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Unformatted text preview: 7 Continued 7-11 Paramagnetism
Occurence: • Free atoms, molecules or lattice defects with odd number of electrons. ⇒ total spin cannot be zero ⇒ permanent magnetic moment. (few compounds with even electron number also paramagnetic.) • Free atoms or ions of transition, rare eath and actinide elements with partially filled inner shells. ⇒ permanent magnetic moment, often also paramagneic in solid state. E.g., rare eath and actinide salts (outer shells screen crystal field). • Metals can be para- or diamagnetic. 7-12 Atoms with N electrons
Only total angular momentum J conserved. J = ∑li + ∑ si
i i N N (7.27) li, si: orbital and spin moment of ith µ J = −g J µB J h (7.28) general case: gJ = gJ(li, si). Interation energy in magnetic field B = (0,0,B):
B EB(J) = -µJB = g J h J z B µ (7.29) Jz = mJ h , mJ = -J,........,J (7.30) (cf. hydrogen atom: ml = -l,.....,l) 7-13 Classical treatment:
EB = -µ B = -µBcosθ ⇒ M = µB<cosθ>natoms all ∠ θ possible T→0 ⇒ θ=0 perfect alignment of µ parallel B. T → ∞ statistical distribution ⇒ <cosθ> = 0
Fig. 7.6 In the classical treatment the angular momentum, and hence the magnetic moment µ, can point to any direction. The magnetization results from the average of cosθ over all atoms in the sample. Quantum mechanical treatment
(7.30) ⇒ only 2J + 1 discreate values of Jz ⇒ splitting of energy levels in 2J + 1 levels. Simplest case: single electron in partially filled shell with l=0* ⇒ J = s = 1/2, gJ = g = 2 EB = ± µBB only 2 possible orientaions * In a closed shell there are always pairs of electrons with l , s and -l ,- s . Therefore the total angular spin and
i i i i (7.31) orbital momentum of closed shells are zero. 7-14 Fig. 7.7 splitting of energy level of a simple spin in a magnetic field B = (0,0,B). In the low energy state the magnetic moment is parallel to the magnetic field. Fig. 7.8 fractional population of a two-level system (Fig.7.7) in thermal equilibrium at temperature T in a magnetic field B. The magnetic moment is proportional to the difference between the two curves. Equilibrium populations of levels (see Fig. 7.8) for N = N1 + N2 atoms N 1, 2 N1 + N 2 = exp[ m µ B B /(kT )] exp[ − µ B B /( kT )] + exp[ + µ B B /( kT )] (7.32) Resultant magnetization:
N 2 − N1 e x − e−x M= µ B = natoms µ B x = n atoms µ tanh x V e + e−x (7.33) x = µBB/(kT) 7-15 1) x<<1 ⇒ tanhx ≈ x
2 n atoms µ B B M≈ kT (7.34) (Curie law) (Curie constant) (7.35) (7.36) ⇒ χ ≈ C/T 2 2 µ 0 g J J ( J + 1) µ B C= 3k for general case (see, e.g., Kittel ), here: J = s = 1/2,gJ = g = 2 2) x >> 1 ⇒ tanh ≈ 1 ⇒ saturation magnetization M = natoms µB (7.37) Requires very strong fields and low temperatures! e.g., B = 5T, T = 300K ⇒ x of order of 10-2: still χ ≈ C/T Fig. 7.9 Cource of magnetization as function of x = µBB/(kT). Note the linear range, which is obeyed for most practical cases, and the occurence of saturation at very low temperatures or extremely strong fields. 7-16 Pauli paramagnetism of conduction electrons
Magnetic moment of a conduction electron: (7.26) ⇒ µz = ±µB (only spin, no orbital angular momentum) (7.38) Fig. 7.10 Pauli paramagnetism at about absolute zero for free electrons. (Left) for B = 0 the electrons with spin up and down have the same energy. (Middle) for B ≠ 0 there is a shift of 2µBB between the electrons of spin up and down. (Left) The numbers of electrons in the ″up″ and ″down″ band will adjust to achive an equal Fermi level (chemical potential). This leads to an excess of moment up electrons in the magnetic field and hence to a net magnetic moment of the conduction electrons. Note that the figure is out of scale. At B = 5T the energy shift is 2µBB ≈ 6×10-4 eV << EF. Therefore, the shift of the Fermi level can be neglected in the calculation: EF ≈EF (B = 0). Mz = (n+ - n-)µB n± : density of electrons with spin up and down
2 ⇒ χ Pauli ≈ µ B D ( E F ) µ 0 (7.39) (7.40) (derivation see practical course) EF(T) ≈ const. in metals ⇒ χPauli(T) ≈ const ! 7-17 Landau diamagnetism of conduction electrons
B-field ⇒ ″condensation″ of conduction electrons on landau tubes: orbits ⊥ B ⇒ diamagnetic contribution.
