1 MTLE-6120: Advanced Electronic Properties of Materials Magnetic properties of materials Reading: ◮ Kasap: 8.1 - 8.8
2 Materials in magnetic fields v × � ◮ Charges circulate around magnetic field due to force q� B ◮ Magnetic dipole moment of infinitesimal current loop (per unit δz ) δµ = 1 � � r × � � d lδI 2 = 1 2 (0 + δxδy ˆ zδI + δyδx ˆ zδI + 0) = δxδy ˆ zδI ◮ Magnetization is density of induced magnetic dipoles: δµ � M = δxδy = ˆ zδI (Per unit )
3 Angular momentum and magnetic moments ◮ Classical charges circulating in magnetic fields ◮ Angular momentum L = mvr ◮ Current I = qv 2 πr r × d � ◮ Magnetic moment µ = 1 � � lI = qvr/ 2 2 ◮ Classical particle µ = q 2 m L ◮ Exactly true for orbital angular momentum µ z = − e 2 mm l � = − m l µ B where µ B ≡ e � 2 m is the Bohr magneton ◮ Similarly for spin: µ z = − g e m s µ B e 2 where g e ≈ 2 . 0023 = 2 + 4 πǫ 0 hc + · · · 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
4 Bound current density due to magnetization ◮ Current within dotted element (per unit δz ) δI y = δI 1 − δI 2 ◮ Corresponding current density: j y = I y δx = δI 1 − δI 2 δx = M z ( x ) − M z ( x + δx ) δx (Per unit ) = − ∂M z ∂x ◮ Generalizing to all directions: � j b = ∇ × � M
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 � B D = ǫ 0 � � E + � � − � P H = M µ 0 P = χ e ǫ 0 � � M = χ m � � E H D = (1 + χ e ) ǫ 0 � � B = (1 + χ m ) µ 0 � � E H ǫ = (1 + χ e ) ǫ 0 µ = (1 + χ m ) µ 0 ◮ Anisotropic magnetism: � χ m · � M = ¯ H (magnetic susceptibility tensor) ◮ Nonlinear magnetism: � M = χ m ( H ) � H (field-dependent susceptibility) ◮ Hysterisis: � M = χ m ( { H ( t ) } ) � H (history-dependent susceptibility)
6 Types of magnetic materials ◮ Distinguish based on magnetic susceptibility χ m and zero-field magnetization � M 0 ◮ Diamagnetic: χ m < 0 and small, � (closed shell, insulators) M 0 = 0 ◮ Paramagnetic: χ m > 0 and small, � (open shell, metals) M 0 = 0 ◮ Ferromagnetism: χ m ≫ 1 , � (certain metals) M 0 � = 0 ◮ Antiferromagnetism: χ m > 0 and small, � (insulators) M 0 = 0 ◮ Ferrimagnetism: χ m ≫ 1 , � (insulators) M 0 � = 0 ◮ 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!
7 Diamagnetism χ m < 0 , | χ m | ≪ 1 ◮ Magnetic moment in field: torque � µ × � T = � B ◮ Angular momentum � µ 2 m L = � gq ◮ But � T = d � L/ d t ⇒ rotation with ω = gqB 2 m (Larmor precession) gq 2 ◮ Corresponding current δI = qω 2 π = 4 πm B ◮ Induced magnetic moment δµ = δI · πr 2 = gq 2 r 2 4 m B (loop radius r ) ◮ Therefore, χ m = − µ 0 n gq 2 r 2 12 m (direction opposite to B , average over x, y, z ) ◮ Typical values n ∼ 0 . 1 ˚ A -3 , r ∼ 1 ˚ A ⇒ χ m ∼ − 6 × 10 − 6 ◮ Example: for Si, χ m = − 5 . 2 × 10 − 6 ◮ Temperature-independent diamagnetic response present in all materials!
8 Paramagnetism: gases and liquids χ m > 0 , | χ m | ≪ 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: χ m = µ 0 N µ 2 x coth x − 1 where x ≡ µB x 2 k B T k B T ◮ µB ≪ k B T for practical magnetic fields, so χ m = µ 0 Nµ 2 3 k B T A -3 , µ ∼ µ B ∼ 10 − 23 J/T ⇒ χ m ∼ +10 − 4 ◮ Typical values N ∼ 0 . 01 ˚ ◮ Unpaired spins ⇒ paramagnetism typically dominates over diamagnetism ◮ Paramagnetic response (Type B) decreases with increasing temperature
9 Paramagnetism: metals χ m > 0 , | χ m | ≪ 1 ◮ Magnetic field changes energy of ↑ vs ↓ by 2 gµ B 2 B ≈ 2 µ B B ◮ Fermi level same for both spins (equilibrium) ◮ Spin imbalance n ↑ − n ↓ = g ( E F ) · 2 µ B B 2 ◮ Magnetization M = ( n ↑ − n ↓ ) gµ B = µ 2 B g ( E F ) B 2 ◮ Therefore susceptibility χ m = µ 0 µ 2 B g ( E F ) ◮ Typical value eg. in Al, χ m ≈ 2 × 10 − 5 ◮ Temperature independent (Type A) ◮ For Cu, Ag, Au, χ m < 0 : why?
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 p x , p y , p z orbitals ◮ One electron: ↑ , 0 , 0 ◮ Two electrons: ↑ , ↑ , 0 or ↑↓ , 0 , 0 ? ◮ Two electrons in p x repel more than p x with p y ◮ Hund’s rule of maximum multiplicity: prefer parallel spins ◮ Exchange interaction between spins − 2 J � S 1 · � S 2 ◮ 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 χ m ∝ µ B g ( E F ) increases by N (because effective µ ∝ N and effective g ∝ 1 /N ) ◮ Other possibility: symmetry breaking and phase transitions!
11 Ferro-, antiferro- and ferri-magnetism ◮ J wants to align (or anti-align) neighboring spins ◮ Entropy ( T ) wants to randomize them ◮ T > T c : entropy wins, paramagnet with χ m ∝ µ 2 T ◮ T < T c : 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 Fe 3 O 4 ) T c = Curie temperature for ferromagnets and Neel temperature for antiferro/ferrimagnets
12 Saturation magnetization ◮ At T = 0 , all spins aligned, maximum magnetization M sat (0) ◮ Increasing T , spins randomized ⇒ reduces M sat ( T ) ◮ Spontaneous magnetization vanishes at T = T c ⇒ paramagnet
13 Magnetic domains ◮ Spins align locally in domains ◮ Spins misaligned along domain walls ◮ Energy cost (and entropy gain) per area of domain wall � B 2 ◮ Gain: reduction in magnetic energy 2 µ 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
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
15 Domain walls ◮ Thick domain wall: slow change in spin ◮ Favorable for minimizing exchange interactions ◮ With thickness δ , energy cost U exchange ∝ δ − 1 ( ≈ π 2 E ex 2 aδ ) ◮ Thin domain wall: rapid change in spin ◮ Favorable for minimizing magnetization along non-easy axes ◮ With thickness δ , energy cost U anisotropy ∝ δ ( ≈ Kδ ) ◮ Total energy U wall = U exchange + U anisotropy ≈ π 2 E ex + Kδ 2 aδ � π 2 E ex ◮ Energy minimized for optimal thickness δ = 2 aK � ◮ Corresponding minimum energy U wall = π 2 E ex K 2 a ◮ For iron, E ex = k B T c ≈ 0 . 1 eV, a ≈ 3 ˚ A and K ≈ 50 kJ/m 3 ⇒ δ ≈ 70 nm and U wall ≈ 7 × 10 − 3 J/m 2
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
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?
Recommend
More recommend
