Insights into Condensed Matter Physics by Prof. Dr. Ir. Paul H. M. van Loosdrecht
Explore magnetism in condensed matter physics with Prof. Dr. Ir. Paul H. M. van Loosdrecht, covering topics like diamagnetism, paramagnetism, ordered magnetism, non-ordered magnetism, crystal field effects, and magnetic properties of materials. Gain knowledge on magnetic moments, susceptibility, temperature dependence, quantum mechanical treatments, spin moments, and more with a focus on key concepts and properties in the field.
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Condensed Matter Physics I Prof. Dr. Ir. Paul H.M. van Loosdrecht II Physikalisches Institut, Room 312 E-mail: pvl@ph2.uni-koeln.de Website: http:/www.loosdrecht.net/ PvL CMP-I WS 14/15
Previously Intro magnetism - Diamagnetism: induced moments, no interaction - Paramagnetism: needs moments, no interaction - Ordered magnetism: needs moments & interaction PvL CMP-I WS 14/15
Magnetism Diamagnetism: Paramagnetism: - Magnetic moments (spin, orbit) - Weak magnetic interactions - Response due to orientation - Magnetization in field direction - No magnetic moments - No magnetic interaction - Response due to induced currents - Magnetization opposite to field - Water - Ideal gases - Superconductors - Metals - odd electron systems - O2, biradicals Ordered magnetism: - Magnetic moments - Strong magnetic interactions - Response due to polarization - Ferro-, antiferro-, ferrimagnetic - Fe, Ni, Co - Cr, high-Tc (CuO systems) PvL CMP-I WS 14/15
Today Non-ordered magnetism Magnetic moments Crystal field effects PvL CMP-I WS 14/15
Magnetization and susceptibility E o Magnetization at T=0: = M H ( ) H M E / kT n e ) H ( n n at finite T: = M H ( ) E / kT n e n M = Magnetic susceptibility: H 2 E o = Only ground state (low T): 2 H PvL CMP-I WS 14/15
Dia- & paramagnetism M= H M Diamagnetism Temperature independent H T M Paramagnetism 1/T dependence H 1/T PvL CMP-I WS 14/15
QM treatment: orbit e + p p A Inclusion of the field in the motion: c 1 = Uniform H-field: A r H Gauge: A = ; 0 2 = + = T V L r p i i 2 2 1 e 1 e = + = T p A p r H i i i i i 2 m c 2 m 2 c i i 2 e 2 i 2 i 2 = + + + T L H x ( y ) H o B 2 PvL CMP-I mc 8 WS 14/15 i H//z
QM treatment: spin = g H S g HS Inclusion spin moment: o B o B z 2 e 2 i 2 i 2 = + + + + = + T L ( g S ) H x ( y ) H T i o B o 0 B 2 mc 8 = + E E E n o , n B 2 n H ' n B = + E n H n B B E E ' n n n ' n 2 ( )n + 2 H n L g S ' n e B o 2 2 i 2 i + + + + H n L g S n H n x y B o 2 E E mc 8 ' n n i n ' n Curie Van vleck Langevin PvL CMP-I WS 14/15
