Optical and Electronic Properties of InGaAs and Nitride Quantum Dots
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(7) Multi-Exciton Spectra in InGaAs Quantum Dots.
(8) Multi-Exciton Spectra in Nitride Quantum Dots. 3X 4X 5X 6X. → 2X → 3X → 4X → 5X. Semiconductor Luminescence Equations. gijξ.
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(11) δ.
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(17) k·p.
(18) k·p. p.
(19) Hξα = εα ξα. Γ. H=−. 2 Δ + Vλ (r) 2mλ.
(20) ΔEλ 0 ; r Vλ (r) = ΔEλ ; λ=e. λ=h. r. = 16. h = 4.0 z0 = 1.6 ΔEe = 350 me = 0.065 m0.
(21) 350 300 250 200 150 100 50 0 0. 1. 2. 3. |m| ω. = 38.578. 4.
(22) 0.3. 0.2 0.15. 0.2. |m| = 0. 0.1. |m| = 1. 0.1 0.05 0. 0. −0.05. −0.1 −0.1. −0.2 0. 5. 10. 15. 20. 25. 30. −0.15 0. 5. 10. 0.14. 20. 25. 30. 0.14. 0.12. 0.12. 0.1. |m| = 2. 0.1. 0.08. 0.08. 0.06. 0.06. 0.04. 0.04. 0.02. 0.02. 0 0. 15. ρ. ρ. 5. 10. 15. 20. 25. 30. 0 0. |m| = 3. 5. 10. 15. 20. 25. ρ. ρ. k.p ∗ ξi,e = ξi,h = ξi. ∗ ξi,e = ξi,h. H. i. 30.
(23) H0. (e,h) εi. H. H0 =. . εei e†iσ eiσ +. iσ. HCoul. . εhi h†iσ hiσ. ,. iσ. 1 ee † † 1 hh † † = Vij,kl eiσ ejσ ekσ elσ + V h h hkσ hlσ 2 ijkl 2 ijkl ij,kl iσ jσ σσ . −. . σσ . he Vij,kl h†iσ e†jσ ekσ hlσ .. ijkl σσ . eiσ e†iσ. σ |i. εei εhi. hiσ h†iσ r|i, λ = ψi,λ (r) λλ 3 Vij,kl = d r d3 r ψiλ∗ (r)ψjλ ∗ (r )V (r − r )ψkλ (r )ψlλ (r) V (r) = e2 /4π 0 r r. λ = e, h. r. ξiλ (r)uλk =0 (r). ξiλ (r). he, Vij,kl. =. . d3 r. uλk (r) . ψiλ. d3 r ψih∗ (r)ψje∗ (r)V (r − r )ψke (r )ψlh (r ) uλk (r). k ·p. λλ. he, Vij,kl . λλ Xij = Vij,ij ∗ ξi,e = ξi,h hh ee Vij,kl = Vkl,ij. he ee and Vij,kl = Vlj,ki. ..
(24) me = 0.065 m0 mh = 0.17 m0. r = 13.69 losc = 5.4 m0. ee Vij,kl ∝ δmi +mj ,mk +ml. . 2. 2 /me losc. ωe =. = 40.20 ωh = 15.37 Ec = e2 /4π 0 r losc = 19.48 s m=0. s. p |i m = ±1. p. (αi , αj , αk , αl ). z p. Vαi αj ,αk αl /Ec. Ec = e2 /4π0 r losc m=0 s m = ±1.
(25) Ne. . |φ = P i. . (e†i )ni. e. i nei =Ne. P j. Nh. (h†j )nj |0 . h. j nh j =Nh. {|φi }. H = H0 + H. φi |H|φj |ψ. I(ω) = |ψi Ei. 2π |ψf |H f. |ψ =. i. αi |φi . |ψi |2 δ(Ei − Ef − ω) .. |ψf . Ef.
(26) H =−. H. n,m. |n dcv. |m. † † Edeh nm en, −σ hm, σ + n|er|m. deh nm dcv n|m E. e. deh nm. n|er|m ∝ δnm P=. . Pσ = d. σ. i,σ. d P z P†. . hi,σ ei,−σ.
(27) s. p. z lze + lzh S2e. Sez. z S2h. Ne , Nh , lze + lzh , S2e , S2h , Sez , Shz Shz.
(28) Ne = Nh = N X. NX. NX NX Sz z. z lz. ,e. + lz. lz =. ,h. s p s. s. p. 2(ωe + ωh ). p. ωe + ωh p 2:1. s.
(29) V = 0. V =0 lz. e ϕe +1 , ϕ−1. ϕe 0. = 0, ±2. lz. = ±1. lz lz lz. =0 = ±2 =0. lz. = ±1. lz. = ±1. lz. =0 lz. = ±1. lz. =0. p. s. ϕh 0. h ϕh +1 , ϕ−1. z. lz. = lz. ,e. + lz. ,h. s. p. |f 1X |0|P|f |2. |f . |0. P Sz. z +Sz. ,e. ,h. =0. lz. ,e. z +lz. ,h. =0.
