetic modulationn are rational fractions of reciprocal lattice vectors. If the propagation vector of a magneticc structure is known, we may express the magnetic moment S„j of atom j in cell n of thee crystal in terms of the magnetic moment S0j of atom j in cell zero by means of the equation
Snjj = e^RS0i (2.28)
wheree R is the vector of translation from magnetic cell zero to cell n. Thus, if the magnetic propagationn vector is known, the problem of deciphering the magnetic structure has reduced too determining the magnetic-moment configuration within one magnetic unit cell.
Theree are two main approaches in the analysis of magnetic-moment configurations fromm symmetry considerations. The classical approach considers the set of crystallographic symmetryy operations augmented by time reversal as an additional symmetry operation [2.48]. Thee physical importance of the operation of time reversal in solids is in the reversal of the magneticc moments. In this way, from the 230 crystallographic space groups 1651 magnetic spacee groups, also termed Shubnikov groups, are constructed [2.49]. The second approach is basedd on the theory of group representations, wherein the magnetic structure is described in termss of the basis functions of irreducible representations of the crystallographic space group [2.50].. The theory of group representations is more general than the Shubnikov-space-group analysis.. It has been demonstrated [2.50; 2.51] that the magnetic structures described by Shubnikovv space groups belong to one-dimensional real representations of the 230 space groups,, whereas representation theory can also deal with those magnetic structures belonging too one-dimensional complex and even to multi-dimensional representations associated with anyy magnetic-propagation vector in or on the first Brillouin zone.
Thiss section deals with a short outline of the theory of group representations, which is usedused to generate all possible magnetic-moment configurations for a given crystal structure and magnetic-propagationn vector. In several parts of this thesis, representation theory will be appliedd to facilitate magnetic-structure analysis from neutron-diffraction experiments and magnetisationn measurements. The theory of group representations is quite general and may be usedd to analyse different properties of the crystalline solid. It can, for example, deal with phonons,, the wave function of an electron in a crystal, crystalline-electric-field splitting, or, in thee case considered here, components of atom spins. An extensive treatise of group theory and magneticc structures is given by Tinkham [2.52] and Izyumov et al. [2.53].
Inn the field of crystallography, the concept of groups is well established. A Bravais latticee is characterised by the specification of all symmetry operations that map the lattice onto itself.. This set of operations is known as the symmetry group or space group of the lattice. In orderr to set the stage for a presentation of the theory of group representations, we state here thee abstract definition of a group. A group G is defined as a set of elements Pg (g = 1,2,...)
Theoreticall aspects 29 9 Pi=e Pi=e Pi Pi P3 3 P4 4 P,=e e e e P2 2 P3 3 P4 4 P2 2 P2 2 e e P4 4 P3 3 P3 3 P3 3 P4 4 e e P2 2 P4 4 P4 4 P3 3 P2 2 e e
Tablee 2.1. Group-multiplication table of the point group belonging to space group P 2Jc (No. 14). 1.. The product of any two elements is in the set; i.e.: the set is closed under group
multiplication n
2.. The associative law holds: (Pi»Pj)»Pk = Pi«(Pj»Pk)
3.. There is a unit element Pi = e such that «?«Pj = P^e = P,
4.. There is in the group an inverse Pf1 to each element Pi such that P^1 «Pi = PfPf1 = e Forr the description of the symmetry of crystallographic structures, we can restrict our attention too finite groups. These contain a finite number h of group elements, where h is said to be the orderr of the group G. A convenient way to characterise a group is by the group-multiplication table.. As an example, the group-multiplication table of the point group belonging to the monoclinicc space group P 2\lc (No. 14) is given in table 2.1.
Too each of the four symmetry operations Pg (1 < g s h = 4) a matrix D(g) can be
ascribed,, the ordinary matrix multiplication serving as group-multiplication operation.
D(l)) = e = (I (I 0 0
lo o
00 o"! 11 000 ij
D(3)) = (-1(-1 0 0\ 0 - 1 0 0 v00 0 - 1 f-If-I 0 Ö) D(2)) = D(4)) = 0 0 0 0 1 1 0 0 0 0 -11 0 00 1 00 0^ 11 0 00 -1 (2.29) )Thee set of matrices, that obeys the group-multiplication table, is called a representation of the group.. The dimensionality of the matrices D(g) is termed the dimensionality 1 of the representation.. Clearly, any group has a plurality of representations of which the irreducible representationss are the most important. The meaning of the irreducible representations and the methodd to obtain them is given below.
Thee matrices of the representation given in 2.29 all have the same block-diagonal form.. When the matrices are multiplied, these blocks are independently multiplied. Hence, the sett of the corresponding blocks satisfies the multiplication rules and this set is also a representationn of the same group, only with a lower dimension. In our example, the three-dimensionall representation given in 2.29 is reducible to three one-dimensional representations (off which the first two are the same).
Rep.1:: D(l) = (l) D(2) = (-l) D(3) = (-l) D(4) = (l)
Rep.2:: D(l) = (l) D(2) = (-l) D(3) = (-l) D(4) = (l) (2.30)
Rep.3:: D(l) = (1) D(2) = (1) D(3) = (-1) D(4) = (-1)
Inn general, a representation is reducible if there exists a unitary transformation that transformss all matrices D(g) to the same block-diagonal form. If no such unitary transformationn exists, the representation is called irreducible. The number of irreducible representationss and their dimensionality are entirely determined by the structure (i.e.: multiplicationn table) of the group. The irreducible representations are used to generate basis functionss of the group. In the following, the generation of basis functions and some essential formulass are described.
