MSE 7025 Magnetic Materials (and Spintronics)cfpai/MSE7025/MSE7025_Lecture3...• The concept of “spinning electron” was proposed by Goudsmit and Uhlenbeck in 1925, to explain

MSE 7025 Magnetic Materials

(and Spintronics)

Chi-Feng Pai [email protected]

Lecture 3: Magnetization, from classical to quantum

mailto:[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)


• Magnetic susceptibility


• Magnetic susceptibility

0

0

Ferromagnetism


• 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”

(O’Handley)

Different “susceptibilities” along different axis even for the same material!

Demagnetizing field Hd

• Microscopic view – Composite of magnetic dipole moments

– Consider a magnetized sample as bellow

(O’Handley)

Hd


• Internal field – Must consider the effect of demagnetization field

– Demagnetization factor N depends on the object shape

(O’Handley)


• 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


• A heuristic approach – Ampere’s current loop concept

~ (Unit)


• 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

• Magnetic potential energy

(Actually, this “re-alignment” picture is not perfectly correct)

(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)



– Einstein & de Haas (magnetic moment ~ angular momentum)

– Gyromagnetic ratio

(This is actually off by a factor of roughly 2, the Lande g-factor of electron)

(present day expression)



– Einstein-de Haas Experiment

(From Wikipedia) (Einstein & de Haas, 1915)



– Einstein-de Haas Experiment / Barnett Effect (rotation M)



– Interesting remarks on the original paper

• Connection to Bohr’s atomic model (but didn’t directly cite it).

V. Ya. Frenkel’ (1979)




• Connection to zero-point-energy (quantum concept)





• 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.





• 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.





“How Experiments End” by P. L. Galison (1987)




“How Experiments End” by P. L. Galison (1987)



– The gyromagnetic anomaly was resolved by quantum mechanics


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

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









– The energy (Hamiltonian) of spin-orbit interaction











(factor of 2 relativistic correction, Thomas precession)



– 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)


• J = L + S

• SOI energy

L (for l=2) S (for s=1/2)


• Total magnetic moment

• Zeeman energy

(μ and J are not parallel!)

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.

Blundell, Magnetism in Condensed Matter (2001)

(j)

Hund’s Rules


Note: Apply strictly to atoms, loosely to localized orbitals in solids, not at all to free electrons

Coulomb interaction

Spin-orbit interaction

Hund’s Rules

• So, how do we determine the ground state?


Hund’s Rules

• Why Hund’s rules are important?


Hund’s Rules


Hund’s Rules



Documents

MSE 7025 Magnetic Materials (and Spintronics)cfpai/MSE7025/MSE7025_Lecture3...• The concept of “spinning electron” was proposed by Goudsmit and Uhlenbeck in 1925, to explain