Everything is (dia)magnetic Langevin diamagnetism: shielding effect (Lenz law) Meissner effect in superconductors = e + p p A QM: inclusion of field Orbit c S = g H g HS Spin o B o B z = + T 0 B 2 e 2 2 i 2 i = + + + L ( g S ) H H x ( y ) i B o B 2 mc 8 PvL CMP-I WS 14/15
Magnetism Diamagnetism: Paramagnetism: - Magnetic moments (spin, orbit) - Weak magnetic interactions - Response due to orientation - Magnetization in field direction - No magnetic moments - No magnetic interaction - Response due to induced currents - Magnetization opposite to field - Ideal gases - Superconductors - Metals - odd electron systems - O2, biradicals Ordered magnetism: - Magnetic moments - Strong magnetic interactions - Response due to polarization - Ferro-, antiferro-, ferrimagnetic - Fe, Ni, Co - Cr, high-Tc (CuO systems) PvL CMP-I WS 14/15
Langevin diamagnetism 2 ( )n + 2 H n L g S ' n e B o 2 2 i 2 i + + + + E H n L g S n H n x y B B o 2 E E mc 8 ' n n i n ' n Low temperature, filled shell ions ( ) = = = J 0 L 0 S 0 0 2 e 2 2 i r E H 0 0 B 2 12 mc i 2 2 2 E e e 2 i r 2 B = = = nZ n n 0 0 r 2 2 2 H mc 6 mc 6 i 2 nZ e 2 o 6 = r ( SI: ) m PvL CMP-I WS 14/15
Paramagnetism M H T 1/T alignment of weakly interacting magnetic moments in a magnetic field Curie law = Magnetic moments = spin, orbit Ground state splitting (Curie) Low lying excited states (van Vleck) Density of states effects (Pauli magnetism) First: Coupling between L and S: Russel-Saunders PvL CMP-I WS 14/15
Spin-Orbit interaction (qualitative) = Magnetic moment electron spin: g S o B Current of the electron in orbit: e v I = ( e R / ) p m eL = = 2 2 2 R mR v 2 R 2 eL = = B / I 2 R 2 3 4 mR Energy of the spin in the field of the orbit: = = e = U m B g S L L S PvL CMP-I WS 14/15 o B 2 3 4 mR
Partially filled shells 1. Russel-Saunders coupling L,S and J commute with Hamiltonian. Quantum numbers L,Lz, S,Sz,J,Jz describe electronic state ) 1 + ( 2 2 2. Hund s rules for n electrons in states (Maximization anti-symmetry + Pauli) I. II. III. Lowest state has highest S Lowest state has highest L( ) Lowest state has minimized LS interaction L+S for more than half filled (L,S opposite) |L-S| for less than half filled (L,S parallel) = z PvL CMP-I WS 14/15 L S J J States are characterized by z
Hunds rules Spectroscopic notation 2S+1XJ L = 0 1 2 3 4 5 6 X = S P D F G H I z S=2 L=2 => 5D0 J=0 -2 -1 0 1 2 = 2 d-shell ( ), 4 electrons Mn3+,Cr2+ z S=5/2 L=0 => 6S5/2 J=5/2 -2 -1 0 1 2 = 2 d-shell ( ), 5 electrons Fe3+,Mn2+ z = 2 d-shell ( ), 6 electrons Fe2+ S=2 L=2 => 5D4 J=4 -2 -1 0 1 2 PvL CMP-I WS 14/15
Rare earth elements 4f Rare earth elements: Ce-Lu 6s2, 5d1 PvL CMP-I WS 14/15
Rare earth elements z -3 -2 -1 0 1 2 3 = 3 f-shell ( ), 7 electrons Gd3+ S=7/2 L=0 => 8S7/2 J=7/2 4f 4f Lu La DOS EF EF Energy PvL CMP-I WS 14/15