(30) |ψa Sz. Sz. ,e. + Sz. |ψb . ,e. ,h. =. lz ,e +lz ,h = 0 Sz ,h = − 12. 1 2. =0. lz Sz. z Sz. ,h. =. |ψc . ,e. ,e. + lz Sz. ,h. Sz. ,e. ,h. +Sz. =0. =0. ,h. Sz. − 12. ,e. =. 1 2. P†. ⎛ ⎜ H=⎜ ⎝. ⎞. εes + εhs − Dss. −Xsp. −Xsp. −Xsp. εep + εhp − Dpp. −Xpp. −Xsp. −Xpp. εep + εhp − Dpp. Xij = Vijij. Dij = Vijji. εep + εhp − Dpp + Xpp. |ψa + |ψb + |ψc + |ψ {|ψa , |ψb , |ψc } {|ψa , |ψb , |ψc } |f |P † |0|2 = |α + β + γ|2 (α, β, γ)t = (0, −1, 1)t. ⎟ ⎟ . ⎠. (0, −1, 1)t. |ψ |ψ . |0|P|f |2 = |f |P † |0|2 P † |0 =. ˜ = α|ψa + β|ψb + γ|ψc |f.
(31) V =0. V = 0. 2.5. X (×2). XX. 200. X. 150. XX. 2. 100 50. NX = 2. 1.5. 0. 120. V =0. V = 0 1. 100 80 60. 0.5. 40 20. NX = 1. 0. 0. 0. 10. 20. 30. ω/. 1X. ωe + ωh BXX = E (2X) − 2E (1X) , −1.90. E (1X). E (2X). 40. 50. 60. 70.
(32) BXX = −1.96 −2 −3. NX N. X. Se = Sh = 1/2 1/2 z. Sz. ,e. + Sz. ,h. Se = S h = 0.
(33) 6X → 5X 5X → 4X 4X → 3X 3X → 2X 2X → 1X 2. 1X → 0X. 1 0. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. ω/. z Sz. ,e. + Sz. ,h. =0. s ω <. s. ω >. p p. 50. s p. S2e , S2h. P.
(34) s. p.
(35) s. p. p. s p. Vij,ji = Dij = Dji H. n ˆ hiσ = h†iσ hiσ. n ˆ eiσ = e†iσ eiσ.
(36) i = j. Vij,ij = Xij = Xji hh. H. n ˆ ei. n ˆ ei↑. +. ee σ = σ. 1 1 e e h εi − Dii n ˆi + εi − Dii n ˆ hi = 2 2 i i e. . 1 ˆi − n ˆ ej − n + Dij n ˆ hi n ˆ hj 2 i,j e e. 1 ˆ i,σ n . − Xij n ˆ j,σ + n ˆ hi,σ n ˆ hj,σ 2 i,j,σ. n ˆ ei↓. n ˆ hi. n ˆ hi↑. i=j +n ˆ hi↓. neiσ = nhi,−σ = nX iσ E=. . X εei + εhi − Dii nX Xij nX i −2 i,σ nj,σ i. .. i<j σ. |i, σ. nX i,σ. EiX = εei + εhi − Dii. s p. s p. p. s s. p.
(37) −−. Dij Dii. Dii Dij. Dii. EiX. |i. Xij Xsp. Xpp ⎞. ⎛ Vijij. Dss Xsp Xsp. i,j. ⎟ ⎜ ⎟ , =⎜ X D X sp pp pp ⎠ ⎝ Xsp Xpp Dpp. i = j EsX EpX Xsp Xpp X X Es +Ep −2Xsp. s p s. z. z.
(38) 6X → 5X 5X → 4X 4X → 3X 3X → 2X 2X → 1X 2. 1X → 0X. 1 0. 0. 10. 20. 30. 40. 50. 60. 70. 80. ω/. p. p s. s Xij s s. p. s p 2Xsp. s s. 90.
(39) 1.5. s. p. 1. 0.5. 0. 0. 10. 20. 30. 40. 50. 60. 70. 80. ω/. 3X → 2X s. p 2Xpp. H ψi |H|ψi . |ψi . 90.
(40) 0.9. p 0.8. 0.7. 0.6. s 0.5. 0.4. 0.3. 0.2. 0.1. 0. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. ω/. 4X → 3X. H. H. p p. p p.