Whenn D^ptg) and D^v(g) are the matrix elements of non-equivalent, irreducible
representationss denoted by T' and rJ, respectively, the following orthogonality relation is valid
S D |ï p( g ) D iv( g ) * = f 51 J5a, 5p vv (2.31)
gg M
wheree h denotes the number of elements in the group, 1; denotes the dimensionality of the \th irreduciblee representation and the summation g runs over all group elements (1 < g < h). The numberr of irreducible representations of a group is related to the order of the group through thee dimensionality theorem
I I ,22 = h (2.32)
i i
Inn the theory of group representations, the concept of basis functions is very important. Forr each irreducible representation V with dimension lj, various sets of 1; degenerate orthogonall basis functions /K' (K = 1 to lj) can be found. From equation 2.31 it can be deduced
thatt the basis functions of non-equivalent representations are also orthogonal. The basis functionss are obtained by using a suitable starting function ƒ describing the physical entity underr consideration. Here, we take ƒ as a set of atom positions in conjunction with a set of pseudo-vectorss describing the magnetic moment at each atom. Application of the projection operator r
PicK=^-EDL(g)'Pgg (2-33)
Theoreticall aspects 31 1 onn the starting function ƒ projects the component of/along the basis function /K'
P L // = fi (2.34)
Thee set of basis functions /K' belonging to a single irreducible representation V describes, in
principle,, a set of magnetic-moment configurations related by symmetry. There are, however, limitationss to this concept, which become apparent in the discussion given below about phase transitionss in magnetic structures.
Inn the Landau theory of second-order phase transitions [2.54], each magnetic structure cann be considered as the result of one or more phase transitions from the initial paramagnetic phasee of the crystal towards magnetically ordered phases with a lower symmetry. As with decreasingg temperature the transition temperature is passed, the appearance of a net magnetic moment,, described by an additional pseudo-vector at the atoms, provides a change of symmetry.. The magnetic-moment configuration S is expressed as a linear combination of the basiss functions of the irreducible representations of the paramagnetic phase
SS = 5 X / K (2.35)
i,K K
Landauu considered the free energy <ï> of the crystal and argued that it can be expanded near the phasee transition as
** = ®o + l ^ l K ? + 0 ( c4) (2.36)
ii K
wheree the A1 are functions of temperature and pressure. In the paramagnetic state, all A1 are positivee and hence for a minimum in the free energy all parameters c^ equal zero. At the transitionn temperature, the change of sign of one of the A' causes the appearance of a non-zero
ccllKK belonging to the ith irreducible representation. Then, equation 2.35 gives us the general
resultt that, at a second-order phase transition from the paramagnetic state, the magnetic order transformss like the basis functions of a single irreducible representation. Thus, through equationn 2.36, to each irreducible representation corresponds an energy, which differs for the variouss representations. However, when the energies of two irreducible representations are equall or are very similar, a degenerate state results, which is described by mixing of the two irreduciblee representations. In addition, higher-order terms in the expansion of O generally containn cross-terms of several clK 's. When these terms are important, the magnetic structure
willl also be determined by mixing of the basis functions of several irreducible representations. Too proceed further, Landau argued that in the analysis the irreducible representations off the symmetry group of the magnetically ordered phase should be taken. This can simplify thee analysis considerably as in the ordered phase the symmetry of the magnetic structure is
Chapterr 2 determinedd by the subgroup Gi of the crystallographic group G which leaves the propagation vectorr q invariant, or which transforms it to an equivalent propagation vector.
Thee hypothesis of a second-order phase transition from the paramagnetic phase to the magneticallyy ordered phase involving a single irreducible representation, may be envisaged as follows.. Near the transition point in a paramagnetic crystal, fluctuations of the magnetic order arisee and these spin fluctuations can be classified in the basis functions of the irreducible representationss of the group G. Upon lowering of the temperature, the correlation length of onee type of the spin fluctuations becomes infinite, resulting in magnetic ordering.
Upp to now, we have investigated the symmetry of magnetic phases arising from a second-orderr phase transition from the paramagnetic state. In some systems, a second phase transitionn takes place from one magnetic state to another. If this transition is of first order, mixingg of irreducible representations is allowed and it is difficult to say anything about the neww structure. If, however, the transition is of second order, the Landau theory can be used to describee the new magnetic structure [2.55]. It is discussed that the new magnetic structure resultss from the old magnetic structure plus the moment configuration corresponding to one additionall irreducible representation [2.56]. In this way, complicated magnetic structures may arise.. Actually, the theory on phase transitions in itinerant-electron systems discussed in sectionn 2.1.2, yields some examples of magnetic phase diagrams that may arise.
Inn case the crystal structure consists of different magnetic atoms or crystallographically non-equivalentt sites, the spin Hamiltonian describing the coupling between the different kinds off atoms generally has the same symmetry as the crystal structure. Then, one has to combine thee resulting moment arrangements from models belonging to the same irreducible representation.. However, if the spin components of the different atoms couple through a lower symmetry,, each spin structure can be described by different irreducible representations. If theree is complete decoupling of the different species, the spin configurations could even belongg to different propagation vectors.
AA complication in the analysis of magnetic structures, that has not yet been discussed, arisess from the determination of the propagation vector q. For a given q, there are several otherr equivalent propagation vectors that are given by the symmetry operations of the crystallographicc point group. This set of equivalent propagation vectors is called the star of q. Forr a magnetic material, the following situations may arise. First, more than one equivalent propagationn vector may contribute to the magnetic structure. Second, the sample may consist off several magnetic domains, the magnetic structure of each domain being described by a differentt propagation vector within the star. Furthermore, mixing of the two above-mentioned casess is possible. In practice, it is difficult to discriminate between the different situations.
Takingg into account the limitations discussed above, the given introduction on the theoryy of group representations supplies us with a sufficient basis of the magnetic-structure analysiss of crystalline solids. In the following chapters, we will apply the theory of group representationss to investigate the magnetic structures of the studied compounds.
