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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
Para-/Diamagnetism of Metal | (Pauli Paramagnetism [More Details]: 3 points + diagram)
* Electron spins respond to ***B***ext (see diagram) * In thermal equilibrium, chemical potential must be equal → uncompensated spins flip * Pauli spin susceptibility *χ*P = const.
26
Para-/Diamagnetism of Metal | (Landau Diamagnetism [Overview]: 2 points)
* To get orbital contribution to magnetization, look at *M* ∝ ∂*F*/∂*B* * _Recall_: Energy of electron depends on ***B***ext (i.e. energy of Landau levels)
27
Para-/Diamagnetism of Metal | (Landau Diamagnetism [Take-Away])
* Landau diamagnetic susceptibility * *χ*L = −1/3*χ*P(m/m* )2
28
Para-/Diamagnetism of Metal | (Total Susceptibility: 2 points)
*χ* = *χ*L + *χ*P ## Footnote * *χ*L and *χ*P same order of magnitude → metals can be para- or diamagnetic
29
Cooperative Magnetism | (Overview: 4 points)
* 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
Cooperative Magnetism | (Dipole-Dipole Interaction: 2 points)
* max *E*dd (≈ 0.1 meV) \<\< *kT* (≈ 25 meV) * Dipole-dipole interaction too weak to explain ordering behavior at room temp
31
Exchange Interaction btwn Localized Electrons | (Exchange Constant: 8 points)
* Energy different between symmetric state (aligned spins) *E*s and anti-symetric state (opposite spins) *E*a gives _exchange constant_ *J*A = *E*s - *E*a * Sign determines type of ordering * *J*A \> 0 → ferromagnetic * *J*A \< 0 → anti-ferromagnetic * Can also analyze via potential *V*(***r***1, ***r***2) = *V*ion(***r***1) + *V*ion(***r***2) + *V*ee(***r***1, ***r***2) * *V*ee(***r***1, ***r***2) \> 0 → favors ferromagnetic behavior * *V*ion(***r***1, ***r***2) \< 0 → facors anti-ferromagnetic behavior * Material behavior depends on which dominates
32
Exchange Interactions btwn Localized Electrons | (Hubbard Model: 4 points)
* Simple model to describe hopping exchange of electrons * Hamiltonian consists of two terms * _Hopping_: hopping intergral *t* * _Repulasion_: potential energy *U*
33
Exchange Interactions btwn Localized Electrons | (Types of Exchange: 7 points + diagram)
* _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
Exchange Interaction of Free Electron Gas | (Take-Away)
There exists an exchange hold around free electrons that casues local density to drop in vicinity of free electron
35
Magnetic Order | (Types: 3 points)
* Ferromagnetism * Ferrimagnetism * Anti-ferromagnetism
36
Magnetic Order | (Stoner Model [Overview]: 5 points + diagram)
* 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
Magnetic Order | (Stoner Model [Take-Away]: 3 points)
* Requiring ∆*E* \< 0 → _Stoner Criterion_ * _​_When met, spin redistribution is energetically favorable * Requires high correlation energy *U* and density of states at *E*F
38
Magnetic Order | (Ferromagnetism [Overview]: 5 points)
* *T* \< *T*C → ferromagnetic * *T* \> *T*C → paragmangetic (thermal fluctuations disturb order) * Second-order phase transition * Magnetization *M* discontinuous * Mangetic susceptibility *χ* continuous
39
Magnetic Order | (Ferromagnetism [Mean-FIeld Theory]: 6 points)
* Reduce magnetic moment interaction many-body problem to one-body problem by using effective field ***B***A to account for exchange interaction * Mean-field constant *γ* * Virtual field * Causes spatial order * Measuring *C*, *T*C allows measurement of *γ* = *T*C/*C* * *​*Allows measurement of ***B***A \>\> lab generated fields
40
Magnetic Order | (Ferromagnetism [Paramagnetic Regime]: 4 points)
* Susceptibility expectation vs reality (see below) * _Curie-Weiss Law_ * *Θ* \> *T*C * *Θ* is measure of echange between paramagnetic momens not accounted for in mean-field theory
41
Magnetic Order | (Ferromagnetism [Susceptibility]: diagram)
42
Magnetic Order | (Ferrimagnetism [Overview]: 2 points)
* Anti-parallel spins, but not all spins are compensated → spontaneous magnetization * Exchange constant *J*AB domiantes → anti-parallel only between *A*, *B* sites