⇒ 1 χ Landau = − χ Pauli 3 independent of T ! (7.38) (involved calculation even for free electrons) ⇒ Total contribution of conductin electrons χ free electrons = 2 χ Pauli 3 paramagnetic (7.39) Deviations from free electron value if m* ≠ m. Order of magnitude χPauli : + 10-6 Some order of magnitude as diamagnetism of closed shells. ⇒ Metals can be dia- or paramagnetic. Noble metals: diamagnetic, alkali metals except for Cs: paramagnetic. 7-18 Fig. 7.11 Characteristic magnetic susceptibilities of diamagnetic and paramagnetic substances. 7-19 Ferromagnetism
Spontaneous magnetization without applied magnetic field. Exchange interaction: Coulomb interaction between electrons depends on spin orientation. : Pauli princilple prevets that electrons approach each other two closely. : Pauli principle does not exclude electrons to be at the same place. Heisenberg model: Exchange interaction described by pairwise nearest-neighbor interaction between spins lokalized at atoms.* Interaction energy: E = -2As1⋅s2 (7.40) A: exchange or coupling constant, depends on overlap of orbitals ⇒ magnetostriction: change in sample dimensions upon change in magnetization.** * There is also a so-called indirect exchange interaction between next-nearest neighbour magnetic ions that are coupled via a diamagnetic ion, e.g., O2-. ** In so-called invar alloys the magnetostriction effect is used to compensate thermal expantion in a certain temperature range since the magnetization at constant field decreases with increasing temperature. 7-20 A > 0 ⇒ parallel spin orientation in ground state, i.e. ferromagnetism. A<0⇒ antiferromagnetism. Collective exchange of conduction electrons Electrons cannot approach each other too closely (Pauli principle). ⇒ Screening of nuclei less effective. ⇒ Lowering of total energy due to increase in Coulombs interaction with nucley. (corresponds to A > 0 in Heinsenberg model). Spontaneous magnetization strongly temperature dependent ! T > TC : Spontaneous magnetization vanishes and crystal becomes paramagnetic. TC: ferromagnetic Curie temperature T >> TC ⇒
C T −θ Curie-Weiss law (7.41) θ: paramagnetic Curie temperature > TC 7-21 Fig. 7.12 Reciprocal of the susceptibility per gram of nickel in the neighborhood of the Curie temperature (358 °C). The density is ρ. The dashed line is a linear extrapolation from high temperatures. Table 7.1 Experimental data for some ferromagnetic materials. TC [K] Fe Co Ni gd Dy EuO Eus 1043 1395 629 289 87 69.4 16.5 Θ [K] 1100 1415 649 302 157 78 19 C [K] 2.22 2.24 0.588 Ms(0) [106 A/m] 1.746 1.446 0.510 2.060 2.920 4.68 3.06 1.930 1.240 7-22 Fig. 7.13 spontaneous magnetization of EuO as a function of temperature. ferromagnetic ↔ paramagnetic 2nd order transition - Magnetization (order parameter) changes gradually. - No heat of transformation. 7-23 Mean field approximation for localized electrons
Exchange insteraction with neighbouring atoms is treated in a mean field approximation as internal ″exchange field″ or ″molecular field″. Calculation for primitive lattice with unpaired electron with l = 0: ⇒ only spin Exchange energy of i-th lattice atom with Z nearest neighbours: E = −2 A∑ s i ⋅ s j