(41) ,P. [H. ] = −E˜ X P P. i |ψ, NX +1 NX + 1 P |ψ, NX + 1 E|ψ,NX = E|ψ,NX +1 − E˜ X a ˆ. H. |ψ, NX |P E˜ X. P. H. E|ψ,NX +1 H. [H. ,a ˆ] = −ˆ a. . |ψ, NX + 1 = [H , P ] + P H |ψ, NX + 1 . X = − E˜ + E|ψ,NX +1 P |ψ, NX + 1 .. |ψ, NX + 1|. 2. P. H h i,σ iσ ei,−σ. .
(42) p. s s. p s s s s. p. s. s. (s). s. p. (p). (sp). Had = Had + Had + Had sp p i, j, k, l p. p s. s. (p). Had (p). Had =. . 1 1 Vij,kj e†iσ ekσ + εei n ˆ hiσ − Vij,kj h†iσ hkσ 2 2 iσ iσ ijkσ ijkσ 1 + Vij,kl e†i,σ el,σ − h†l,σ hi,σ e†j,σ ek,σ − h†k,σ hj,σ , 2 ijkl εei n ˆ eiσ −. σσ . s sp (s) Had. =. εes. e 2 1 1 1 e h ˆs − n − Dss n ˆ hs , ˆ s + εs − Dss n ˆ hs + Dss n 2 2 2. e e. (sp) ˆs − n ˆp − n n ˆ es,σ n Had = Dsp n ˆ hs n ˆ hp − Xsp ˆ ep,σ + n ˆ hs,σ n ˆ hp,σ , σ. n ˆ e,h s,p. s (εes +εhs −Dss )nX s,σ −2Xsp σ n ˆX ˆX s,σ n p,σ. p.
(43) i=k. Vijkj ∝ δmi +mj ,mk +mj = δmi ,mk Vijkj ∝ δi,k. εe,h i p. i. (p) Had. =. (p). Had. 1 1 1 1 e h − Dpp − Xpp n ˆ p + εp − Dpp − Xpp n ˆ hp 2 2 2 2 1 + Vij,kl e†i,σ el,σ − h†l,σ hi,σ e†j,σ ek,σ − h†k,σ hj,σ . 2 ijkl. εep. σσ . Had s 2Xsp p. H Had. Pp. p p. e†i,σ ej,σ − h†j,σ hi,σ , Pp = 0 Pp Pp p. Pp. . n ˆ e,h p , Pp = −Pp Pp. s. e,h n ˆ s , Pp = 0 p s. ne,h s,σ = 1. [Had , Pp ] = − εep + εhp − Dpp − Xpp − 2Xsp Pp. . p.
(44) 6X → 5X 5X → 4X 4X → 3X 3X → 2X 2X → 1X 2. 1X → 0X. 1 0. 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. ω/. EpX = εep + εhp − Dpp − Xpp − 2Xsp = EpX − Xpp − 2Xsp. ,. EpX = 82.5 meV p s.
(45) = εe. H. 1 1 Wi n ˆ e + εe n ˆh − Wi n ˆh 2 i 2 i 1 + Vij,kl e†i,σ el,σ − h†l,σ hi,σ e†j,σ ek,σ − h†k,σ hj,σ 2 ijkl n ˆe. −. σσ . p. [P. ,H. n ˆe n ˆ ei,↑ + n ˆ ei,↓. n ˆe ] (εe. + εh. i ¯ W. =. 1 N. i. −. )P p. . Wi i. j. Wi e†i h†i. Vijij. Wi. E˜ X P Wi Wi. N ˜ E. ≈ εe. + εh. ¯ −W. . ¯ W. s ¯ W. p. = Dpp − Xpp −2Xsp s. −2Xsp. p. = Dss. p. s s p p sp. p.
(46) d ±2 s d. p. p Pp† s. p. {εci }i. cv {εei }i. {εvi }i P. S−. eh. s. {εhi }i.
(47) n. μ p μ. 1 2.3.
(48) XX. 3 2 1. Slope (X) = 0.95 + 0.05. 4. 3. Slope (XX) = 2.31 + 0.05. 2 1. 0. Z. quasi−resonant excitation. 2 1. 2. Y XX. X. 0 1.334 1.336 1.338 1.340 1.342. Energy (eV). 400. 3. 2. log Power (W/cm ). 1.31. 4. Cross-Corr. 1.4. Z. Y. 1.2 1.0. 300. 0.8 Poisson. 200. 0.50 -20 -10 0. 10 20 30 40 50. 0.6 0.4. Delay τ (ns). μ p. IXX ∝ I 2. (2). X. Correlation Counts n(τ). PL Intensity (10 3 arb. units). 4. Off-resonant Excitation 5. Correlation Function g (τ). off−resonant excitation. log Intensity (arb. units). 5.
(49) τX ≈ 1.0. τXX ≈ 500. ΔEX−XX = 4.2 ± 0.1. p. 50/50. ω. ω τ. n(τ ) τ τ 20 5 15. τ = −15.
