MSE 7025 Magnetic Materials (and Spintronics) Lecture 3: Magnetization, from classical to quantum Chi-Feng Pai [email protected] Course Outline • Time Table Week Date Lecture 1 Feb 24 Introduction 2 March 2 Magnetic units and basic E&M 3 March 9 Magnetization: From classical to quantum 4 March 16 No class (APS March Meeting, Baltimore) 5 March 23 Category of magnetism 6 March 30 From atom to atoms: Interactions I (oxides) 7 April 6 From atom to atoms: Interactions II (metals) 8 April 13 Magnetic anisotropy 9 April 20 Mid-term exam 10 April 27 Domain and domain walls Course Outline • Time Table Week Date Lecture 11 May 4 Magnetization process (SW or Kondorsky) 12 May 11 Characterization: VSM, MOKE 13 May 18 Characterization: FMR 14 May 25 Transport measurements in materials I: Hall effect 15 June 1 Transport measurements in materials II: MR 16 June 8 MRAM: TMR and spin transfer torque 17 June 15 Guest lecture (TBA) 18 June 22 Final exam Units of B-field and H-field • “Magnetic flux density” and “magnetic field strength” Notation Unit SI or cgs? H (B) kG = 1000 G = 1000 gauss cgs (symbol misused) H oersted (Oe) cgs H ampere/meter (A/m) SI B gauss (G) cgs B tesla (T) SI μ0H tesla (T) SI Magnetization “M” and magnetic susceptibility “χ” • Magnetization (per unit volume) Magnetization “M” and magnetic susceptibility “χ” • Magnetic susceptibility Magnetization “M” and magnetic susceptibility “χ” • Magnetic susceptibility Ferromagnetism 0 0 Magnetization “M” and magnetic susceptibility “χ” • Magnetic susceptibility (T-dependence) Diamagnetism M vs H in reality • Magnetization curves – Trends depends on object shape and field direction – Existence of “easy axis” and “hard axis” Different “susceptibilities” along different axis even for the same material! (O’Handley) Demagnetizing field Hd • Microscopic view – Composite of magnetic dipole moments – Consider a magnetized sample as bellow Hd (O’Handley) Demagnetizing field Hd • Internal field – Must consider the effect of demagnetization field – Demagnetization factor N depends on the object shape (O’Handley) Demagnetizing field Hd • Internal field – Must consider the effect of demagnetization field – Demagnetization factor N depends on the object shape • Magnetic susceptibility (experimental) (O’Handley) Magnetic dipole moment • Similar to the concept of electrical dipole moment, but... • Magnetism comes from moving charges Magnetic dipole moment • A heuristic approach – Ampere’s current loop concept (Unit) ~ Magnetic dipole moment • Bohr-van Leeuwen theorem – Non-existence of magnetization under classical picture – Statistical mechanics yields that the total magnetization must be zero in a classical system! – Even when you apply an external magnetic field, and the electrons form circular orbits, the net moment should still be zero… Magnetic (dipole) moment • Magnetic moment unit conversion • Magnetization unit conversion emu/cm3 = 103 A/m /Volume Torque, energy, and force • Dipole moment in a magnetic field (Actually, this “re-alignment” picture is not perfectly correct) • Magnetic potential energy (a.k.a. Zeeman energy) Torque, energy, and force • Force experienced by the moment • Since the moment is typically independent of position • Dimension analysis Microscopic origin of magnetization • From Ampere to Einstein (1915): Angular momentum – Ampere’s molecular current – Einstein & de Haas (magnetic moment) (angular momentum of an orbiting electron) Microscopic origin of magnetization • From Ampere to Einstein (1915): Angular momentum – Einstein & de Haas (magnetic moment ~ angular momentum) (This is actually off by a factor of roughly 2, the Lande g-factor of electron) (present day expression) – Gyromagnetic ratio Microscopic origin of magnetization • From Ampere to Einstein (1915): Angular momentum – Einstein-de Haas Experiment (From Wikipedia) (Einstein & de Haas, 1915) Microscopic origin of magnetization • From Ampere to Einstein (1915): Angular momentum – Einstein-de Haas Experiment / Barnett Effect (rotation M) Microscopic origin of magnetization • From Ampere to Einstein (1915): Angular momentum – Interesting remarks on the original paper • Connection to Bohr’s atomic model (but didn’t directly cite it). V. Ya. Frenkel’ (1979) Microscopic origin of magnetization • From Ampere to Einstein (1915): Angular momentum – Interesting remarks on the original paper • Connection to zero-point-energy (quantum concept) V. Ya. Frenkel’ (1979) Microscopic origin of magnetization • From Ampere to Einstein (1915): Angular momentum – Interesting remarks on the original paper • The reported value was actually a factor of 2 off, but Einstein and de Haas claimed that the number is in good agreement with theoretical prediction. V. Ya. Frenkel’ (1979) Microscopic origin of magnetization • From Ampere to Einstein (1915): Angular momentum – Interesting remarks on the original paper • The reported value was actually a factor of 2 off, but Einstein and de Haas claimed that the number is in good agreement with theoretical prediction. V. Ya. Frenkel’ (1979) Microscopic origin of