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OXIDE and SEMICONDUCTOR MAGNETISM

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OXIDE and SEMICONDUCTOR MAGNETISM OXIDE and SEMICONDUCTOR MAGNETISM J. M. D. Coey Physics Department, Trinity College, Dublin 2 Ireland. 1. Single-ion effects 2. Collective Effects 3. Examples www.tcd.ie/Physics/Magnetism Boulder July 2003 1 Some references: Magnetic oxides: D. J. Craik (editor) Wiley-Interscience 1975 Magnetism and the chemical bond: J. B. Goodenough Wiley-Interscience 1963 Introduction to ligand field theory. C. Ballhausen McGraw Hill 1962 Mineralogical applications of crystal-field theory. R. G. Burns CUP 1993 (2nd ed.) Point charge calculations of energy levels of magnetic ions in crystalline electric fields. M. T. Hutchings Solid State Physics 16 227 - 73 (1964) Boulder July 2003 2 1. Single-ion effects 1.1 Ubiquity of oxides. Oxide structures. Octahedral and tetrahedral sites. Magnetic ions – 3d, 4d, 4f. 1.2. Electronic structure of free ions (summary). Hund’s rules. g- factors. Paramagnetic susceptibility. 1.3. Ions in solids. Crystal field. Crystal field Hamiltonian. One-electron states. 3d t2g and eg states. Notation. 2p-3d hybridization. One- electron energy-level diagrams in different symmetry. Quenching of orbital moment. Many-electron states. Orgel and Tanabe-Sugano diagrams. 1.4. Crystal field and anisotropy. Single-ion anisotropy. Determination m of Bn . Boulder July 2003 3 1.1 Ubiquity of Oxides Earth’s crust is composed almost entirely of oxides — rocks, economic minerals, water. Composition in atomic % 2- Oxygen (O ) is most abundant Fe O followed by silicon (Si4+) and Si aluminium (Al3+). Al3+ Al Crust is mostly composed of Fe aluminosilicates. Mg Iron (Fe2+/Fe3+) is most O2- Si4+ Ca abundant magnetic element. It is K 40 times as abundant as all other magnetic elements Na together. H Others Boulder July 2003 4 Ion Abundance (at%) Configuration O2- 60.7 2p6 Si4+ 20.6 2p6 Al3+ 6.1 2p6 Na+ 2.6 2p6 Fe2+/3+ 2.1 3d6/5 H+ 2.1 1s0 Ca2+ 1.9 3p6 Mg2+ 1.8 2p6 K+ 1.5 3p6 Electronic configuration of 92% of the ions in the crust is the same 2p6 ! The 2p6 closed shell is very stable. Boulder July 2003 5 Abundances of magnetic ions Price scales roughly inversely with abundance. Boulder July 2003 6 Boulder July 2003 7 Ionic structures 1/2 1/2 Roct = (2 -1)rO = 58 pm Rtet = ((3/2) - 1)rO = 0.32 pm Boulder July 2003 8 Cation radii in oxides: low spin values are in parentheses. 4-fold pm 6-fold pm 6-fold pm 12-fold pm tetrahedral octahedral octahedral substitutional Mg2+ 53 Cr4+ 3d2 55 Ti3+ 3d1 67 Ca2+ 134 Zn2+ 60 Mn4+ 3d3 53 V3+ 3d2 64 Sr2+ 144 Al3+ 42 Cr3+ 3d3 62 Ba2+ 161 Fe3+ 3d5 52 Mn2+ 3d5 83 Mn3+ 3d4 65 Pb2+ 149 Fe2+ 3d6 78 (61) Fe3+ 3d5 64 Y3+ 119 Co2+ 3d7 75 (65) Co3+ 3d6 61 (56) La3+ 136 Ni2+ 3d8 69 Ni3+ 3d7 60 Gd3+ 122 The radius of the O2- anion is 140 pm Boulder July 2003 9 Gd Co Gd Co As metallic atoms, the transition metals occupy one third of the volume of the rare earths. As ions they occupy less than one tenth. Co2+ Co0 Gd Boulder July 2003 10 1.2 Electronic structure of free ions Consider a single electron in a central potential. A hydrogenic atom is composed of a nucleus of charge Ze at the origin and an electron at r,q,f. First, consider a single electron in a central potential fe = Ze/4π r e0 z 2 2 2 -e H = - (¨ /2m)— - Ze /4πe0r l r In polar coordinates: q —2 = ∂2/∂r2 +(2/r)∂/∂r + 1/r2{∂2/∂q2 + cotq∂/∂q + (1/sin2q)∂2/∂2f} l f Ze The term in parentheses is -l 2. Schrödinger’s equation is H y = Ey The wave function y means that the probability of finding the electron in a small volume dV ar r is y*(r)y(r)dV. (y* is the complex conjugate of y). Eigenfunctions of the Schrödinger equation are of the form y(r,q,f) = R(r)Q(q)F(f). ml ® The angular part Q(q)F(f) is written as Yl (q,f). ml The spherical harmonics Yl (q,f) depend on two integers l, ml, where l is ≥ 0 and |ml| ≤ l. F(f) = exp(imlf) where ml = 0, ±1, ±2 ....... The z-component of orbital angular momentum, represented by the operator l z = -i¨∂/∂f, has eigenvalues <F| lz |F> = ml¨. ml Q(q) = Pl (cosq), are the associated Legendre polynomials with l ≥ |ml|, so ml = 0, ±1, ±2,....