(50) t = t+τ. ω. ω n(τ ). t. 50/50 n(τ ). τ. n. p s. eh eh. X. XX.
(51) s. p X − XX −. X + XX +. +. X +. XX+. XX X+. −. XX. −. X. −. XX. −. X. 2. XX. X. 1. 0. 31. XX. 2. 32. 33. X. 1. 0 21. 22. ω/. 23. 24. 25. ω/. X. le eh. =6. XX. , lh. =7.
(52) X −∗. XX. 2. −. XX. X. 1 −*. X. 0 28. 29. 30. 31. 32. 33. 34. ω/ XX − → X −∗ → e. s. p. > 96% A EA = 2EsX +εep −Xsp εep. s. EsX Xsp. p s. p.
(53) X −*. XX −. A. EB = EsX + εep − Xsp. XX − → X −∗ → e B. EC = εep. s ≈2. p. ≈ 1.2. EA = 2EsX − Δ + εep − Xsp. Δ. EXX − = EA − EB = EsX − Δ EX −∗ = EB − EC = EsX − Xsp . ≈ 4.9. Xsp Δ. eh C. Xsp − Δ EX −∗ < EXX −. sp. p.
(54) sp −∗. B1. B2 B. s. B1. p B1. B2. eh B1. B1 B2. A B. S e = 1, Sze = 1. B1 e. S =0. p. B e S = 1, Sze = 0. B2 B1. B2 sp.
(55) p p p Xsp −∗. −∗.
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(57) d. d 24 = 16. NX −1. s d. ωe + ωh p 3(ωe + ωh ). 2(ωe + ωh ) 1:2:3.
(58) |g lz. |d, |e, |f . |b, |c. |a. |a. |h ,e. + lz. ,h. = 0 Sz. |g. |f . ,e. =. 1 2. Sz. ,h. |g. |h. |f . = − 12. |a. |h. |f . |a. s p s. d d. p p. d. 120. 8×8 5. d. p. 4. 1. 0.1. s. 3. 0.01. 2. 0. 100. 50. 150. 200. 1. 0 20. 40. 60. 80. 100. 120. 140. ω/. s p. d. 160. 180.
(59) |a. |h. 3×3. (ωe + ωh ) , 2(ωe + ωh ) , 2(ωe + ωh ) , 3(ωe + ωh ), 3(ωe + ωh ) , 3(ωe + ωh ) , ωe + 3ωh , 3ωe + ωh .. V0 =. π 2. Ec =. π. e2 2 4πεr ε0 lk. ⎡. H. 1. ⎢ ⎢ 1 ⎢ ⎢ 4 ⎢ ⎢ 1 ⎢ 4 ⎢ ⎢ ⎢ 3 ⎢ 32 = −V0 ⎢ ⎢ 3 ⎢ ⎢ 32 ⎢ ⎢ 3 ⎢ 16 ⎢ ⎢ ⎢ −1 ⎢ 4 ⎣ − 14. t = ωe + ωh. 1 4. 1 4. 3 32. 3 32. 3 16. − 14. − 14. 11 16. 3 16. 31 128. 15 128. 7 64. 1 16. 1 16. 3 16. 11 16. 15 128. 31 128. 7 64. 1 16. 1 16. 31 128. 15 128. 585 1024. 105 1024. 57 512. 9 128. 9 128. 15 128. 31 128. 105 1024. 585 1024. 57 512. 9 128. 9 128. 7 64. 7 64. 57 512. 57 512. 153 256. 1 16. 1 16. 9 128. 9 128. 7 − 64. 11 16. 3 16. 1 16. 1 16. 9 128. 9 128. 7 − 64. 3 16. 11 16. +. |φn |P |0|. 2. 7 7 − 64 − 64. ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦. 8×8.
(60) |φn |A = |a, 1 |B = √ (|b + |c), 2 1 |C = √ (|b − |c), 2 1 |D = √ (|d + |e + |f ), 3 1 |E = √ (|d − |e), 2 1 |F = √ (|d + |e − 2|f ), 6 1 |G = √ (|g − |h), 2 1 |H = √ (|g + |h). 2 [A, B, H, D, F, G, C, E] ⎡. t−1. ⎢ √ ⎢ ⎢ − 42 ⎢ √ ⎢ ⎢ 42 ⎢ √ ⎢ ⎢ − 3 8 ⎢ H=⎢ √ 6 ⎢ ⎢ 32 ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎣ 0. −. √ 2 4. √. 2 4. 2t−. 7 8. − 18. − 18 2t− √. 7 8. √. √. 3 8 √ − 5326 √ − 966. −. 6 32 √ − 3643 √ − 231923 √ 3 2 256. 0. 0. 0. 0. 0. 0. ΔE. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. √ − 5326 √ − 3643. √ − 231923. √ 3 2 256. 0. ΔE. 0. 0. 0. 0. 0. 0. 0. 2t−. 0. 0. 0. 0. 0. − 18. − 966. 3t−. 51 64. 3t−. ΔE = ωh − ωe 6×6. ΔE = 0. 243 512. 2t−. 6×6. 2×2 5×5. 2×2 2×2. P + |0 = |A +. √. 2|B +. √. 1 2. 1 2. ⎤. − 18 3t−. 15 32. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ , ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦. V0 1×1 P + |0. 3|D |C. |E. |A, |B.
