The Quantum Theory of Magnetism

Transcription

1 The Quantum Theory of Magnetism Norberto Mains McGill University, Canada I: 0 World Scientific Singapore NewJersey London Hong Kong

2 Contents 1 Paramagnetism 1.1 Introduction 1.2 Quantum mechanics of atoms L-S ( Russel-Saunders ) coupling Hund's rules Spin-orbit splitting 1.3 The quantum theory of paramagnetism 1.4 Crystal-field corrections Effects of crystal-field symmetry Stevens Operator equivalents 1.5 Quenching of L 1.6 Time reversal and spin Kramers degeneracy 1.7 Effective Spin Hamiltonian Effective gyromagnetic ratio Single-ion anisotropy energy Interacting Spins 2.1 Weiss model of ferromagnetism Critical behaviour of Weiss model 2.2 Microscopic basis of magnetism The direct exchange interaction The superexchange mechanism The RKKY interaction vii

3 viii CONTENTS 3 Mean Field Approximation Helmholtz Free Energy Mean Field Susceptibility Specific Heat of Ferromagnet The Oguchi method Modulated Phases MFA for Antiferromagnetism Longitudinal susceptibility Transverse susceptibility Spin-flop and other transitions Helimagnetism Goldstone's theorem 99 4 Spin waves Introduction Holstein-Primakoff transformation Linear spin-wave theory Semiclassical picture Macroscopic magnon theory Thermal properties Total spin deviation Non-linear corrections The Heisenberg antiferromagnet Introduction AFM spin-waves Sublattice magnetization Ground state energy of AFM Green's functions methods Definitions Spectral representation RPA for spin 1/2 ferromagnet Comparison of RPA and MFA RPA for arbitrary spin RPA for ferromagnets Paramagnetic phase Linear FM chain 152

4 CONTENTS ix Square FM lattice FM with a finite applied field RPA for antiferromagnet Spin 1/2 AFM Arbitrary spin AFM Zero-point spin deviation Correlation length RPA susceptibility of AFM Spin-flop transition xii at low T Transverse susceptibility Single-site anisotropy Dynamic linear response Energy absorbed from external field Susceptibility of FM Corrections to RPA Dipole-Dipole interactions Dipolar Hamiltonian Dipole-exchange spin-waves Uniform precession (k = 0) mode Eingenmodes for k Demagnetization factors Ellipticity of spin precession Effect of magnons an total spin Magnetostatic modes Itinerant Magnetism Introduction Hubbard Hamiltonian Instability of paramagnetic phase Magnons in the Stoner model The RPA susceptibility Singularities of the susceptibility in Stoner model Metals with degenerate bands Spin-density waves 239

5 x CONTENTS 7.8 Hartree Fock description of SDW Effects of correlations Kinetic exchange interaction Indirect Exchange Introduction Effective s-d exchange interaction Indirect exchange Hamiltonian Range function and band structure Semiconductors Intrinsic semiconductors, high T Intrinsic semiconductors, low T Degenerate semiconductors Magnetic multilayer systems Low dimensions Introduction Mermin-Wagner theorem B ogoliub ov inequality Application to the Heisenberg model Dipolar interactions in low dimensions Dipole-exchange cross-over One dimensional instabilities Antiferromagnetic chain RPA for the AFM chain Exchange dominated regime Dipolar dominated regime Dipolar interaction in layers Monolayer Bilayer Surface magnetism Introduction MFA treatment of surfaces Surface excitations LRPA method Wave functions for bulk and surface 343

6 CONTENTS xi 10.6 Surface density of magnon states Surface phase-transitions Dipolar surface effects Surface magnetism in metals Two-magnon eigenstates Introduction Green's function formalism One dimension Two dimension Summary of results Anisotropy effects Other interactions Introduction Two-magnon interaction Three-magnon processes Magnon-phonon interaction Bilinear magnon-phonon interaction 395 A Group theory 401 A.1 Definition of group 401 A.2 Group representations 403 A.2.1 Reducibility 403 A.3 Orthogonality relations 405 A.4 Projection operators 406 A.5 Coordinate transformations 406 A.6 Wigner-Eckart theorem 407 A.7 Space groups 409 A.8 Bloch's theorem 411 B Appendix B Time Reversal 413 B.1 Antilinear operators 413 B.2 Anti-unitary operators 415 B.3 Time reversal 416 Index 419