1 / 23

[ r ] =  dEG n /2 p,

In summary. We use this in Ch. 10. [ r ] =  dEG n /2 p,. . Where G = [EI – H - S ] -1 , [A] = G G G +. , G n = G S in G +. . . . We will use this in this Chapter. . . . . . . . . Already used in Chapter 1. g 1 f 1 + g 2 f 2. . . N = dE D(E-U).

harrisalbert
Download Presentation

[ r ] =  dEG n /2 p,

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. In summary We use this in Ch. 10 [r] = dEGn/2p,  Where G = [EI – H - S]-1, [A] = GGG+ , Gn = GSinG+    We will use this in this Chapter         Already used in Chapter 1 g1f1 +g2f2   N = dE D(E-U) f1 - f2 g1f1 +g2f2 I1=dET(E) [r]= dE G G+/2p g1 + g2     2q g1g2   I1 = dETr[S1inA-G1Gn] h g1 + g2 2q I1= dE D(E-U)[f1-f2] h where T(E) = Tr[G1GG2G+]

  2. Onto practical applications       f1 - f2 g1f1 +g2f2 I1=dET(E) [r]= dE G G+/2p       where T(E) = Tr[G1GG2G+] What do these equations predict for coherent quantum transport?

  3. G1(1,1) = ħv1/a, G2(N,N) = ħv2/a 1-D wire -t e0 v2 v1 {y} = [G]{S} Only S(1) exists, equal to iħv1/a t = y(N) = G(N,1)S(1) T = |t|2v2/v1 = (ħv1/a)(ħv2/a)|G(1,N)|2 Thus T = Tr[G1GG2G+]

  4. 1-D wire -t e0 What do we expect for DOS, Transmission ? H = e0 S1 = S2 = -teika G1 = i(S1-S1+)= 2tsinka = G2 = G/2 G = (EI-H-S1-S2)-1 = 1/(E-e0+2teika) E = e0 - 2tcoska  ħv = dE/dk = 2tasinka G = (EI-H-S1-S2)-1 = i/2tsinka D = GGG+/2p = 1/2ptsinka = a/pħv T = G1GG2G+ = 1

  5. 1-D wire Like Exam 2

  6. 1-D wire EXPT Halbritter PRB ’04

  7. 1-D wire with weak coupling to contacts -t e0 v2 v1 S1(1,1) = S2(N,N) = -[t’2/t]eika t’ < t isolates wire and makes its levels discrete (‘molecule’) Discrete levels create quantum resonances (standing waves between ends)

  8. 1-D wire with weak coupling to contacts Ignoring Laplace term (‘screening’)

  9. Reed 1-D wire with weak coupling to contacts

  10. 1-D wire with a defect -t e0 e1 H = e1 S1 = S2 = -teika G1 = i(S1-S1+)= 2tsinka = G2 = G/2 G = (EI-H-S1-S2)-1 = 1/(e0-e1-2itsinka) E = e0 - 2tcoska  ħv = dE/dk = 2tasinka G = (EI-H-S1-S2)-1 D = GGG+/2p = ħv/p[ħ2v2 + (De)2] T = G1GG2G+ = 1/[1 + (De/ħv)2]

  11. Analytical Derivation -t e0 e1 y|+ - y|- = 0  t-(1+r) = 0 dy/dz|+ - dy/dz|- = [2mU0/ħ2]y|0  ik[t-(1-r)] = 2mU0t/ħ2 T = |t|2 = ħ2v2/(ħ2v2+U02)

  12. Defect, e0-e1=2 T < 1 More importantly, split off level outside band (that’s how dopants work!)

  13. e0 -t 0 0 Hon = Hoff = e0 -t t 0 g-1 = a – bgb+, a = EI-Hon, b = -Hoff S1 = S2 = bgb+ 1-D wire with a basis -t e0

  14. 1-D wire with a basis

  15. 2-D wire e0 0 -t -t Hon = Hoff = e0 -t 0 -t g-1 = a – bgb+, a = EI-Hon, b = -Hoff S1 = S2 = bgb+ e0 -t e0 -t

  16. Multiple Modes EI = Hon – Hoffeika – Hoff+e-ika  Bands E1 = 2t(1-coska)+t, E2=2t(1-coska)-t

  17. 2-D wire e0 -t e0 -t g-1 = a – bgb+, a = EI-Hon, b = -Hoff S1 = S2 = bgb+

  18. Multiple Modes Bands E1 = 2t(1-coska)-2t, E2,3 = 2t(1-coska)+t

  19. 1-D wire with a barrier -t e0

  20. 1-D wire with a barrier -t e0 Slight subtleties – 2D cross section  sum over transverse modes (f  F)

  21. Double barrier Resonant Tunneling Diode e0

  22. gd=0.1 ech=0 ed=0.5 t= 1 EF=-0.5 g0ch = 1/(EF-ech + igch) g0d = 1/(EF-ed + qVG + igd) 1-D wire with a scatterer g-1ch = (g0ch)-1 – t2g0d Dch = i(gch-gch+)/(2p)

  23. Fano interference between a channel and a quantum dot 1-D wire with a scatterer

More Related