(61) |D. +. |f |P |0|. |f . 2. 1, 2. 3 5×5. 3×3. 1 [ g|H0|g + g|H0|h − h|H0 |g − h|H0|h ] 2 1 = [ g|H0|g − h|H0|h ] 2 1 = [ (3ωh + ωe ) − (ωh + 3ωe )) ] = ωh − ωe = ΔE . 2. G|H0|H =. 5×5. 3×3. 5. 4. 1. 3. 0.1. 2. 0.01 0. 100. 50. 150. 200. 1. 0 20. 40. 60. 80. 100. 120. 140. 160. 180. ω/. ωh = ωe = t/2. t. ωh. ωe. ΔE = 0. s d p |g. |h. |b. |c.
(62) Sz. = 0. ω < 40. s 90 ω > 130. p d. p d p. s. p d. s p. p d p. d d.
(63) 12X → 11X 11X → 10X. 10X → 9X 9X → 8X 8X → 7X 7X → 6X 6X → 5X 5X → 4X 4X → 3X 3X → 2X 2X → 1X 1X → 0X. 1 0 0. 20. 40. 60. 80. 100. 120. 140. ω/ s p d =0. Sz s. p. d. s. d. p.
(64) 6X → 5X. 5X → 4X. 4X → 3X 3X → 2X 2X → 1X 1 0. 1X → 0X 0. 20. 40. 60. 80. ω/. 3X. d sp. spd. 3X. |ψa .
(65) Hes/t =. . ee εes + εep + Dsp. ee −Xsp. ee −Xsp. ee εes + εep + Dsp. ee ee )1− Xsp = (εes + εep + Dsp. 0 1. ee ee Es/t = εes + εep + Dsp ± Xsp. ee 2Xsp. hh 2Xsp. .. 1 0. ee 2Xsp. ts. st. hh 2Xsp. ee Xsp. d |ψa . |ψb . |ψc d. E. 1+A. A. |ψa . |ψb . |ψc . |ψa 3X → 2X |ψb . d. d. |ψc . |ψa . 4X s.
(66) ←3 ←2. Pp. Ps. ←3. p. ← 3×3 ← 2×2. s. s p 3 + 3 × 3 × 4 = 39. 2×3 = 6. s. s. d p. s. d. p. d d. s. p. s p.
(67) p d. p. 12X → 11X. 11X → 10X. 10X → 9X. 9X → 8X 8X → 7X 2 1 0. 7X → 6X 0. 20. 40. 60. 80. 100. ω/ Sz. =0 p. d p s p s. s p. 120. 140.
(68) s 2Xsp p. Xpd Xp+ d− Xp+ d0 p− lz = +2, 0, −2. +, 0, −. z +, −. d. m = ±2. m=0. Xp+ d+ lz = +1, −1. p. d 2Xp+ d− + 2Xp+ d0 + 2Xp+ d+ = 18.16. d d. p [H, Pp ]. Pp s. p. d. p d. (s). (p). (sd). (pd). (d). Had = Had + Had + Had + Had + Had. s. p d. (s) Had. [Had , Pd ] s p. (p) Had. Pd. d (sd) Had. d (sd). Had =. . s. e e Dsdj n ˆ hs (ˆ ˆ hdj ) − ˆs − n ndj − n. j∈{+,0,−}. . Xsdj (ˆ nes,σ n ˆ edj ,σ + n ˆ hs,σ n ˆ hdj ,σ ) .. j∈{+,0,−},σ. s Pd. Xs,d+ Xsdj. n ˆ es = Xs,d− = Xs,d0. e e. ¯ sd n ˆ d,σ + n ˆ hs,σ n ˆ hd,σ X σ ˆ s,σ n.
(69) Pd d s d. p s. p i ∈ {+, −}. Xpi,dj. j ∈ {+, 0, −}. d (d) Had. (d). d. 1 1 Wi n ˆ e + εe n ˆh − Wi n ˆh 2 i 2 i 1 + Vij,kl e†i,σ el,σ − h†l,σ hi,σ e†j,σ ek,σ − h†k,σ hj,σ 2 ijkl. Had = εe n ˆe −. Wi i. σσ . j. (d). Vijij. Wi e†i h†i. d. d. s. W+ = W− = W0 × Pd p. [Pd , Had ] (εed + εhd )Pd −. d d Xsdj. Xpi ,dj. ¯ pd X. j ∈ {+, 0, −} Wi d. ¯ sd X ¯ W. ¯ − 2(X ¯ sd + 2X ¯ pd ) EdX ≈ εed + εhd − W ¯ sd + 2X ¯ pd ) −2(X p. ωe + ωh. p. s EdX. ≈ 133.
