Detemining n and μ: The Hall Effect V x, E x + + + + + + + + + + + --------- E y I, J x F = qe + qv B F y = ev D B z F y = ee y B z In steady state, E Y = v D B Z = E H, the Hall Field Since v D =-J x /en, R H = ne E H = J x B Z = R H J X B Z ne σ = neμ
Expeimental Hall Results on Metals Valence= metals look like fee-electon Dude metals Valence= and 3, magnitude and sign suggest poblems Metal Valence -/R H nec Li Na K Rb Cs Cu Ag Au Be Mg In Al 3 3 0.8...0 0.9.5.3.5 0. -0.4-0.3 0.3 Hall coefficients of selected elements in modeate to high fields* * These ae oughly the limiting values assumed by R n as the field becomes vey lage (of ode 0 4 G), and the tempeatue vey low, in caefully pepaed specimens. The data ae quoted in the fom n 0 :n. Whee n 0 is the density fo which the Dude fom (.) agees with the measued R u :n 0 = -/R H ec. Evidently the alkali metals obey the Dude esult easonably well, the noble metals (Cu. Ag, Au) less well, and the emaining enties, not at all. Table by MIT OpenCouseWae.
Response of fee e- to AC Electic Fields Micoscopic pictue ty e - E = E e iωt Z O B=0 in conducto, dp(t) p(t) iωt = ee 0 e and F(E) >> F(B) dt τ p(t) = p 0 e iωt iωp 0 = p 0 ee 0 τ ee 0 p = ee 0 ω>>/τ, p out of phase with E 0 iω p 0 = iω ω, p 0 τ ω<</τ, p in phase with E p 0 = ee 0 τ
What if ωτ>>? When will J = σe beak down? It depends on electons undegoing many collisions, on the aveage a collision time τ apat. As long as thee ae many collisions pe cycle of the AC field (ωτ<<), the AC σ will be the DC σ. x x x xx x x x x x x x x x x x x x x x x But conside the othe limit: ωτ>>. Now thee will be many cycles of the field between collisions. In this limit, the electons will behave moe like electons in vacuum, and the elation between J and E will be diffeent x x
Complex Repesentation of Waves sin(kx-ωt), cos(kx-ωt), and e -i(kx-ωt) ae all waves e -i(kx-ωt) is the complex one and is the most geneal imaginay A θ Acosθ iasinθ eal e iθ =cosθ+isinθ
Response of e- to AC Electic Fields Momentum epesented in the complex plane imaginay p (ω >>/τ ) p p (ω <</τ ) eal Instead of a complex momentum, we can go back to macoscopic and ceate a complex J and σ iωτ nep 0 ne J (t) = J 0 e J 0 = nev = = m m( iω ) τ σ 0 ne τ σ =,σ 0 = iωτ m E 0
Response of e- to AC Electic Fields Low fequency (ω<</τ) electon has many collisions befoe diection change Ohm s Law: J follows E, σ eal High fequency (ω>>/τ) electon has nealy collision o less when diection is changed J imaginay and 90 degees out of phase with E, σ is imaginay Qualitatively: ωτ<<, electons in phase, e-iadiate, E i =E +E t, eflection ωτ>>, electons out of phase, electons too slow, less inteaction,tansmission ε=ε ε 0 ε = τ 0 4 sec,νλ = c,ν = 3x00 cm / sec 0 4 Hz 5000x0 8 cm E-fields with fequencies geate than visible light fequency expected to be beyond influence of fee electons
Response of light to inteaction with mateial Need Maxwell s equations fom expeiments: Gauss, Faaday, Ampee s laws second tem in Ampee s is fom the unification electomagnetic waves! SI Units (MKS) Gaussian Units (CGS) D = ρ D = 4πρ B = 0 B = 0 B B xe = xe = c t t 4π D D xh = J + xh = J + c c t t D = E + 4πP D = ε 0 E + P = εe B = H + 4πM B = μ 0 H + μ 0 M = μh μ = μ μ 0 ;ε = ε ε 0
Waves in Mateials Non-magnetic mateial, μ =μ 0 Polaization non-existent o swamped by fee electons, P=0 B xe = t E xb = μ 0 J + μ 0 ε 0 t xb x( xe) = t Fo a typical wave, E = E e i(k ϖt ) = E e ik e iϖt = E()e iϖt 0 0 E() = iϖμ 0 σe() μ 0 ε 0 ω E() E() = ω ε (ω )E() c iσ ε (ω ) = + ε 0 ω E = E [μ 0 J + μ 0 ε 0 ] t t E() = E 0 e ik E = μ ω 0 σ E E + μ 0 ε 0 t t k = ε (ω ) c ω c v = = k ε (ω ) Wave Equation
Waves in Mateials Waves slow down in mateials (depends on ε(ω)) Wavelength deceases (depends on ε(ω)) Fequency dependence in ε(ω) ε (ω) = + iσ iσ = + 0 ε 0 ω ε 0 ω( iωτ ) ε (ω) = + ω p = ne ε 0 m iω τ p ω iω τ Plasma Fequency Fo ωτ>>>, ε(ω) goes to Fo an excellent conducto (σ 0 lage), ignoe, look at case fo ωτ<< ε (ω) iω τ p ω iω τ iω τ p ω
Waves in Mateials ω ω σ 0 k = ε (ω ) = i c c ωε 0 ω + i σ σ 0 ω σ 0 ω k = 0 = + i c ωε 0 ε 0 c ε 0 c Fo a wave E = E 0 e i(kz ωt ) Let k=k eal +k imaginay =k +ik i The skin depth can be defined by E = E 0 e i[k z ωt ] e k i z ε c o δ = = = σω σ μ ω k i o o o δ
Waves in Mateials Fo a mateial with any σ 0, look at case fo ωτ>> ω p ε ( ω )= ω ω<ω p, ε is negative, k=k i, wave eflected ω>ω p, ε is positive, k=k, wave popagates R ω p ω
Success and Failue of Fee e- Pictue Success K/σ=themal conduct./electical conduct.~ct Metal conductivity Κ = c Hall effect valence= v v them τ 3 Skin Depth c v = = nk b ;v them = Wiedmann-Fanz law T v m E 3 3k b T Examples of Failue Κ = 3 3k nkb b T 3 nk τ = b Tτ Insulatos, Semiconductos 3 m m Hall effect valence> ne τ σ = Themoelectic effect m Colos of metals Κ 3 k b Theefoe : σ = e T Luck: c veal =c vclass /00; v eal =v class *00 ~C!
Wiedmann-Fanz Success 73K 373K Element Li Na K Rb Cu Ag Au Be Mg Nb Fe Zn Cd Al In Tl Sn Pb Bi Sb k (watt cm-k) 0.7.38.0 0.6 3.85 4.8 3..3 k σ T (watt-ohm K ). x 0 8..3.4.0.3.3.36 k (watt cm-k) 0.73 3.8 4.7 3..7 k σ T (watt-ohm K ).43 x 0 8.9.38.36.4.5 0.5 0.80.3.0.38.4.90.6.8.49.4.5 0.54 0.73..0.30.5.78.88.30.9 0.88 0.5 0.64 0.38 0.09 0.8.58.75.48.64 3.53.57 0.80 0.45 0.60 0.35 0.08 0.7.60.75.54.53 3.35.69 Expeimental themal conductivities and Loenz numbes of selected metals Themoelectic Effect Exposed Failue when cv and v ae not both in popety E = Q T c v nk b nk b Themopowe Q is Q = = = 3ne 3ne e 3 Table by MIT OpenCouseWae. Themopowe is about 00 times too lage!