SLIDE 1
MTLE-6120: Advanced Electronic Properties of Materials Magnetic properties of materials
Contents: ◮ Diamagnetism ◮ Paramagnetism type B (dipole rotation) ◮ Paramagnetism type A (metals) ◮ Ordering: ferro-, antiferro- and ferri-magnetism ◮ Domain structure and hysteresis Reading: ◮ Kasap: 8.1 - 8.8
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SLIDE 2 Materials in magnetic fields
◮ Charges circulate around magnetic field due to force q v × B ◮ Magnetic dipole moment of infinitesimal current loop (per unit δz)
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dlδI = 1 2 (0 + δxδyˆ zδI + δyδxˆ zδI + 0) = δxδyˆ zδI ◮ Magnetization is density of induced magnetic dipoles:
δµ δxδy = ˆ zδI
(Per unit )
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SLIDE 3 Angular momentum and magnetic moments
◮ Classical charges circulating in magnetic fields ◮ Angular momentum L = mvr ◮ Current I =
qv 2πr
◮ Magnetic moment µ = 1
2
lI = qvr/2 ◮ Classical particle µ =
q 2mL
◮ Exactly true for orbital angular momentum µz = −e 2mml = −mlµB where µB ≡ e
2m is the Bohr magneton
◮ Similarly for spin: µz = −gemsµB where ge ≈ 2.0023 = 2 +
e2 4πǫ0hc + · · · is called the gyromagnetic ratio
◮ Both components produce and interact with magnetic fields the same way ◮ M is the total density of orbital and spin magnetic moments
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SLIDE 4 Bound current density due to magnetization
◮ Current within dotted element (per unit δz) δIy = δI1 − δI2 ◮ Corresponding current density: jy = Iy δx = δI1 − δI2 δx = Mz(x) − Mz(x + δx) δx = −∂Mz ∂x ◮ Generalizing to all directions:
M
(Per unit )
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SLIDE 5 Constitutive relations
◮ Material determines how P (and hence D) depends on E ◮ Material determines how M (and hence H) depends on B ◮ Simplest case: linear isotropic dielectric
E + P
µ0 − M
E
H
E
H ǫ = (1 + χe)ǫ0 µ = (1 + χm)µ0 ◮ Anisotropic magnetism: M = ¯ χm · H (magnetic susceptibility tensor) ◮ Nonlinear magnetism: M = χm(H) H (field-dependent susceptibility) ◮ Hysterisis: M = χm({H(t)}) H (history-dependent susceptibility)
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SLIDE 6
Types of magnetic materials
◮ Distinguish based on magnetic susceptibility χm and zero-field magnetization M0
◮ Diamagnetic: χm < 0 and small, M0 = 0 (closed shell, insulators) ◮ Paramagnetic: χm > 0 and small, M0 = 0 (open shell, metals) ◮ Ferromagnetism: χm ≫ 1, M0 = 0 (certain metals) ◮ Antiferromagnetism: χm > 0 and small, M0 = 0 (insulators) ◮ Ferrimagnetism: χm ≫ 1, M0 = 0 (insulators)
◮ Bohr-van Leeuwen theorem: Classical statistical mechanics of charged particles ⇒ M = 0 ◮ All magnetism is quantum mechanical despite our picture of current loops ◮ In fact, can mostly ignore orbital component; it’s all spin!
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SLIDE 7 Diamagnetism
χm < 0, |χm| ≪ 1 ◮ Magnetic moment in field: torque T = µ × B ◮ Angular momentum L = µ 2m
gq
◮ But T = d L/dt ⇒ rotation with ω = gqB
2m (Larmor precession)
◮ Corresponding current δI = qω
2π = gq2 4πmB
◮ Induced magnetic moment δµ = δI · πr2 = gq2r2
4m B (loop radius r)
◮ Therefore, χm = −µ0n gq2r2
12m (direction opposite to B, average over x, y, z)
◮ Typical values n ∼ 0.1˚ A-3, r ∼ 1˚ A ⇒ χm ∼ −6 × 10−6 ◮ Example: for Si, χm = −5.2 × 10−6 ◮ Temperature-independent diamagnetic response present in all materials!