43
Magnetic Order | (Ferrimagnetism [Susceptibility]: 5 points)
* Use near-field approximation * Two Curie temperatures *C*A*, C*B and mean-field constants *γ*AA = *γ*BB = 0 * Critical temperature *T*C = |*γ*AB|(*C*A*C*B)1/2 * Susceptibility *χ* = *µ*o*∂*(***M***A + ***M***B)/*∂**B***ext
44
Magnetic Order | (Ferrimagnetism[Susceptibility]: graph)
45
Magnetic Order | (Anti-Ferromagnetism [Motivation]: 3 points)
* 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* \< *T*N
46
Magnetic Order | (Anti-Ferromagnetism [Neel Temperature]: 5 points)
* Use same near-field appraoch as ferrimagnetism with: * *C*A = *CB* = *C* * *γ*AA = *γ*BB \< 0 * ***M***A = -***M***B * ​_Neel ordering temperature_ given by
47
Magnetic Order | (Anti-Ferromagnetism [Paramagnetic Regime]: 2 points)
* *T* \> *T*N → susceptibility (see below) * Curie-Weiss Temperature *Θ* = −|*γ*AB + *γ*AA|
48
Magnetic Order | (Anti-ferromagnetism [AFM Regime]: 5 points + diagram)
* Consider ***B*** perpendicular and parallel to spins * _Perpendicular_: *χ*⊥ = |*γ*AB|-1 * _Parallel_: * *χ*||(*T*=0) = 0 * Increasies toward *χ*||(*T*=*T*N) = *χ*⊥
49
Magnetic Order | (Susceptibility [Summary]: diagram)
50
Magnetic Anisotropy | (Overview)
Experiments show preferred direction of magnetization along *_easy axis_* and rejects mangetization along *_hard axis_*
51
Magnetic Anisotropy | (Anisotropy Energy: 8 points)
* 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
Magnetic Domains | (Exp vs Reality: 3 points)
* _Expectation_: * For *T* \<\< *T*C → *M* = *M*S * _Reality_: * *M* \<\< *M*S because of domains * *Mdomain *= *M*S
53
Magnetic Domains | (Imaging: 2 points)
* _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
Magnetic Domains | (Stray Fields: 4 points + diagram)
* 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
Magnetic Domains | (Wall Types: 4 points + diagram)
* _Bloch Wall_: * Magnetization rotates out of the plane of the domain wall * _Neel Wall_: * Magnetization rotates in the plane of the domain wall
56
Magnetization Curve | (3 points + diagram)
* 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
Magnetization Dynamics | (Goal)
Determine how mangetization responds to external field
58
Magnetization Dynamics | (Assumptions: 2 points)
* Rigid spin coupling → homogeneous mode *q* = 0 (*λ* = ∞) * _Equilibrium_: * Magnetization points along *B*eff
59
Magnetization Dynamics | (Out of Equilibrium: 3 points + 2 equations)
* Push magnetization out of equilibrium → system now feels torque ***τ*** = ***vM*** × ***B***eff → precession of ***M*** around ***B***eff * Can relate angular momentum ***L*** to magnetic moment and torque (see below) * gyromagnetic ratio *γ* = *gµ*B/(hbar)
60
Magnetization Dynamics | (Take-Away)
Landau-Lipschitz equation
61
Ferromagnetic Resonance | (Overview: 2 points + diagram)
* If ***B***ext oscillates at resonance frequency *ω*o = *γB*eff → leads to absorption * No damping, because _resonance condition_ met ***M*** x *dt**M*** = -*MB*1cos*θ*
62
Ferromagnetic Resonance | (Spectroscopy: 2 points + plot)
* Measure miscrowave absorption of thin film as function of DC field (see below) * Can measure gyromagnetic ratio, anisotropy field, damping constant, ...
63
Spin Waves | (Overview: 3 points)
* _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 ***B***A becomes relevant
64
Spin Waves | (Magnon)
Spin waves are quantized quasi-particles
65
Spin Waves | (Different Modes: 2 points)
* _Exchange Mode_: small *λ* → ***B***A domaintes ***B***eff * _Dipolar Mode_: large *λ* → ***B***Ani dominates at some point
66
Spin Waves | (Exchange Modes [Dispersion]: 5 points + 2 equations)
* _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
68
Spin Waves | (Exchange Mode [DIspersion]: graph)
69
How to measure magnetic susceptibility *χ* | (2 points + 2 diagrams)
* Faraday's (left) Guoy's Scale (right) * SQUID

Advanced Experimental Physics II (3 decks)

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