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Advanced Experimental Physics II > Chapter 3: Magnetic Properties > Flashcards

Chapter 3: Magnetic Properties Flashcards

1
Q

Magnetic Properties

(Overview: 5 points)

A
  • Consider interation between solid and magnetic field
    • Linear response regime
  • Distinguish between quasi-bound and free electrons
    • Quasi-Bound: magnetic properties are that of lattice atom
    • Free: magnetic properties described by Fermi statistics
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2
Q

Macroscopic Quantities

(Assumptions)

A
  • Consider insulting solid (i.e. no shielding currents) exposed to magnetic field
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3
Q

Macroscopic Quantities

(Magnetization and Magnetic Susceptibility: 2 points)

A
  • External field leads to magnetization M
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4
Q

Macroscopic Quanities

(Magnetic Moment: 2 points)

A
  • Classically given as (see below)
    • When current due to single electron, I = -e/T with T = 2πr /v
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5
Q

Macroscopic Quantities

(Magnetic Moment [Bohr Magneton]: 2 points)

A
  • Consider Bohn quantization of orbital momentum L = movr ≡ (\hbar) → Bohr magneton
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6
Q

Macroscopic Quantities

(Magnetic Moment [For Solid])

A
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7
Q

Macroscopic Quantities

(Magnetic Permeability)

A
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8
Q

Macroscopic Quantities

(Local Magnetic Field Hloc: 4 points)

A
  • Similar as for local electric field
    • Demagnetization field HN = -NM
    • Lorentz field HL = M /3
      • HL very small in para-/diamagnetic material because χ << 1
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9
Q

Macroscopic Quantities

(Demagnetization and Stray Field [Consider]: 2 points)

A
  • Thin disk of ferromagnetic material
  • Homogenous mangetization along normal of disk N = 1
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10
Q

Macroscopic Quantities

(Demagnetization and Stray Field [Take-Away]: 4 points and diagram)

A
  • M induces B
  • Inside disk: demagnetization field HN = B /µo - M
  • Outside disk: stray field Hs = B /µo
  • Amperes law only holds if Hs and HN in opposite directions
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11
Q

Macroscopic Quantity

(Magnetostatic Self-Energy: 4 points)

A
  • Magnetostatic self-energy Em arises between each atomic magnetic moment interacts with the magnetic field create by all other moments in solid
  • Energy of one magnetic moment µ in Hloc is Eµ = -µoµHloc
  • Integrate over entire volume to get total self-energy (see below)
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12
Q

Microscopic Theory of Magnetic Properties

(Diamagnetic Solids: 2 points)

A
  • Hext = 0 → no magnetic moments
  • Hext > 0 → finite magnetization from induced magnetic moment opposite to applied field M = χdiH for χdi < 0
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13
Q

Microscopic Theory of Magnetic Properties

(Diamagnetic Solids [Types]: 6 points)

A
  • Two types
    • Larmor Diamagnetism:
      • atomic diamagnetism in insulator
      • magnetic moments due to atoms or tightly bound electrons
    • Landau Diamagnetism:
      • Magnetization of free electrons in metals
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14
Q

Microscopic Theory of Magnetic Properties

(Paramagnetic Solids: 3 points)

A
  • Magnetic moments even for Hext = 0
    • Due to orbital electron motion µL or spin of crystal electrons µS
  • ​Hext > 0 → magnetic moments align with external field M = χpaH with χpa > 0
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15
Q

Microscopic Theory of Magnetic Properties

(Russel-Saunders Coupling: 2 points)

A
  • Orbital momenta couple to total orbital momenum L = ∑ili and spin couple to total spin S = ∑isi ​
    • Total angular momentum given by Russel-Saunders Coupling J = L + S
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16
Q

Microscopic Theory of Magnetic Properties

(Paramagnetic Solids [Types]: 4 points)

A
  • Langevin Paramagnetism:
    • due to atomic paramagnetism in insulators
  • Pauli Paramagnetism:
    • due to conducting electrons in metals
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17
Q

Microscopic Theory of Magnetic Properties

(Ferromagnetic Materials: 5 points)

A
  • For T < Tc → spontaneous magnetization without external field
    • Results from exchange interaction causing spatial ordering of permanent magnetic moments
    • Exchange interactions purely quantum mechanical:
      • Pauli Principle
      • Coulomb interaction
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18
Q