Spectroscopic splitting factor Level splitting in a field ' = L ( B + g S ) H g J H o j B B + = L ( g S ) g J o j + = L ( g S ) J g J J o j 1 ( + + + = ) 1 + L L g L ) o S g S S g j ( j o j = L S J ( + J L L S S / ) 2 ) 1 + 1 ( + ) 1 + 1 ( ) 1 + 1 ( g j ( j ) o g l ( l ) o g s ( s ) o = g j PvL CMP-I WS 14/15 ) 1 + j ( j 2
Spectroscopic splitting E s shell, 1 electron: 2S1/2 (S=1/2,L=0,J=1/2) => gj=go=2 BH H Mercury 3S1 3P0 transition (6s17s1 6s16p1, G.S. 6s25d10) mJ=1 mJ=0 gJ=2 3S1 mJ=-1 gJ=0 3P0 PvL CMP-I mJ=0 WS 14/15
1 0 3S1 -1 3P0 Experimental spectrum of the 4046.6 , 7 3S1 --> 6 3P0 transitions of atomic Hg with (a) zero magnetic field (b) a magnetic field B = 29.0 kG. PvL CMP-I WS 14/15
Crystal field splitting Rare earth s: 4f shell s small ( inner electrons) Iron group: 3d shell s on the outside => decoupling of L and S, J no longer good QN => splitting of the 2L+1 orbital states => Quenching of the orbital angular momentum (Lz ) PvL CMP-I WS 14/15
2D p states in a 2 fold potential = VCF Q cos( 2 ) = i im = Y ( , ) cos( e ) e p-states in 2D: , l m i = = p e ) r ( R or , 1 PvL CMP-I WS 14/15
i i = = p V p d e Q cos( 2 ) e 0 , 1 CF , 1 i i = = p V p d e Q cos( 2 ) e 0 , 1 , 1 CF i i = = p V p d e Q cos( 2 ) e Q , 1 CF , 1 i i = = p V p d e Q cos( 2 ) e Q , 1 , 1 CF , 1 p , 1 p E Q 0 0 = + H H V o o CF p p Q E 0 0 o , 1 , 1 = E 0 0 E Q , 1 p , 1 p o 0 0 Q E p p o , 1 , 1 = 0 E E Q + + : p p 1 1 2Q PvL CMP-I WS 14/15 : p p 1 1
With LS coupling J ( J 2 / = ) 1 + L ( L ) 1 + ) 1 + L S S ( S = / 1 = p L S p 15 ( / 4 2 3 / / ) 4 2 2 , 1 , 1 / 1 = p L S p 2 , 1 , 1 = = p L S p 3 ( / 4 2 3 / / ) 4 2 1 , 1 , 1 = = p L S p 3 ( / 4 2 3 / / ) 4 2 1 , 1 , 1 , 1 p , 1 p / 2 Q 0 0 = + + H H V L S o CF p p Q 0 0 , 1 , 1 = E 0 0 / 2 Q , 1 p , 1 p / 0 0 Q p p , 1 , 1 3 1 8 2 = + u + + : u p v p 2 2 2 + 1 1 9 / 16 Q 2 2 = + E / 4 9 / 16 Q 1 3 / 8 PvL CMP-I : v p u p WS 14/15 2 = v 1 1 2 2 2 + 9 / 16 Q
2 1.5 1 0.5 E/Q 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 0 0.5 1 1.5 2 2.5 3 /Q PvL CMP-I WS 14/15
Angular momentum 2 2 = + J u , 1 p J , 1 p v p J p + + , 1 , 1 , , 1 3 / 8 3 1 3 / 8 1 = + + 2 2 2 2 2 2 2 2 + + 9 / 16 Q 9 / 16 Q 1 3 / 4 = + 1.6 1.4 2 2 2 + 9 / 16 Q 3/2 1.2 1 1 3 / 4 = J 0.8 J + + , , 2 2 2 0.6 + 9 / 16 Q 0.4 0.2 0 -1/2 -0.2 -0.4 PvL CMP-I WS 14/15 -0.6 0 0.5 1 1.5 2 2.5 3 3.5 4 /Q
Magnetic moment 1 = + = + = , 1 p m , 1 p L ( gS ) 1 ( 2 ) 2 z z z B B B 2 1 = = = , 1 p m , 1 p L ( gS ) 1 ( 2 ) 0 z z z B B 2 3 / 4 2 2 = + = + m u , 1 p m , 1 p v p m p 1 z z z B + + , 1 , 1 , , 2 2 + 9 / 16 Q 3 / 4 = m 1 z B + + , , 2 2 + 9 / 16 Q 2 2 1.8 1.6 1.4 1.2 m/ B 1 0.8 0.6 0.4 PvL CMP-I 0.2 0 WS 14/15 0 0 0.5 1 1.5 2 2.5 3 3.5 4 /Q