j =1 Z (NNN interaction neglected) (7.42) = −2 Az < s j > s i , <sj>: time average
M =− n atoms gµ B < s j > ⇒ < s j >= − M n atoms gµ B / h (7.43) h (7.44) 2 Az ⇒ E = (− gµ B s i / h ) ⋅ M 2 14243 natoms g 2 µ B 44 −µ 14 244 4 3
B MF (7.45) 7-24 B MF 2 zA M = µ 0γ M = 22 natoms g µ B mean field (exchange field) (7.46) γ= 1 2 zA 2 µ 0 natoms g 2 µ B molecular field constant (7.47) Effective magnetic field: Beff = B + BMF Energy of electron spins in magnetic field: E± = ± gµBBeff (7.48) n± natoms e − E± / kT = − E+ / kT e + e − E− / kT + : ↑ sz −:↓ B eff (7.49) M= 1 gµB(n+ 2 - n -) = 1 g µ B B eff 1 2 gµBnatomstanh kT 2 (7.50) 7-25 Saturation magnetization: Ms(0) = 2 gµBnatoms all spins oriented 1 No applied field:
1 gµ B B A zA M = M s (0) tanh M = M s (0) tanh 2 ⋅ n gµ B kT kT atoms (7.47) (7.51) ⇒ T M (T ) M (T ) = tanh c T M ( 0) M s ( 0) s TC = zA 2k (7.52) Note: TC ∝ z ⇒ TC decreases at surface. (lower coordination number) 7-26 M (T ) ≡ m M s (0) m = tanh( m / t ) T ≡t TC t < 1, i.e., T > TC solution only for m = 0 ⇒ No spontaneous magnetization for T > TC. Fig. 7.14 Graphical solution of eq.(7.52). The resulting couse of m = M(T)/Ms(0) as function of temperature is shown in Fig. 7.15 for Ni. 7-27 Fig. 7.15 saturation magnetization of nickel as a function of temperature, together with the theoretical curve for S = 1/2 on the mean field theory. Fit with mean field theory ⇒ J = 1/2 , i.e.,
1 Ms(0) = 2 gµBnatoms ⇒ g = 1.2 ! ? (see below) Ni: band ferromagnet ⇒ mean field theory for localized electrons inaequate ! 7-28 Behaviour near TC
1 M tanh x ≈ x − x 3 + ... T → TC ⇒ M << 1 ⇒ 3 s (7.53) (7.52) ⇒ M (T ) T ≈ 3 1 − T M s ( 0) C γ 1/ 2 (7.54) T M ∝ 1 − T Experiment: C , γ = 1/3, (7.55) γ: critical exponent Reason for γ ≠ 1/2: only long-range order, i.e., magnetization vanishes at TC, certain local ordering remains. T >> T ⇒ tanh x ≈ x C 1 gµ B Beff 1 (7.50) M (T ) = gµ B natoms tanh 2 2 kT 4kTC Beff = BA + B = M +B 22 natoms g µ B ↑ (7.57 )
B: applied field ⇒ 7-29 2 g 2 µ B n atoms 1 M (T ) = B T − TC 4k (7.56) ⇒ χ= C T − TC with
2 µ 0 µ B g J J ( J + 1) C= n atoms 3k (7.57) (7.58) Currie constant for Currie-Weiss law with J = S =1/2 θ = TC (7.59) (7.47), (7.52), (7.58) ⇒ TC = Cγ E.g., EuO: γ = TC / C = 69.4K / 4.68k = 15 BMF = µ0γ
1.930×10 6 A / m M0 12(3)
s = 36T >> experimental fields 7-30 • External fields hardly affect magnetization of ferromagnets well below Tc. • BMF >> dipole field of neigbouring atoms Order of magnitude:
µ0 µ B ≈ 0.04T 4πa 3 a: lattice parameter BMF no magnetic field ! (due to quantummechanical exchange interaction) ⇒ BMF does not appear in Maxwell eqs. Mean field approximation describes exchange interaction between localized electrons. Good model for: - lanthanides (partially filled 4f-shells) - many ionic compounds of d- and f- transition metals Does not apply for 3d-transition metal ferromagnets Ni, Co, Fe. 7-31 Band ferromagnetism of Ni, Co and Fe
