5.2. Electrons in Crystals rge number of electrons in a so ndependent electron model Periodic ionic potential exists Home Page 5.2.1. Free Electron gas model Title Page For alkali metals such as Li Na. K. noble metals Cu. Ag. Au Contents (r)=0, h2 2m 2(r)=Ev(r) vk(r)=2-1/2eik 2.1 Go Back Full Screen E(k)=h2k2/2m 22) Close Fermi surface EF=E(kF), (523)
Home Page Title Page Contents JJ II J I Page 14 of 41 Go Back Full Screen Close Quit 5.2. Electrons in Crystals A large number of electrons in a solid Independent electron model Periodic ionic potential exists 5.2.1. Free Electron Gas Model For alkali metals, such as Li, Na, K, noble metals Cu, Ag, Au V (r) = 0, − ~ 2 2m ∇2ψ(r) = Eψ(r) ψk(r) = Ω −1/2 e ik·r (5.2.1) E(k) = ~ 2 k 2 /2m (5.2.2) Fermi surface EF = E(kF), (5.2.3)
Fermi wavevector kF 22 dk (2r)3 (524) ∫dk=4π3/3 1/3 Home Page (52.5) Title Page 2/3 Contents 3 (5.2.6) Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 15 of 41 Go Back Full Screen Close Quit Fermi wavevector kF N = X k = 2Ω (2π) 3 Z dk, (5.2.4) R dk = 4πk3 F /3 kF = 3π 2N Ω 1/3 (5.2.5) EF = ~ 2 2m 3π 2N Ω 2/3 . (5.2.6)
Fermi wavevector kF 22 dk (2r)3 (524) ∫dk=4π3/3 1/3 Home Page (52.5) Title Page 2/3 Contents 3 (5.2.6) En(k),vnk(r),-(丌/a)<k<(丌/a) Go Back Fig.5.2.1 Full Screen Close
Home Page Title Page Contents JJ II J I Page 15 of 41 Go Back Full Screen Close Quit Fermi wavevector kF N = X k = 2Ω (2π) 3 Z dk, (5.2.4) R dk = 4πk3 F /3 kF = 3π 2N Ω 1/3 (5.2.5) EF = ~ 2 2m 3π 2N Ω 2/3 . (5.2.6) En(k), ψnk(r), −(π/a) < k < (π/a) Fig. 5.2.1
E E Home Page Title Page Contents 3T/a -2Tla -t/a0 tla 2la 3/a k -/a Figure 5.2.1 Dispersion curves of one-dimensional free electron gas for (a) age I6 of 4. extended-zone scheme, and(b) reduced-zone scheme Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 16 of 41 Go Back Full Screen Close Quit -3π/a -2π/a -π/a 0 π/a 2π/a 3π/a k k E E -π/a π/a Figure 5.2.1 Dispersion curves of one-dimensional free electron gas for (a) extended-zone scheme, and (b) reduced-zone scheme
5.2.2. Nearly-Free Electron Model V(r) as a perturbation (k) (0) Home Page E的E)+(8+>2=k(52n Title Page Contents k wnk=vnk+ n'k/ (528) En '(k)-ent(k') one-dimensional case. n age I7 of 4. E1(k)≈E(k)+ (529) Go Back Full Screen V(a)e/adx Close
Home Page Title Page Contents JJ II J I Page 17 of 41 Go Back Full Screen Close Quit 5.2.2. Nearly-Free Electron Model V (r) as a perturbation ψ (0) nk , E (0) n (k) En(k) = E (0) n (k) + D ψ (0) nk |V | ψ (0) nk E + X n0k0 0 D ψ (0) n0k0 |V | ψ (0) nk E 2 E (0) n (k) − E (0) n0 (k0 ) (5.2.7) ψnk = ψ (0) nk + X n0k0 0 D ψ (0) n0k0 |V | ψ (0) nk E E (0) n (k) − E (0) n0 (k0) ψ (0) n0k0. (5.2.8) one-dimensional case, n = 1, n 0 = 2 E1(k) ' E (0) 1 (k) + |V−2π/a| 2 E (0) 1 (k) − E (0) 2 (k) , (5.2.9) V−2π/a = 1 L Z V (x)ei2πx/adx