5.2.3. Tight-Binding Electron Model 3d bands in transition metals V(r) very strong Wannier function (r-l -ik. L wnkr (5214) Home Page Title Page ank(r)=N-1/2\ ik lwn(r (5 2.15 Contents Wannier functions. orthogonalized for different n and l l) is a localized function Take a bloch function in a band age 23 of 4. vk(r)=N-1/(r)eik (5.2.16) Go Back assume same u(r) Full Screen For cubic lattice. a. Wannier function at the origin Close sin(r /a)sin(Ty/a)sin(Tz/a) u(r (rr/a(ry/a( (5217)
Home Page Title Page Contents JJ II J I Page 23 of 41 Go Back Full Screen Close Quit 5.2.3. Tight-Binding Electron Model 3d bands in transition metals V (r) very strong Wannier function wn(r − l) = N −1/2X k e −ik·lψnk(r) (5.2.14) ψnk(r) = N −1/2X l e ik·lwn(r − l) (5.2.15) Wannier functions, orthogonalized for different n and l, wn(r − l) is a localized function Take a Bloch function in a band ψk(r) = N −1/2 u(r)eik·r , (5.2.16) assume same u(r) For cubic lattice, a, Wannier function at the origin w(r) = sin(πx/a) sin(πy/a) sin(πz/a) (πx/a)(πy/a)(πz/a) u(r) (5.2.17)
decreasing oscillate When a not small. to use atomic wavefunction u(r-U)≈a(r-U) k(r)=N-12∑ca(r-l) (5.2.18) Home Page Title Page kV+V(r)wrdr Contents E(k)= ∫ kynar (5.2.19) /%(+h)h2 V+V(r)Spa(r)di (5.220) age 24 of 4. h2 V+va(r)da(r)=Faca(r) Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 24 of 41 Go Back Full Screen Close Quit decreasing oscillated When a not small, to use atomic wavefunction w(r − l) ' φa(r − l) ψk(r) = N −1/2X l e ik·lφa(r − l) (5.2.18) E(k) = R ψ ∗ k n − ~ 2 2m ∇2 + V (r) o ψkdr R ψ ∗ kψkdr ' X h e ik·hEh (5.2.19) Eh = 1 Ωc Z φ ∗ a (r + h) − ~ 2 2m ∇ 2 + V (r) φa(r)dr (5.2.20) − ~ 2 2m ∇2 + va(r) φa(r) = Eaφa(r) (5.2.21)
decreasing oscillated When a not small. to use atomic wavefunction u(x-)≈oa(r-U) k(r)=N-1/ 1/∑c(r-D) (5218) Home Page h2 Titie +v(r)kdr E(k) ∫vv (52.19) Contents Q a(r+ 2v2+v(r)oala 2m (5220) 2 V+va(r) Oa(r)= Eapa(r) (5.2.21) age 24 of 4. Go Back For a simple cubic lattice with h=(a, 0,0),(0, a, 0) and(0, 0, a) Full Screen E(k)a Ea+ 2 E1oo(cos akr cos aky +cos akz) .2.22 Close BZ is a cube, bandwidth 12 Flool, lowest energy Ea +6E100 Fig.5.2.4
Home Page Title Page Contents JJ II J I Page 24 of 41 Go Back Full Screen Close Quit decreasing oscillated When a not small, to use atomic wavefunction w(r − l) ' φa(r − l) ψk(r) = N −1/2X l e ik·lφa(r − l) (5.2.18) E(k) = R ψ ∗ k n − ~ 2 2m ∇2 + V (r) o ψkdr R ψ ∗ kψkdr ' X h e ik·hEh (5.2.19) Eh = 1 Ωc Z φ ∗ a (r + h) − ~ 2 2m ∇2 + V (r) φa(r)dr (5.2.20) − ~ 2 2m ∇2 + va(r) φa(r) = Eaφa(r) (5.2.21) For a simple cubic lattice with h = (a, 0, 0), (0, a, 0) and (0, 0, a) E(k) ' Ea + 2E100(cos akx + cos aky + cos akz). (5.2.22) BZ is a cube, bandwidth 12|E100|, lowest energy Ea + 6E100. Fig. 5.2.4
Home Page Title Page Contents age 25 of 4. Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 25 of 41 Go Back Full Screen Close Quit
E Home Page Title Page Contents age 26 of 4. Go Back Full Screen Close atom SO Figure 5.2. 4 Atomic levels spreading into bands as lattice separation decreases
Home Page Title Page Contents JJ II J I Page 26 of 41 Go Back Full Screen Close Quit atom solid E 1/a Figure 5.2.4 Atomic levels spreading into bands as lattice separation decreases