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(73) 2 0.7.
(74) s. s. p. p dij = ψie |er|ψjh . dij ∝ δi,j. s p. s p. p.
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(77) k·p.
(78) V |α, R. α R , α | −. R. t. i−1. 2 Δ2 2m. +V. (r)|R, α. t. i+1. i. t. α |Rα. Ri , αi | −. ε δij + t δi,j±1. t. 2 Δ2 2m. +V. ε H=. . ε|ii| + t. i. |ψl =. i 2πl m N. e. El. |m ,. m. l m. N 0. N −1. |i + 1i| + |i − 1i|. i. |ψl . . 2π l . El = ε + 2t cos N. i (r)|Rj , αj .
(79) N →∞. 2π l N. k. k=0. El. El ≈ ε + 2t − 2tk 2 t. ε + 2|t| ε ε − 2|t|. −N/2. +N/2. 0 l. t<0. s p N N ×N. N. tA. tA. tB. tB. ×N. ×N. tA. tA.
(80) εA − εB. εi. α=s. p. ΔV. sp3 |α, R α = p x , py , pz. s R. EαR,α R = αR|H. |αR .. H EαR,α R ∼ 10 meV. 16 × 16 N. H (k) ×N EαR,α R Γ. ψi (r). k = 4 × 4 = 16.
(81) z. φαR (r) = r|α, R ψi (r) =. . R. ciRα. ciRα φαR (r).. Rα. . α , R|H|α, RciRα − E i ciR α = 0,. Rα. Ei α , R|H|α, R. 0.5. 106 ×106.
(82) c. P P φp Vp (r) = −eφp (r) Δ φp = − 01r ρp. P = −ρp. ρp. r P P. P P P. =P. ez P. ∼ ez. 5.5 MV/cm. e ψ1,2,3. c 1.609. 1.6259. s. a. a c. =. p. 8/3 ≈ 1.633.
(83) electrons E1e = 1.7320 eV px : 0.051 py : 0.051 pz : 0.144 s : 0.754. E2e = 1.9621 eV px : 0.041 py : 0.136 pz : 0.118 s : 0.705. E3e = 1.9621 eV px : 0.136 py : 0.041 pz : 0.118 s : 0.705. Y X. holes E1h = 0.6256 eV px : 0.499 py : 0.499 pz : 0.001 s : 0.000. E2h = 0.6158 eV px : 0.144 py : 0.834 pz : 0.016 s : 0.006. E3h = 0.6158 eV px : 0.834 py : 0.144 pz : 0.016 s : 0.006. 10 % p. E1,2,3. 50 %.
(84) 5% R, R Vijkl ≈. . j∗ k l ci∗ Rα cR β cR β cRα. RR αβ. α, β e20 . 4πε0 εr |R − R |. i, j, k, l i, l. e0. j, k. ε0. εr = 8.4 e h deh ij = e0 ψi |r|ψj .
(85) C∞v z z xy C∞v k·p. C3v C2v C3v C2v.
(86) y. x sp3. c. a. z. z sp3. IC2x (x, y, z) → (−x, y, z). x. π x.
(87) sp3. z. z. sp3. z C2z z. π 2. IC4z.
(88) y 2π 3. z. 4π 3. C3v. 2π 3. z. C3z IC2y. 2π. x. y. C3v. A. B. H A [H, B] = 0. A B H A. B. H. [A, B] = 0 H. A. B p J2. Jx Jy. Jz.
(89) x. M yR. x. y. x. = My. y. =. x. y. =. x. y. =R. y. (x, y, z) → (x, −y, z). J2. (2j + 1). p p. C3v C2v. p. C3z. =RM y. y. My. R. 2 j(j + 1). x.
(90) C3v. C3v. C2v. D2d. D2d IC4z. (T ψ)(r) = ψ(T −1 r) (Rψ)(r) = ψ(R−1 r) Rψ −1 ψ R r. T. R−1 r. r. s. p |φ =. r. . |sR a(R) +. R. r|sR = ψs (r − R) r|piR = ψpi (r − R) p. R . |pj R bj (R) ,. R,j. s i ∈ {x, y, z}. R p. p± =. √1 (px 2. R ± ipy ) R. y MR 2π. M y |p+ = |p− ,. 2π. M y |p− = |p+ .. R|p+ = e−i 3 |p+ , R|p− = e+i 3 |p− ,.
