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)