Microscopic Theory of Magnetic Properties

(Langevin Paramagnetism: 4 points)

A
  • Classical description (good for large J)
  • Look at thermodynamic value of magnetization M = nV<µz>
  • For high Bext → <µz> = µ
    • ​Saturation magnetization Ms = nVµ
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19
Q

Microscopic Theory of Magnetic Properties

(Langevin Paramagnetism [Curie Law])

A
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20
Q

Microscopic Theory of Magnetic Properties

(Langevin Paramagnetism: diagram)

A
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21
Q

Microscopic Theory of Magnetic Properties

(Langevin Paramagnetism [QM Curie Law]: 2 points)

A
  • two-level system
  • similar description as classical Cure Law (up to factor 1/3)
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22
Q

Para-/Diamagnetism of Metal

(Overview: 3 points)

A
  • Recall: Effect of B-field on free electrons → Landau Diamagnetism
  • Add contribution from electron spin → Pauli Paramagnetism
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23
Q

Para-/Diamagnetism of Metal

(Pauli Paramagnetism [Overview]: 2 points)

A
  • Electron spin magnetic µs moment can take two values µs = ±µB
  • Magnetization M = (n+ − n-)µB is function of spin-up/-down densities
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24
Q

Para-/Diamagnetism of Metal

(Pauli Paramagnetism [Curie Law]: 6 points)

A
  • Expectation:
    • Curie-Law M = C /T
  • Reality:
    • ​Curie Constant is no longer constant C = C(T)
  • Explanation:
    • Only electrons near EF can respond to field, and this population increases with T
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25
Q

Para-/Diamagnetism of Metal

(Pauli Paramagnetism [More Details]: 3 points + diagram)

A
  • Electron spins respond to Bext (see diagram)
  • In thermal equilibrium, chemical potential must be equal → uncompensated spins flip
  • Pauli spin susceptibility χP = const.
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26
Q

Para-/Diamagnetism of Metal

(Landau Diamagnetism [Overview]: 2 points)

A
  • To get orbital contribution to magnetization, look at M ∝ ∂F/∂B
  • Recall: Energy of electron depends on Bext (i.e. energy of Landau levels)
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27
Q

Para-/Diamagnetism of Metal

(Landau Diamagnetism [Take-Away])

A
  • Landau diamagnetic susceptibility
    • χL = −1/3χP(m/m* )2
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28
Q

Para-/Diamagnetism of Metal

(Total Susceptibility: 2 points)

A

χ = χL + χP

  • χL and χP same order of magnitude → metals can be para- or diamagnetic
29
Q

Cooperative Magnetism

(Overview: 4 points)

A
  • Some materials have spontaneous polarization below characteristic frequency
  • Caused by cooperative magnetism
    • ​Finite interactions betewen atomic magnetic moments cause alignment
  • Per usual, must distinguish between bound and free electrons
30
Q

Cooperative Magnetism

(Dipole-Dipole Interaction: 2 points)

A
  • max Edd (≈ 0.1 meV) << kT (≈ 25 meV)
    • Dipole-dipole interaction too weak to explain ordering behavior at room temp
31
Q

Exchange Interaction btwn Localized Electrons

(Exchange Constant: 8 points)

A
  • Energy different between symmetric state (aligned spins) Es and anti-symetric state (opposite spins) Ea gives exchange constant JA = Es - Ea
  • Sign determines type of ordering
    • JA > 0 → ferromagnetic
    • JA < 0 → anti-ferromagnetic
  • Can also analyze via potential V(r1, r2) = Vion(r1) + Vion(r2) + Vee(r1, r2)
    • Vee(r1, r2) > 0 → favors ferromagnetic behavior
    • Vion(r1, r2) < 0 → facors anti-ferromagnetic behavior
    • Material behavior depends on which dominates
32
Q

Exchange Interactions btwn Localized Electrons

(Hubbard Model: 4 points)

A
  • Simple model to describe hopping exchange of electrons
  • Hamiltonian consists of two terms
    • Hopping: hopping intergral t
    • Repulasion: potential energy U
33
Q

Exchange Interactions btwn Localized Electrons

(Types of Exchange: 7 points + diagram)