(91) C3v s |φs (r) s. pz |φp± (r). p± px. |φpy (r). . |φpx (r). py. a(R). |φs (r) =. |φp± (r). {bi (R)}3i=1 s |φs (r). ∗ |sR α(R) + |pz R βz (R) + |p+ R ZR β(R) + |p− R ZR β ∗ (R). R. R. α(R) βz (R). β(R). C3v ZR. β(R) . 2 XR + YR2 eiφR R φR. XR + iYR. XR. YR. R s α(R) = α(R) |p+ . s C3v. 2π. e−i 3. pz α(R) βz (R) C3z ∗ ZR. . +i 2π 3. β(R ) = β(R). ∗ ZR e. ∗ ZR . R = R−1 R . 2 2π 2 iφR ∗ 2 = XR + Y R e = XR + YR2 e−i(φR − 3 ) = |p− . s s. |φs (r).
(92) s |s. |pz . s. ∗ 2 2|ZR β(R)|2 = 2(XR + YR2 )|β(R)|2 ∗ ZR β(R) |p+ R → 0. β(R). |φp± (r) px. py. |φpx (r). 2 3 p |φpx (r). . |φp± (r) ∝ |φpx (r) ± i|φpy (r) E1 = 1.7320 eV. E2,3 = 1.9621 eV. E1 = 0.6256 eV. E2,3 = 0.6158 eV. |φpx (r)+i|φpy (r) |φpx (r) |φpy (r) √1 |φpx (r) − i|φpy (r) 2 C3z √1 2. |φpx (r). . |φpx (r). p |φpy (r). Epx. . H(R|φpx (r)) = H cos θ|φpx (r) + sin θ|φpy (r) = Epx cos θ|φpx (r) + sin θ H|φpy (r) . . H R|φpx (r) = R H|φpx (r) = Epx cos θ|φpx (r) + sin θ|φpy (r) . 2π 3.
(93) H|φpy (r) = Epx |φpy (r) Epx = Epy. C2z R|φpx (r) = −|φpx (r) Epx Epy. |φpy (r). edeh ij. e. xy. e z√ e = 1/ 2(1, 1, 0) deh ij. deh ss s i, j ∈ {x, y}. deh pi pj. ⎛ deh ij. 0 ⎜ eh ⎜ = ⎝ dp + s deh p− s. deh sp+. deh sp−. 0. 0. 0. 0. ⎞ ⎟ ⎟ . ⎠. s p λz |x ± iy|λz . 2π 3. p. s 2π. λz = 0. s. λz = ±1. e−iλz 3 p±. −1 −1 λz |x + iy|λz = λz |R+ R+ (x + iy)R+ R+ |λz . −1 = e+i(λz −λz )θ λz |R+ (x + iy)R+ |λz . = e+i(λz −λz −1)θ λz |x + iy|λz , R+. θ=. 2π 3. −1 R+ (x + iy)R+ =.
(94) eiφ (x+iy). λz |x+iy|λz . (λz − λz − 1)θ λz −. λz. λz − λz − 1 = 0. deh s,pi C3z. 2π z. θ. + 1 = 0 λz |x − iy|λz |λz − λz | = 1. λz |x ± iy|λz = 0. eh deh ss dpi ,pj. deh pi ,s. deh p± ,s. deh λz ,λz i, j ∈ {±}. |λz − λz | 3 = 1 eh dss = dp+ p+ = dp− p− = 0 deh deh deh s,p± p+ p− = 0 p− p+ = 0. deh ij. s s s C∞v deh, = 0. deh, = 0. J j=. 3 2. mj = ± 32.
(95) s |φh+ (r) =. . |p+ ↑R β(R) ,. R. β(R). |φh− (r) =. . |p− ↓R β ∗ (R) ,. R. p. θ e±iθ. j =. 1 2. j = s. s C∞v. |p− . ↑. C∞v. ↓. s β(R). C∞z. Δj = 1. 1 2. p. |p+ . 3 2.
(96) cc Vijkl /. (i, j, k, l). |φs (r). ±. |φp± (r). C∞v. θ = 2π 3 |r−r| ee ee Vijkl = ei(λi +λj −λk −λl )θ Vijkl . z. Vijkl C∞z. z. z. z. (λzi + λzj − λzk − λzl ). (s, px , py ) (s, p+ , p− ). 3 = 0.
(97)
(98)
(99) s x. y.
(100) p+. p. p+ , p−. ∝ eimφ. 2π 3. R |s. |s ,. = +i 2π 3. |p+ ,. −i 2π 3. |p− .. R |p+ = e R |p− = e 0, +1, −1. z. λz. p−. 3. λz = 0, +1, −1 λz. C3v. C∞v. C3v. (1, 1, 0) deh s,p±. deh p± ,s. k·p P P=P P P. +P. ,. eh∗ = deh ps ep+ , σ hs, −σ + dps σ eh∗ = deh sp es, σ hp , −σ + dsp σ. +. ep , σ hs, −σ , − es, σ hp , −σ . −.
