In the case k e T/ a, E(0)=h2k2/2m, E0)=h2(k-2T/a)2/2m -+ degenerate perturbation theory E4()=1E9(8)+E()+(-9(E()+4 Home Page (52.10 Title Page Contents Eg=E+(k)-E(k)=2|V2/ (52.1) Fig.5.2.2 age 18 of 4. Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 18 of 41 Go Back Full Screen Close Quit In the case k ' π/a, E (0) 1 = ~ 2k 2/2m, E(0) 2 = ~ 2 (k − 2π/a) 2/2m −→ E (0) 1 ' E (0) 2 , degenerate perturbation theory E±(k) = 1 2 ( E (0) 1 (k) + E (0) 2 (k) ± E (0) 2 (k) − E (0) 1 (k) 2 + 4|V−2π/a| 2 1/2 ) (5.2.10) Eg = E+(k) − E−(k) = 2|V−2π/a| (5.2.11) Fig. 5.2.2
E E Home Page Titie Contents 3cla-2tla -la o la 2la tla k -ta o /a (a) age 19 of 4. Figure 5.2.2 Bands and gaps in one-dimensional nearly-free electron model for(a) Go Back extended zone scheme, and(b)reduced zone scheme Full Screen Close
Home Page Title Page Contents JJ II J I Page 19 of 41 Go Back Full Screen Close Quit -3π/a k -2π/a -π/a 0 π/a 2π/a 3π/a -π/a k 0 π/a E E (a) (b) Figure 5.2.2 Bands and gaps in one-dimensional nearly-free electron model for (a) extended zone scheme, and (b) reduced zone scheme
Bragg diffraction for -T/a<k<T/a 1k xF()-到等 (52.12) right travelling ng wave v,(0)=[-/2 exp(ik 1k left travelling wave v22= L-1/2 exp[i(k-2r/a). i] Home Page wavefunctions at bz boundaries Title Page Contents v(x)=√2L n(x)±v2x( eirr/a±e-inx/a √2L (52.13) standing waves v+=(2/L)1/2 cos(ma a)and v_=(2/L)1/2 sin(Ta /a) Fig.5.2.3 age 20 of 4. Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 20 of 41 Go Back Full Screen Close Quit Bragg diffraction for −π/a < k < π/a ψ1k = ψ (0) 1k + V−2π/a E (0) 1 (k) − E (0) 2 (k) ψ (0) 2k , (5.2.12) right travelling wave ψ (0) 1k = L −1/2 exp(ikx) left travelling wave ψ (0) 2k = L −1/2 exp[i(k − 2π/a)x] wavefunctions at BZ boundaries ψ±(x) = 1 √ 2L h ψ (0) 1,π/a(x) ± ψ (0) 2,π/a(x) i = 1 √ 2L e iπx/a ± e −iπx/a (5.2.13) standing waves ψ+ = (2/L) 1/2 cos(πx/a) and ψ− = (2/L) 1/2 sin(πx/a) Fig. 5.2.3
y-2 Home Page Title Page Contents v(x) Figure 5.2.3 Bragg Reflection of electron in a periodic structure Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 21 of 41 Go Back Full Screen Close Quit V (x) | |2 | | ψ− 2 ψ+ x Figure 5.2.3 Bragg Reflection of electron in a periodic structure
viewpoint of scattering Periodic potential Strong scattering at k= T/a Gaps open at bz boundaries electronic band structure In high-dimensions, Bragg condition, energy gaps at BZ also create Quantitative details depend on the specific periodical potential various energy band structures Home Page Title Page Contents age 22 of 4. Go Back Full Screen Close
Home Page Title Page Contents JJ II J I Page 22 of 41 Go Back Full Screen Close Quit viewpoint of scattering: Periodic potential → Strong scattering at k = π/a Gaps open at BZ boundaries → electronic band structure In high-dimensions, Bragg condition, energy gaps at BZ also create Quantitative details depend on the specific periodical potential → various energy band structures