Metal: a free electron gas model. Drude theory: simplest model for metals Sommerfeld theory: classical mechanics quantum mechanics

Transcription

1 Metal: a free eletron gas model Drude theory: simplest model for metals Sommerfeld theory: lassial mehanis quantum mehanis

2 Drude model in a nutshell Simplest model for metal Consider kinetis for eletrons only One of most suessful models for metal Why? Before the birth of quantum mehanis and atomi physis 897,J.J. Thomson disovered eletron 9,Drude proposed his theory Annalen de Physik, 566 9, ibid., 69 9.

3 Desription of ondution Ohm s law I Differential form V R ondutivity V or ρ l, I l a R l R ρ a a resistivity

4 A model of eletron gas Although there eist at least two types of harged partiles in a metal, ions and eletrons, Drude assumed that only eletrons ontribute to the eletri ondutivity. Valene eletrons vs. ore eletrons

5 Density and average distane n V N r s ~ n m 4πn 4π r / s

6 Basi assumptions Ignore eletron-eletron interation independent eletron approimation, ignore eletron-ion interation free eletron approimation, homogeneous applied field eletron Kineti theory Collisions are instantaneous, ome from the sattering between eletrons and ions impure atoms; Sattering rate is /τ,τ is eletron relaation time; letrons are assumed to ahieve thermal equilibrium only through ollisions, e.g., eletrons memory will be erased immediately after the ollision and be distributed randomly following statistial mehanis.

7 Mean free path Mean free path: The average distane that an eletron travels between two ollision. The length sale for transport. l vτ Aording to Bloh theorem, eletrons are sattering-free in a perfet period potential, thus the mean-free-path ahieve rystal sample size. mv k B T v v when T << T / F F F k B Classial Quantum

8 letri ondutivity Definition of urrent ne v ne p t m Newton s law Non-ollision probability or dp t f t p t + dt p t + f t dt dt P NC dt p t + dt τ d p t p t + dt τ dt τ [ p t + f t dt] f t frition fore

9 DC ondutivity p m ternal M field f t e + and does not vary with time steady state solution d p t dt p t e p t τ eτ ne p t m ne τ m ne τ m Drude formula

10 all oeffient, zˆ, dp dt dp dt y e e y e p m e + p m y p τ p τ y Let e be ylotron frequeny m y τ y τ + + y

11 all oeffiient y, y τ ne y all oeffiient: R y ne It depends on arrier density and harge only A method to measure the sign of harge

12 AC ondutivity AC field: Re t i e t an be negleted ~ ~ v k t t e t p dt t p d τ Re Let t i e p t p τ τ τ i define i m ne m p ne e p p i AC Drude ondutivity

13 AC ondutivity m ne i τ τ, Im Re τ τ τ + + Q: ow to measure real and imaginary part? Length sales:,, λ r r l >>, ',, ', λ r r r dr r l <<

14 Propagation of M waves Mawell equation: t t + 4 π t i e solution i i i i + π π 4 4 ompare with ε π ε i 4 +

15 Propagation of M waves 4πi 4π Im ε i 4π Re Refration and damping p τ >> ε, p 4πne m, iτ ne τ m < Reε <, Imε p M wave an not propagate

16 p : another physial Plasma osillation meaning Consider the propagation of harge density wave ρ iρ t 4πρ r, r, i t e iρ 4π ρ 4πi + p Condition for the propagation of a harge density wave

17 Thermal ondutivity T T > T Thermal energy heat flow eat urrent:thermal energy per unit time rossing unit area perpendiular to flow diretion. Fourier s law: q κ T thermal ondutivity Wiedemann-Franz empirial law: κ T onstant Lorentz number

18 Thermal ondutivity Loal equilibrium: eletron average energy ε ε T ε T Q: ow does the energy of eletron hange mirosopially when move from point to, where temperatures are different? Classial kineti theory: ε m < v > ~ k B T has to hange when going through regions with different T Mehanis: Collision with ions, with other eletrons, et, where energy transfer an take plae. < v Length sale to distinguish different : > >> l vτ

19 Thermal ondutivity D model: n n q vε T vτ vε T + vτ n average left right going eletron number per unit volume v average veloity ε average energy arried by eletrons If temperature varies slowly with on sale of mean free path l q dε nv τ dt dt d By definition, the speifi heat is given by C V dε n dt q C V v τ dt d

20 Thermal ondutivity4 D model: v v v v v z y T v C V q τ Thermal ondutivity: C V vlc V v τ κ Lorentz number: 8 /. K ohm W e k T ne T k nk T ne mv C T B B B V κ

21 Thermal ondutivity5 Rather good agreement with eperiments! κ T C V ne mv T nk B ne T k B T k e B. 8 W ohm / K Three mistakes made by Drude: a fator of two,..; wrong speifi heat, the right one should be times smaller; wrong v, the right one shoud be times larger. The last two errors anel eah other to give rise to orret results!

22 Thermopower Seebek effet T T > T Q T thermopower q To estimate Q, we onsider the mean eletroni veloity due to T D model: v Q [ v vτ v + vτ ] τv d v τ d dt τ d d dt v dv d D model: v Q d τ dt v T d τ dt v 6 T

23 Thermopower On the other hand, the mean eletroni veloity due to the eletri field is v eτ eτq T m m v Q v Vanishing urrent + eq m + d v 6 dt d Q e dt mv CV ne kb 4 Q.4 V / e K Observed thermopower in metals at room temperaure ~ -6 V/K Drude model fails here!

24 Generalization of Drude model Classial mehanis Quantum mehanis: Sommerfeld theory Free eletron approimation Involve eletron-ion interation: Band theory Independent eletron approimation Involve eletron-eletron interation: Fermi-Liquid theory Loal thermal equilibrium Smaller length sale, non-equilibirum: Mesosopi physis

25 Summary The onepts of relaation time and mean free path Drude formula ne τ m iτ, all oeffiient R y ne Comple dieletri onstant 4πi p ε + τ >> The physial meaning of p Thermal ondutivity κ v τ C V vlc V Wiedemann-Franz law Seebek effet and thermopower