Half-metallic ferromagnets: From band structure to many-body effects M.I. Katsnelson∗ Institute for Molecules and Materials, Radboud University of Nijmegen, NL-6525 ED Nijmegen,, Netherlands V.Yu. Irkhin Institute of Metal Physics, 620219 Ekaterinburg, Russia L. Chioncel Institute of Theoretical Physics, Graz University of Technology, A-8010 Graz, Austria and Department of Physics, University of Oradea, 410087 Oradea, Romania A.I. Lichtenstein Institute of Theoretical Physics, University of Hamburg, 20355 Hamburg, Germany R.A. de Groot Institute for Molecules and Materials, Radboud University of Nijmegen, The Netherlands and Zernicke Institute for Advanced Materials, NL-9747 AG Groningen, The Netherlands arXiv:0711.0872v1 [cond-mat.mtrl-sci] 6 Nov 2007 1 Abstract A review of new developments in theoretical and experimental electronic structure investigations of half-metallic ferromagnets (HMF) is presented. Being semicon- ductors for one spin projection and metals for another ones, these substances are promising magnetic materials for applications in spintronics (i.e., spin-dependent electronics). Classification of HMF by the peculiarities of their electronic structure and chemical bonding is discussed. Effects of electron-magnon interaction in HMF and their manifestations in magnetic, spectral, thermodynamic, and transport prop- erties are considered. Especial attention is paid to appearance of non-quasiparticle states in the energy gap, which provide an instructive example of essentially many- body features in the electronic structure. State-of-art electronic calculations for correlated d-systems is discussed, and results for specific HMF (Heusler alloys, zinc-blende structure compounds, CrO2, Fe3O4) are reviewed. ∗Electronic address: [email protected] 2 Contents I. Introduction 6 II. Classes of half-metallic ferromagnets 10 A. Heusler alloys and zinc-blende structure compounds 10 1. Heusler C1b alloys 10 2. Half-metals with zinc-blende structure 16 3. Heusler L21 alloys 17 B. Strongly magnetic half-metals with minority spin gap 19 1. Chromium dioxide 19 2. The colossal magnetoresistance materials 21 C. Weakly magnetic half-metals with majority spin gap 22 1. The double perovskites 22 2. Magnetite 23 D. Strongly magnetic half-metals with majority spin gap 24 1. Anionogenic ferromagnets 24 E. Sulphides 25 1. Pyrites 26 2. Spinels 26 F. Miscellaneous 27 1. Ruthenates 27 2. Organic half-metals 28 III. Model theoretical approaches 29 A. Electron spectrum and strong itinerant ferromagnetism in the Hubbard model 29 B. Electron spectrum in the s-d exchange model: The non-quasiparticle density of states 36 C. The problem of spin polarization 45 D. The tunneling conductance and spin-polarized STM 48 E. Spin waves 51 F. Magnetization and local moments 55 G. Nuclear magnetic relaxation 58 3 H. Thermodynamic properties 61 I. Transport properties 64 J. X-ray absorption and emission spectra. Resonant x-ray scattering 70 IV. Modern first-principle calculations 72 A. Different functional schemes 72 B. LDA+DMFT: the Quantum Monte Carlo solution of the impurityproblem 78 C. Spin-polarized T -matrix fluctuating exchange approximation 83 V. Electronic structure of specific half-metallic compounds 87 A. Heusler alloys. 87 1. NiMnSb: electronic structure and correlations 87 2. Impurities in HMF: lanthanides in NiMnSb 91 3. FeMnSb: a ferrimagnetic half-metal 94 4. Co2MnSi: a full-Heusler ferromagnet 96 B. Half-metallic materials with zinc-blende structure 97 1. CrAs: tunable spin transport 97 2. VAs: correlation-induced half-metallic ferromagnetism? 101 C. Half-metallic transition metal oxides 105 1. CrO2: a rutile structure half-metallic ferromagnet 105 VI. Exchange interactions and critical temperatures in half-metallic compounds 108 A. The Green’s function formalism 109 B. The frozen-magnon approach and DFT calculations of spin spirals 109 C. First-principle calculations 110 1. Semi-Heusler C1b alloys 110 2. Full-Heusler L21 alloys 111 3. Zinc-blende half-metals 111 VII. Conclusions 112 Acknowledgments 114 References 114 4 Figures 130 Tables 161 5 I. INTRODUCTION Twenty-five years ago the unusual magnetooptical properties of several Heusler alloys motivated the study of their electronic structure. It yielded an unexpected result: some of these alloys showed the properties of metals as well as insulators at the same time in the same material depending on the spin direction. This property was baptized half-metallic magnetism (de Groot et al., 1983b). Although it is not exactly clear how many half-metals are known to exist at this moment, half-metallic magnetism as phenomenon has been gen- erally accepted. Formally the expected 100% spin polarization of the charge carriers in a half-metallic ferromagnet (HMF) is a hypothetical situation that can only be approached in the limit of vanishing temperature and neglecting spin-orbital interactions. However, at low temperatures (as compared with the Curie temperature which exceeds 1000K for some HMF’s) and minor spin-orbit interactions a half-metal deviates so markedly from a normal material, that the treatment as a special