Introduction. Free Electron Fermi Gas. Energy Levels in One Dimension

Transcription

1 ree Electro er Gas Eergy Levels Oe Deso Effect of eperature o the er-drac Dstrbuto ree Electro Gas hree Desos Heat Capacty of the Electro Gas Electrcal Coductvty ad Oh s Law Moto Magetc elds heral Coductvty of Metals Itroducto ree electro odel: Wors best for alal etals (Group I: L, Na, K, Cs, Rb) Na: oc radus ~.98A,.. dst ~.8A. Successes of classcal odel: Oh s law. σ / κ alures of classcal odel: Heat capacty. Magetc susceptblty. Mea free path. Quatu odel ~ Drude odel Eergy Levels Oe Deso Paul-excluso prcple: No two electros ca occupy the sae quatu state. d ψ Hψ ψ dx Orbtal: soluto of a -e Schrodger equato 0 0 oudary codtos: ψ ( ) ψ ( ) π ψ A s x π A s x L λ,, λ L L Partcle a box Quatu ubers for free electros: (, s ) s, Degeeracy: uber of orbtals havg the sae eergy. er eergy eergy of topost flled orbtal whe syste s groud state. N free electros: π L N π L 4

2 Effect of eperature o the er-drac Dstrbuto ree Electro Gas hree Desos er-drac dstrbuto : f ( ) e β + β Hψ d ψ d ψ d ψ dx dy dz ( r) + + ψ ( r) Checal potetal μ μ() s detered by N d g ( ) f ( ) At 0: f ( ) ( ) or all : 0 < for 0 > ( ) f or >> μ : ( ) f e β (oltza dstrbuto) g desty of states 5 D e-gas Partcle a box (fxed) boudary codtos: ( y z) ( L y z) ( x z) ( x L z) ( x y ) ( x y L) ψ 0,, ψ,, ψ,0, ψ,, ψ,,0 ψ,, 0 xπ yπ zπ ψ A s xs ys z L L L Perodc boudary codtos: ( x, y, z) ( x + L, y, z) ( x, y + L, z) ( x, y, z + L) ψ ψ ψ ψ ψ A e r,, π 0, ±, ±, L Stadg waves ravelg waves 6 pψ ψ p ψ v V 4π N free electros: N 8 π π N V ψ s a oetu egestate wth egevalue. / / π N V / π N v V 7 8

3 Desty of states: D( ) D V ds 8 π ( ) V N π D / V ( ) π N V ds 8 π V 4π V 4 π / π V π / Heat Capacty of the Electro Gas (Classcal) partto theore: etc eergy per partcle (/). N free electros: Ce N ( orders of agtude too large at roo tep) Paul excluso prcple Ce ~ N ~ 0 4 K for etal ( ) ( ) f ( ) U d D f e β + N β D( ) θ ( ) ( ) D N Usg the Soerfeld expaso forula ( ) ( ) ( ) ( ) ( ) ( )( ) d H f d H + ζ d H d free electros 9 U d D + ζ + D + O d 4 ( ) ( )( ) ( ) ( ) 4 N d D ( ) + ζ ( )( ) + O( ) 0 d N d D + ζ + O d 4 ( ) ( )( ) ( ) d D D ζ O d 4 ( ) + ( ) ( ) + ( )( ) + ( ) ζ ( )( ) D d d D D d for -D e-gas d D D d for -D e-gas N d D ( ) + 0 d ( ) D ( ) ζ ( )( ) ζ ( )( ) U d D + ζ + D + O d 4 ( ) ( )( ) ( ) ( ) D d d D D d for -D e-gas d D D ζ D O d 4 ( ) + ( ) ( ) + ( )( ) + ( ) + ( ) 4 ( ) + ( )( ) ( ) + ( ) d D ζ D O π ζ ( ) 6 U π CV D( ) N π CV N -D e-gas

4 Experetal Heat Capacty of Metals or << θ ad << : C γ + A el + ph C γ + A π C D ( ) D( ) N N π N V / Devato fro e-gas value s descrbed by th : γ ( ) th obs γ e gas γ ( ) th obs γ e gas Possble causes: e-ph teracto e-e teracto Heavy fero: th ~ Ue, CeAl, CeCu S. Electrcal Coductvty ad Oh s Law dp dt ree partcle costat E feld Heseberg pcture: [ p, H ] [, q ] p E r q E dp dt Loretz force o free electro: e E + v c d dt dp dt q E Experetal Electrcal Resstvty of Metals Doat echass hgh : e-ph collso. low : e-purty collso. No collso: ( t) ( 0) e t E δ ( t) Matthesse s rule: + ph + ph p p p dep of (collso freq addtve) phoo purty Resdual resstvty: ( ) 0 p Saple depedet K ph ( ) ( ) Saple depedet p Collso te : j qv ( e) ( ) δ e E Oh s law e σ 5 Resstvty rato: ( ) roo p p ~ oh-c per atoc percet of purty 6

5 Cosder Cu wth resstvty rato of 000: ro able, we have L ( K ) ( 95K ) Ipurty cocetrato: p.7 0 oh-c resstvty rato c Very pure Cu saple: σ ( 4K ) 0 σ ( 00K ) 7 pp 9 8 ( 4K ) 0 s v.57 0 c s ( ) ( ) 95.7 oh-c p ~ oh-c per atoc percet of purty l 4K v 4K 0.c or > θ : 7 8 Moto Magetc elds Equato of oto wth relaxato te : d + δ dt q E + v c d + v q E + v dt c q e for electros Let { e ˆ, e, e// } be a rght-haded orthogoal bass Hall Effect j 0 y q E q E y c Ex q ω q vx Ex q ω v 0 q y c x q vz Ez 0 q E c x d + v q E + v dt c d + v q E v dt c d + v q E dt // // Steady state: q q v E + ω v q q q v E ωc v q c q v// E// q ω c cyclotro frequecy 9 c electros Hall coeffcet: R E y H jx q q E x c Ex qc 0

6 heral Coductvty of Metals ro Chap 5: er gas: K C v l K π π el N v v N I pure etal, K el >> K ph for all. Wedea-raz Law: π N K σ q π q Lorez uber: K L σ π q.45 0 watt-oh/deg 8 for free electros L watt-oh/deg for free electros