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SLIDE 8 Paramagnetism: gases and liquids
χm > 0, |χm| ≪ 1 ◮ Closed-shell molecule: every orbital has ↑↓; no net spin, µ = 0 ◮ Open-shell molecule: some unpaired ↑ / ↓; can have net spin, µ = 0 ◮ With no applied field, µ in random directions ⇒ M = 0 ◮ With field, energy of one magnetic dipole is −µ · B ◮ Therefore, magnetic susceptibility is: χm = µ0N µ2 kBT x coth x − 1 x2 where x ≡ µB kBT ◮ µB ≪ kBT for practical magnetic fields, so χm = µ0Nµ2
3kBT
◮ Typical values N ∼ 0.01˚ A-3, µ ∼ µB ∼ 10−23 J/T ⇒ χm ∼ +10−4 ◮ Unpaired spins ⇒ paramagnetism typically dominates over diamagnetism ◮ Paramagnetic response (Type B) decreases with increasing temperature
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SLIDE 9 Paramagnetism: metals
χm > 0, |χm| ≪ 1 ◮ Magnetic field changes energy of ↑ vs ↓ by 2 gµB
2 B ≈ 2µBB
◮ Fermi level same for both spins (equilibrium) ◮ Spin imbalance n↑ − n↓ = g(EF )
2
· 2µBB ◮ Magnetization M = (n↑ − n↓) gµB
2
= µ2
Bg(EF )B
◮ Therefore susceptibility χm = µ0µ2
Bg(EF )
◮ Typical value eg. in Al, χm ≈ 2 × 10−5 ◮ Temperature independent (Type A) ◮ For Cu, Ag, Au, χm < 0: why?
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SLIDE 10 Hund’s rule and exchange interaction
◮ So far, treated spins independently. How would spins interact? ◮ Due to their magnetic field i.e. dipole-dipole: typically weak ◮ Consider filling up electrons in degenerate px, py, pz orbitals ◮ One electron: ↑, 0, 0 ◮ Two electrons: ↑, ↑, 0 or ↑↓, 0, 0? ◮ Two electrons in px repel more than px with py ◮ Hund’s rule of maximum multiplicity: prefer parallel spins ◮ Exchange interaction between spins −2J S1 · S2 ◮ Very sensitive to distance and can flip sign! (Kasap Figure 8.20) ◮ Next: materials with strong exchange interactions between adjacent atoms ◮ If N spins response to magnetic field together, then χm ∝ µ2
Bg(EF )
increases by N (because effective µ ∝ N and effective g ∝ 1/N) ◮ Other possibility: symmetry breaking and phase transitions!
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SLIDE 11 Ferro-, antiferro- and ferri-magnetism
◮ J wants to align (or anti-align) neighboring spins ◮ Entropy (T) wants to randomize them ◮ T > Tc: entropy wins, paramagnet with χm ∝ µ2
T
◮ T < Tc: J wins, three ordering possibilities:
- 1. J > 0: parallel spins ⇒ M = 0,
ferromagnet (eg. Fe, Co, Ni)
- 2. J < 0: anti-parallel spins ⇒ M = 0,
antiferromagnet (many oxides)
- 3. J < 0: anti-parallel dissimilar spins ⇒ M = 0,
ferrimagnet (eg. ferrite Fe3O4) Tc = Curie temperature for ferromagnets and Neel temperature for antiferro/ferrimagnets
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SLIDE 12
Saturation magnetization
◮ At T = 0, all spins aligned, maximum magnetization Msat(0) ◮ Increasing T, spins randomized ⇒ reduces Msat(T) ◮ Spontaneous magnetization vanishes at T = Tc ⇒ paramagnet
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SLIDE 13 Magnetic domains
◮ Spins align locally in domains ◮ Spins misaligned along domain walls ◮ Energy cost (and entropy gain) per area of domain wall ◮ Gain: reduction in magnetic energy B2
2µ reduced
◮ Random domains: unmagnetized state (M = 0, B = 0) ◮ Magnetized state: domains aligned which costs magnetic energy ◮ Will magnetization disappear automatically? ◮ Not necessarily: barrier to domain rotation
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SLIDE 14
Magneto-crystalline anisotropy
◮ Exchange interaction anisotropic, so M has preferred directions ◮ eg. In Fe, strong J along (100) directions: easy axis ◮ Weaker J along (111) directions: hard axis ◮ Domains tend to snap to easy axes, barrier to rotate through hard axes ◮ Difference in energies: magnetocrystalline anisotropy energy K