category of materials is justified. The confusion on the number of well-established half-metals originates from the fact that there is no “smok- ing gun” experiment to prove or disprove half-metallicity. The most direct measurement is spin-resolved positron annihilation (Hanssen and Mijnarends, 1986), but this is a tedious, expensive technique requiring very dedicated equipment. The only proven HMF so far is NiMnSb to the precision of the experiment, which was better than one hundredth of an electron (Hanssen et al., 1990). This number also sets the scale for concerns of temperature- induced depolarization and spin-orbit effects, detrimental for half-metallicity. The half-metallicity in a specific compound should not be confused with the ability to pick up 100% polarized electrons from a HMF. The latter process involves electrons crossing a surface or interface into some medium where their degree of spin polarization is analyzed. This is clearly not an intrinsic materials property. The richness but also the complications of surfaces and interfaces are still not fully appreciated. Because of all these experimental complications, it is not surprising that electronic struc- ture calculations continue to play an important role in the search for new HMF as well as the introduction of new concepts like half-metallic antiferromagnetism. However, electronic structure calculations have weaknesses as well. Most of the calculations are based on density functional theory in the LDA or GGA approximation. It is well known that these methods underestimate the band gap for many semiconductors and insulators, typically by 30%. It 6 has been assumed that these problems do not occur in half-metals since their dielectric re- sponse is that of a metal. This assumption was disproved recently. A calculation on the HMF La0.7Sr0.3MnO3 employing the GW approximation (that gives a correct description of band gaps in many semi-conductors) leads to a half-metallic band gap 2 eV in excess to the DFT value (Kino et al., 2003). The consequences of this result are possibly dramatic: if it were valid in half-metallic magnetism in general, it would imply that many of the materials, showing band gaps in DFT based calculations of insufficient size to encompass the Fermi energy, are actually in reality bona fide half-metals. Clearly much more work is needed in this area. The strength of a computational approach is that it does not need samples: even nonex- istent materials can be calculated! But in such an endeavour a clear goal should be kept in mind. Certainly, computational studies can help in the design of new materials, but the challenge is not so much in finding exotic physics in materials that have no chance of ever being realized. Such studies can serve didactical purposes, in which case they will be in- cluded in this review. However, the main attention will be devoted to materials that either exist or are (meta)stable enough to have a fair chance of realization. This review will cover half-metals and will not discuss the thriving area of magnetic semi- conductors. Some overlap exists, however. The much older field of magnetic semiconductors started with semiconductors like the europium monochalcogenides and cadmium-chromium chalcogenides (Nagaev, 1983). Later, the attention changed to the so-called diluted mag- netic semiconductors (Delves and Lewis, 1963). These are regular (i.e. III-V or II-VI) semi- conductors, where magnetism is introduced by partial substitution of the cation by some (magnetic) 3d transition element. The resulting Curie temperatures remained unsatisfac- tory, however. The next step in the development was the elimination of the non-magnetic transition element altogether. This way HMF’s can be realized provided that the remaining transition-metal pnictides could be stabilized in the zinc-blende or related structures. The review will treat not the (diluted) magnetic semiconductors as such, but some aspects of metastable zinc-blende HMF’s. HMF’s form a quite diverse collection of materials with very different chemical and phys- ical properties and even the origin of the half-metallicity can be quite distinct. For this reason it is required to discuss the origin of the band gap in terms of two ingredients that define a solid: the crystal structure and the chemical composition. Two aspects are of im- 7 portance in this context. The first one is “strong magnetism” versus “weak magnetism”. In a strong magnet, the magnetic moment will not increase if the exchange splitting is increased hypothetically. Thus the size of the magnetic moment is not determined by the strength of the exchange interaction, but is limited instead by the availability of electron states. In practice this implies that either the minority spin sub shell(s) responsible for the magnetism is (are) empty or the relevant majority channel(s) is (are) completely filled. In the case of weak magnetism the magnetic moment is determined by a subtle compromise between the energy gain of an increase in magnetic moment (the exchange energy) and the (band) en- ergy the increase of the magnetic moment costs.