3d-transition metals Experiment: J = 1/2 ⇒ magnetization due to electron spins Ms(0) = (1/2)gµBnatoms ⇒ g = 1.2 ≠ 2 ! for Ni Theoretical explanation analogous to Pauli paramagnetism: 10 electrons in overlapping 3d- and 4s-bands effective moment per atom: µeff = 0.6µB ⇒ gµeff = 1.2µB Fig. 7.16 Calculated density of states of Ni. Due to the quantummechanical exchange interaction there is a preferred orientation of the spins, which gives rise to an external exchange field. Orientation of the spins parallel to the exchange field to an energy reduction while antiparallel orientation shifts the density of states to higher values. This lends to a net number of 0.6 electrons per atom with spin orientation parallel to the exchange field at 0K and hence an effective magnetic moment of 0.6µB per atom. This so-called band ferromagnetism is analogous to the Pauli paramagnetism if one envisions the exchange field as an extreemly strong magnetic field. The strong magnetization is also caused by the high density of stated of the Fermi level in the d-band.
7-32 Domain structure
Ferromagnetic sample generally has no net magnetic moment! Reason: domain structure ⇒ reduction of magnetic energy. Fig. 7.18 Illustration of the reduction of magnetic energy by formation of domains of uniform spin orientation. The domains are generated by so-called Bloch walls, where the spin orientation changes gradually within a distance of the order of 40nm (for Fe). Typical domain sites are 1 - 10 µm. The microstructure of the domain results from a competition between reduction in magnetic energy and the energy required to form Bloch walls, which originates from the rotation of the spins away from the direction of easiest magnetization. Magnetic anisotropy: prefered direction of magnetization,e.g., <100> in Fe. Reason: orbits of electrons of lattice atoms are affected by electrostatic field of neighbouring atoms ⇒ anisotropy. Anysotropy in orbital moments affects spin orientation via spin-orbit coupling. * * The orbital moment of an electron around the central nucleus gives rise to an orbital magnetic moment. The correponding magnetic field acts on the magnetic moment arising from the electron spin. 7-33 Bloch walls Fig. 7.18 Illustration of the spin rotation in a 180° Bloch wall. The extension of the wall is a balance of the exchange energy, which trys to avoid large missorientation of neighboring spins, and the magnetic anisotropy energy, which trys to minimize the number of spins being oriented in a direction different from that of easiest magnetization. Fe: Bloch wall thickness ≈ 40mm 7-34 Hysteresis loop Fig. 7.19 Sketch of a magnetization curve of a ferromagnetic material (d) and corresponding changes in the domain structure(a-c). B small: Domains with favourable orientation to applied field grow at expense of others (Fig.7.19b). B large: rotation of the magnetization, B → ∞ ⇒ M || B Remanence Mr: remaining magnetization at B = 0 due to irreversible Bloch wall displacements. Coercitivity BC : required field to remove remaining magnetization.
lim Ms: saturation magnetization B→∞ M ( B) 7-35 ...
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This note was uploaded on 01/19/2010 for the course MATERIALS M504 taught by Professor Adelung during the Spring '02 term at Uni Kiel.
- Spring '02