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SLIDE 15 Domain walls
◮ Thick domain wall: slow change in spin ◮ Favorable for minimizing exchange interactions ◮ With thickness δ, energy cost Uexchange ∝ δ−1 (≈ π2Eex
2aδ )
◮ Thin domain wall: rapid change in spin ◮ Favorable for minimizing magnetization along non-easy axes ◮ With thickness δ, energy cost Uanisotropy ∝ δ (≈ Kδ) ◮ Total energy Uwall = Uexchange + Uanisotropy ≈ π2Eex
2aδ
+ Kδ ◮ Energy minimized for optimal thickness δ =
2aK
◮ Corresponding minimum energy Uwall =
2a
◮ For iron, Eex = kBTc ≈ 0.1 eV, a ≈ 3 ˚ A and K ≈ 50 kJ/m3 ⇒ δ ≈ 70 nm and Uwall ≈ 7 × 10−3 J/m2
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SLIDE 16 Crystal grains vs magnetic domains
◮ Domain wall thickness sets typical magnetic domain size ◮ Therefore, two regimes in polycrystalline materials:
- 1. Grain size smaller than domain wall thickness
◮ Single magnetic domain per grain ◮ Adjacent domains have different easy / hard directions
- 2. Grain size larger than domain wall thickness
◮ Many magnetic domains per grain ◮ Adjacent domains within grain have same anisotropy ◮ Anisotropy directions change along grain boundaries
◮ Grain-size distribution: combination of grains in both regimes
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SLIDE 17
Magnetostriction
◮ Bond lengths along and perpendicular to spin differ ◮ Consequence: spin polarization produces anisotropic strain ◮ Magnetostrictive strain λ: value along magnetization ◮ Iron λ > 0 while nickel λ < 0 ◮ Couples oscillating magnetic fields to mechanical oscillations ⇒ losses ◮ Iron-nickel alloys reduce electrostriction with cancelling contributions ◮ Transverse strain typically opposite sign (volume conservation) ◮ High fields: overall compression (minimizes field energy) ◮ Analogous effect in dielectrics: electrostriction ◮ How is this different from piezo-electricity?
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SLIDE 18
Magnetization to saturation
◮ Increase field: domains align to field ◮ Reversible: smooth domain wall motion within grains ◮ Irreversible: domain walls pinned by defects, and broken free ◮ Barkhausen effect / noise: magnetization increases in jumps (magnitude of jumps span several orders of magnitude ⇒ noise) ◮ Domain wall motion till grains align along easy axis closest to field ◮ Beyond that, field overcomes K so as to align M to H in each grain ⇒ Msat saturation magnetization
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SLIDE 19
Reversing magnetization
◮ Reduce field: domains return to easy axis ◮ But domains along easy axis closest to field direction ◮ Residual magnetization Mr at zero field ◮ Apply reverse magnetic field to randomize domains ◮ Coercive field strength Hc zeros magnetic field ◮ Subsequent cycles follow outer loop ◮ Energy loss per cycle = area inside M-H curve
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SLIDE 20
Magnetization patterns
◮ Smaller M-H (or B-H) loops for smaller driving fields ◮ Lower peak magnetization, but lower losses per cycle ◮ Arbitrary pattern of H: trajectory stays within full-saturation loop ◮ To restore M to zero, spiral in by reducing oscillations of H
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SLIDE 21
Hard vs. soft magnetic materials
◮ Hard materials: large Hc and Mr ◮ Useful for permanent magnets which need high Mr and Br ◮ Metal alloys like alnico (0.9 T), SmCo (1.1 T) and NdFe (1.2 T) ◮ Soft materials: low Hc and Mr (still high µ) ◮ Can cycle magnetic field with lower energy loss ◮ Useful for transformers and electromagnets ◮ Pure metals, metallic glasses, certain alloys and ferrites
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