A
  • Direct Exchange:
    • Overlapping orbits (e.g. covalent bonds)
  • Super Exchange:
    • Indirect change via diamagnetic atom (e.g. via O2+ in MnO)
    • Considered virtual hopping
  • Double Exchange:
    • Combined hopping of electrons
34
Q

Exchange Interaction of Free Electron Gas

(Take-Away)

A

There exists an exchange hold around free electrons that casues local density to drop in vicinity of free electron

35
Q

Magnetic Order

(Types: 3 points)

A
  • Ferromagnetism
  • Ferrimagnetism
  • Anti-ferromagnetism
36
Q

Magnetic Order

(Stoner Model [Overview]: 5 points + diagram)

A
  • Simplest model for understanding ferromagnetism in metals
  • Exchange interaction facors parallel spin alignment
    • Some electrons spontaneous redistribute from spin-down to spin-up
    • To be spontaneous, redistribution must be energetically favorable
      • i.e. decrease in potential energy must overcompensate increase in kinetic energy
37
Q

Magnetic Order

(Stoner Model [Take-Away]: 3 points)

A
  • Requiring ∆E < 0 → Stoner Criterion
    • _​_When met, spin redistribution is energetically favorable
    • Requires high correlation energy U and density of states at EF
38
Q

Magnetic Order

(Ferromagnetism [Overview]: 5 points)

A
  • T < TC → ferromagnetic
  • T > TC → paragmangetic (thermal fluctuations disturb order)
  • Second-order phase transition
    • Magnetization M discontinuous
    • Mangetic susceptibility χ continuous
39
Q

Magnetic Order

(Ferromagnetism [Mean-FIeld Theory]: 6 points)

A
  • Reduce magnetic moment interaction many-body problem to one-body problem by using effective field BA to account for exchange interaction
    • Mean-field constant γ
    • Virtual field
    • Causes spatial order
    • Measuring C, TC allows measurement of γ = TC/C
      • ​Allows measurement of BA >> lab generated fields
40
Q

Magnetic Order

(Ferromagnetism [Paramagnetic Regime]: 4 points)

A
  • Susceptibility expectation vs reality (see below)
    • Curie-Weiss Law
    • Θ > TC
    • Θ is measure of echange between paramagnetic momens not accounted for in mean-field theory
41
Q

Magnetic Order

(Ferromagnetism [Susceptibility]: diagram)

A
42
Q

Magnetic Order

(Ferrimagnetism [Overview]: 2 points)

A
  • Anti-parallel spins, but not all spins are compensated → spontaneous magnetization
  • Exchange constant JAB domiantes → anti-parallel only between A, B sites
43
Q

Magnetic Order

(Ferrimagnetism [Susceptibility]: 5 points)

A
  • Use near-field approximation
    • Two Curie temperatures CA, CB<em> </em>and mean-field constants γAA = γBB = 0
    • Critical temperature TC = |γAB|(CACB)1/2
  • Susceptibility χ = µo∂(MA + MB)/∂Bext
44
Q

Magnetic Order

(Ferrimagnetism[Susceptibility]: graph)

A
45
Q

Magnetic Order

(Anti-Ferromagnetism [Motivation]: 3 points)

A
  • MnO exhibits new Bragg peaks at low temperature in neutron scattering experiments → suggests double for unit cell
  • X-ray scattering shows no new peaks → structure stays the same
  • Suggests anti-ferromagnetic order for T < TN
46
Q

Magnetic Order

(Anti-Ferromagnetism [Neel Temperature]: 5 points)

A
  • Use same near-field appraoch as ferrimagnetism with:
    • CA = C<span>B</span> = C
    • γAA = γBB < 0
    • MA = -MB
  • ​Neel ordering temperature given by
47
Q

Magnetic Order

(Anti-Ferromagnetism [Paramagnetic Regime]: 2 points)

A
  • T > TN → susceptibility (see below)
  • Curie-Weiss Temperature Θ = −|γAB + γAA|
48
Q

Magnetic Order

(Anti-ferromagnetism [AFM Regime]: 5 points + diagram)

A
  • Consider B perpendicular and parallel to spins
    • Perpendicular: χ⊥ = |γAB|-1
    • Parallel:
      • χ||(T=0) = 0
      • Increasies toward χ||(T=TN) = χ⊥
49
Q

Magnetic Order

(Susceptibility [Summary]: diagram)