±l. Boulder July 2003 11 z The square of the orbital angular momentum l2 has eigenvalues l(l+1)¨. The orbital angular momentum has magnitude ÷[l(l+1)]¨ and its projection ml¨ 2 along z can have any value from -l¨ to +l¨. The quantities lz and l can be measured simultaneously (the operators commute). In the vector model, ÷[l(l+1)]¨ The total angular momentum is a vector which precesses around z. 0 Spherical harmonics. s Y0 = ÷(1/4π) 0 ±1 ±if p Y1 = ÷(3/4p) cos q Y1 = ±÷(3/8p) sin q e 0 2 ±1 ±if d Y2 = ÷(5/16p)(3cos q - 1) Y2 = ±÷(15/8p) sinq cosq e ±2 2 ±2if Y2 = ÷(15/32p) sin q e 0 3 ±1 2 ±if f Y3 = ÷(7/16p)(5cos q - 3cosq) Y3 = ±÷(21/64p)(5cos q - 1)sinq e ±2 2 ±2if ±3 3 ±3if Y3 = ÷(105/32p) sin qcosq e Y3 = ±÷(35/64p) sin q e •The radial part R(r) depends on l and also on n, the total quantum number; n > l; hence l = 0, 1, ......(n-1). l R(r) = Vn (Zr/na0)exp[-(Zr/na0)] 0 2 2 V1 = 1. Here a0 = 4πe0¨ /me = 52.9 pm is the first Bohr radius, the basic length scale in atomic physics. The energy levels of the 1-electron atom are 4 2 2 2 2 E = -Zme /8h e0 n = -ZR/n 4 2 The quantity R = me /8h e0 = 13.6 eV is the Rydberg, the basic energy in atomic physics. For the central Coulomb potential fe, the potential energy V(r) depends only on r, not on q or f. E depends only on n. Boulder July 2003 12 The three quantum numbers n, l, ml denote an orbital, a spatial distribution of electronic charge. Orbitals are denoted nx, x = s, p, d, f for l = 0, 1, 2, 3. Each orbital can accommodate up to two electrons with spin ms = ±1/2. No two electrons can be in a state with the same four quantum numbers (Pauli exclusion principle). The hydrogenic orbitals are listed in the table n l ml ms No of states 1s 1 0 0 ±1/2 2 2s 2 0 0 ±1/2 2 2p 2 1 0,±1 ±1/2 6 3s 3 0 0 ±1/2 2 3p 3 1 0,±1 ±1/2 6 3d 3 2 0,±1,±2 ±1/2 10 4s 4 0 0 ±1/2 2 4p 4 1 0,±1 ±1/2 6 4d 4 2 0,±1,±2 ±1/2 10 4f 4 3 0,±1,±2,±3 ±1/2 14 •The Pauli principle states that no two electrons can have the same four quantum numbers. Each orbital can be occupied by at most two electrons with opposite spin. Boulder July 2003 13 n =1 s l = 0 X, Y, Z n =2 p l = 1 X, Y, Z X, Y, Z ml =0 ml =±1 n =3 d l = 2 X, Y, Z X, Y, Z ml =0 ml =±1 ml =±2 X, Y, Z Boulder July 2003 14 n =1 l = 0 n =2 l = 1 ml = 0 ml = ±1 n =3 l = 2 Boulder July 2003 15 The many-electron atom. 2 In the many-electron atom, terms like e /4πe0rij, must be added to the Hamiltonian. One way of dealing with the extra Coulomb interactions is to suppose that each electron sees a different spherical charge distribution, which produces a different central potential for each one. The potential with many electrons is not a simple Coulomb potential well; the degeneracy of electrons with different l is raised. The 4s shell, for example, is then lower in energy than the 3d shell, which defines the shape of the periodic table. The quantities VI(r) must be determined self-consistently (the Hartree-Foch approximation) When several electrons are present on the same atom, at most two of them having opposite spin can occupy the same orbital (Pauli principle). Their spin and orbital angular momenta add to give resultants S = Ssi, MS = Smsi, L = Sli, ML = Smli. 1s 2s 2p ML MS Consider the six-electron carbon atom; ↑Ø + ↑Ø + ↑ Ø ↑ Ø 2 0 2 0 2 2 2 1s 2s 2p . The 15 states fall into three ↑ ↑ 2 1 groups, or terms. ↑ Ø 1 0 Ø ↑ 1 0 Ø Ø 1 -1 The notation for terms is to denote L = 1. ↑ ↑ 1 1 ↑ Ø 0 0 2. 3. ….by S, P, D, …. and to include the ↑Ø 0 0 spin multiplicity 2S + 1 as a superscript. Ø ↑ 0 0 Ø Ø 0 -1 The energy splitting of the terms is of Ø Ø 0 -1 order 1 eV. ↑ Ø -1 0 Ø ↑ -1 0 2S+1 L ↑ ↑ -1 1 ↑Ø -1 0 In spectroscopy, the energy unit cm-1 is used. Handy conversions are:1 eV ≡ 11605 K and 1 cm-1 ≡ 1.44 K Boulder July 2003 16 Term L S (ML, MS) 1S 0 0 (0,0) 3 P 1 1 (1,1)(1,0)(1,-1)(0,1)(0,0)(0.-1)(-1,1)(-1,0)(-1,-1) S J 1D 2 0 (2,0)(1,0)(2,1)(2,0)(0,0)(-1,0)(-2,0) Finally we need to couple the spin and orbital angular momentum L to form a resultant J.
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