magnetization • From Ampere to Einstein (1915): Angular momentum – Interesting remarks on the original paper “How Experiments End” by P. L. Galison (1987) Microscopic origin of magnetization • From Ampere to Einstein (1915): Angular momentum – Interesting remarks on the original paper “How Experiments End” by P. L. Galison (1987) Microscopic origin of magnetization • From Ampere to Einstein (1915): Angular momentum – The gyromagnetic anomaly was resolved by quantum mechanics V. Ya. Frenkel’ (1979) Magnetization as angular momentum: Precession • Torque is the time variation rate of angular momentum (From wiki) One page quantum mechanics • Plank constant • Uncertainty principle • Schrodinger equation (wave mechanics) • Heisenberg’s formulation (matrix mechanics) One page quantum mechanics • Bohr model • Hydrogen atom, modern representation – The principle quantum number (n=1,2,3…) – The orbital quantum number (l=0,1,2,3,…n-1) – The magnetic quantum number (ml=-l,-l+1,…,0,…l-1,l) – The spin quantum number (ms=-s,-s+1,..,0,…,s-1,s) Electron Spin • The concept of “spinning electron” was proposed by Goudsmit and Uhlenbeck in 1925, to explain the anomalous Zeeman effect. (Note that this is a purely quantum concept) – The electron should possess angular momentum of – The spin quantum number is – Consider a quantum state (modern representation) of a electron Electron Spin • Spin angular momentum – Generally speaking – For electron spin • Dirac’s theory (Dirac equation) • Quantum electrodynamics – Gyromagnetic ratio Electron Spin • Spin angular momentum – The expectation value of magnetic moment (Bohr magneton) – Typical ferromagnetic elements • Fe • Co • Ni Electron Spin • Stern-Gerlach experiment (1922) – 2-3 years before Goudsmit and Uhlenbeck’s theory – Originally not designed for verifying “spin” “Right Experiment, Wrong Theory: The Stern-Gerlach Experiment” Additional reading: http://plato.stanford.edu/entries/physics-experiment/app5.html Electron Spin • Stern-Gerlach experiment (1922) – 2-3 years before Goudsmit and Uhlenbeck’s theory – Still got Nobel prize anyway… Between angular and spin – Spin-Orbit Interaction • Spin-orbit Interaction – Interaction between L and S – The orbiting charge (nucleus) creates a magnetic field B acting upon the moment of electron spin Between angular and spin – Spin-Orbit Interaction • Spin-orbit Interaction – Interaction between L and S – The orbiting charge (nucleus) creates a magnetic field B acting upon the moment of electron spin Between angular and spin – Spin-Orbit Interaction • Spin-orbit Interaction – Interaction between L and S – The orbiting charge (nucleus) creates a magnetic field B acting upon the moment of electron spin – The energy (Hamiltonian) of spin-orbit interaction Between angular and spin – Spin-Orbit Interaction • Spin-orbit Interaction – Interaction between L and S – The orbiting charge (nucleus) creates a magnetic field B acting upon the moment of electron spin – The energy (Hamiltonian) of spin-orbit interaction Between angular and spin – Spin-Orbit Interaction • Spin-orbit Interaction – Interaction between L and S – The orbiting charge (nucleus) creates a magnetic field B acting upon the moment of electron spin – The energy (Hamiltonian) of spin-orbit interaction (factor of 2 relativistic correction, Thomas precession) Between angular and spin – Spin-Orbit Interaction • Spin-orbit Interaction – The expectation value of this Hamiltonian Energy – Proportional to atomic number to the fourth (Z4) – Consider a n=2, l=1 state of the hydrogen atom, what’s the order of magnitude of this spin-orbit interaction energy ESO? What’s the magnitude of the magnetic field B acting on the spin moment S? (potential energy) Total angular momentum = orbital + spin • J = L + S – L = orbital angular momentum – S = spin angular momentum – Coupled due to SOI – J = L + S has a simpler behavior L (for l=2) S (for s=1/2) Total angular momentum = orbital + spin • J = L + S L (for l=2) S (for s=1/2) • SOI energy Total angular momentum = orbital + spin • Total magnetic moment (μ and J are not parallel!) • Zeeman energy Lande g-factor LS coupling and jj coupling • LS coupling – In light atoms (Z<30) – SOI is weak – For weak magnetic fields (Zeeman effect) • jj coupling – In heavier atoms – SOI is strong Hund’s Rules • So, how do we determine the ground state? – For a given atom with multiple electrons, the total orbital angular momentum L and spin angular momentum S can have (2l+1)(2s+1) combinations. (j) Blundell, Magnetism in Condensed Matter (2001) Hund’s Rules Coulomb interaction Spin-orbit interaction Note: Apply strictly to atoms, loosely to localized orbitals in solids, not at all to free electrons Blundell, Magnetism in Condensed Matter (2001) Hund’s Rules • So, how do we determine the ground state? Blundell, Magnetism in Condensed Matter (2001) Hund’s Rules • Why Hund’s rules are important? Blundell, Magnetism in Condensed Matter (2001) Hund’s Rules • Why Hund’s rules are important? Hund’s Rules • Why Hund’s rules are important? Blundell, Magnetism in Condensed Matter (2001) .