A
50
Q

Magnetic Anisotropy

(Overview)

A

Experiments show preferred direction of magnetization along easy axis and rejects mangetization along hard axis

51
Q

Magnetic Anisotropy

(Anisotropy Energy: 8 points)

A
  • Energy required to turn from easy- to hard-axis (see below)
    • Magneto-crystalline:
      • Due to spin-orbital coupling
        • Causes tilted spins
    • Shape:
      • From demanetization tensor N being highly dependent on shape
    • Induced:
      • Elastic tension and exchange anisotropy at interfaces
52
Q

Magnetic Domains

(Exp vs Reality: 3 points)

A
  • Expectation:
    • For T << TC → M = MS
  • Reality:
    • M << MS because of domains
      • M<span>domain </span>= MS
53
Q

Magnetic Domains

(Imaging: 2 points)

A
  • Magnetic Force Microscopy:
    • Sharp magnetic tip scans magnetic material and responds to magnetic structure of sample
  • X-Ray Dichroism:
    • Compare x-ray absorption spectrum to left- and rightcircularly polarized light
54
Q

Magnetic Domains

(Stray Fields: 4 points + diagram)

A
  • Magnetic domains reduce stray fields (see below)
    • Edge domains minimize stray fields (right-most below)
    • Magnetic field energy decreases
    • Anisotropy and domain wall energy increases
55
Q

Magnetic Domains

(Wall Types: 4 points + diagram)

A
  • Bloch Wall:
    • Magnetization rotates out of the plane of the domain wall
  • Neel Wall:
    • Magnetization rotates in the plane of the domain wall
56
Q

Magnetization Curve

(3 points + diagram)

A
  • Energy density ∝ area of hysteresis loop
  • Beginning of dashed initial-line → reversible domain wall movement
  • End of dashed initial-line → irreversible domain wall movement
57
Q

Magnetization Dynamics

(Goal)

A

Determine how mangetization responds to external field

58
Q

Magnetization Dynamics

(Assumptions: 2 points)

A
  • Rigid spin coupling → homogeneous mode q = 0 (λ = ∞)
  • Equilibrium:
    • Magnetization points along Beff
59
Q

Magnetization Dynamics

(Out of Equilibrium: 3 points + 2 equations)

A
  • Push magnetization out of equilibrium → system now feels torque τ = vM × Beff → precession of M around Beff
  • Can relate angular momentum L to magnetic moment and torque (see below)
    • gyromagnetic ratio γ = gµB/(hbar)
60
Q

Magnetization Dynamics

(Take-Away)

A

Landau-Lipschitz equation

61
Q

Ferromagnetic Resonance

(Overview: 2 points + diagram)

A
  • If Bext oscillates at resonance frequency ωo = γBeff → leads to absorption
    • No damping, because resonance condition met M x dtM = -MB1cosθ
62
Q

Ferromagnetic Resonance

(Spectroscopy: 2 points + plot)

A
  • Measure miscrowave absorption of thin film as function of DC field (see below)
    • Can measure gyromagnetic ratio, anisotropy field, damping constant, …
63
Q

Spin Waves

(Overview: 3 points)

A
  • Recall: So far, considered spin-flip to be minimum excitation (q = 0)
  • Now, consider colletive motion of spins with q > 0 → spin-waves emerge
    • Angle between spins no longer zero → echange field BA becomes relevant
64
Q

Spin Waves

(Magnon)

A

Spin waves are quantized quasi-particles

65
Q

Spin Waves

(Different Modes: 2 points)

A
  • Exchange Mode: small λ → BA domaintes Beff
  • Dipolar Mode: large λ → BAni dominates at some point
66
Q

Spin Waves

(Exchange Modes [Dispersion]: 5 points + 2 equations)

A
  • Case: B = 0 and qa << 1
  • Depends on (anti-)ferromagnetism (see below)
  • Measure with neutron scattering spectroscopy or Raman spectroscopy
  • Decay by Stoner excitations
    • Single electron excitations
67
Q
A
68
Q

Spin Waves

(Exchange Mode [DIspersion]: graph)

A
69
Q

How to measure magnetic susceptibility χ

(2 points + 2 diagrams)

A
  • Faraday’s (left) Guoy’s Scale (right)
  • SQUID

Advanced Experimental Physics II (3 decks)

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