(101) s. p. p P. p. + 0. s. s p. 0, +, −. −. −. 0. +. P z. z.
(102) sp. ss αβ. ps α s. ps. pp β. p sp. εep + εhs. εes + εhp. 13 12 11 10 9 8 7 6 5 4 3 2 1 0. 0X → 1X. 1.150. 1.200. 1.250. 1.300. 1.350. 1.400. 1.450. 1.500. ω/. s. p s. Sze = + 12. − 21 Λz Λz P. †. p. Λz = 0 3 = ±1 3 = +1. Λz |ψa . 3 = −1 |ψb P†. Szh = z.
(103) |ψa |ψa . |ψb . |ψb . Λz = +1, +1. |ψc . |ψc −2. ⎛ ⎜ H=⎜ ⎝. eh Vijkl. eh εes + εhp − Dsp. eh Vss,−+. eh Vs+,−−. eh Vss,−+. eh εep + εhs − Dps. eh V++,s−. eh Vs+,−−. eh V++,s−. eh εep + εhp − Dpp. Dijeh Λz = −2. |ψa . eh Vs+,−−. ⎞ ⎟ ⎟ . ⎠. eh Vijji. eh V++,s−. Λz = 1. |ψc . |ψb . 2×2 Λz. Λz. 2 |deh sp |. Λz. 2 |deh ps |. 3 eh εes + εhp − Dsp. eh εep + εhs − Dps.
(104) deh ss s. p p. 6X → 5X. 5X → 4X 4X → 3X 3X → 2X 2X → 1X 1X → 0X. 1 0 1.150. 1.200. 1.250. 1.300. ω/. Sztot = 0. 1.350. 1.400. 1.450. 1.500.
(105) p. p. p. p s. s p. 5X → 4X.
(106) εep +εhs. εes +εhp. |A ΔEA→B = εee¯ + εhh¯ − Deeh ¯ + ¯h +. ΔEA→B. |B . e eh h ee e Deee ¯i ni − De¯i ni − Xe¯i ni. ¯ i =e¯,h. . h eh e hh h Dihh . n − D n − X n ¯ ¯ ¯ i h i ih i ih. ¯ i =e¯,h. ¯ h. e¯. λλ Vijji. . e¯. Dijλλ. ¯ h. i . Xijλλ ¯ e¯ h. λλ Vijij. i εee¯. εhh¯. −Deeh ¯ ¯h e¯. ¯ h. 3X eh hh ee ee eh hh eh hh eh E3X→2X = εep + εhs − Dps − Xsp − Xsp + 2Dsp − Dps + Dps − Dpp + Dss − 2Dss .. E3X→2X = εep + εhs + Dsp − Dpp − Dss − 2Xsp P.
(107) E4X→3X = E3X→2X + ΔE. ee hh − Xpp − Xsp ,. E5X→4X = E4X→3X + ΔE. ,. E6X→5X = E5X→4X + ΔE. . ee eh hh eh (Dpp − Dpp ) + (Dps − Dps ). ΔE ee Xpp. Vpee + p− p+ p− 4X. 3X. e↔h. 6X → 5X. 5X → 4X. 4X → 3X 3X → 2X 2X → 1X 1X → 0X. 1 0 1.150. 1.200. 1.250. 1.300. ω/. 1.350. 1.400. 1.450. 1.500.
(108) 3X → 2X. 1. 3X → 2X. 0 1.150. 1.200. 1.250. 1.300. 1.350. 1.400. 1.450. 1.500. ω/. Sztot = 0. |A. |A. p 3X. 3X. |B p + ↔ p−.
(109) P 2X, i|P. |A2X, i|P |B∗ |2X, i|P. P. Λz |2X, i |A + |B |2 |2X, i. |A. |A |B P |B. i. −. |A. |B 3X. |A. 2X. P. ts. Sztot = 0. ss ee 2Xsp. 2X hh 2Xsp. eh deh sp /dps. 4X → 3X 4X → 3X 3X → 2X. P. |B. P. z. P. |A.
(110) P. −−→. ±. | 3X sp. 2X. P. 2X. 3X → 2X 3. 2. 4X → 3X. 1. 0 1.150. 1.200. 1.250. 1.300. 1.350. 1.400. 1.450. 1.500. ω/. Sztot = 0. 4X. P. 3X 3X Λz = +1. Szh = − 12. Λz 3X. Λz = −1 3 = ±1 Sze = + 12.
(111) P. −−→ |. ;. 4X. 3X. sp. H ⎡ H=E. ee Xpp. ⎢ ⎢ 1 − ⎢ Xspee ⎣ Xspee. Xspee. Xspee. Xspee. ee Xpp. ee Xpp. Xspee. ⎤ ⎥ ⎥ ⎥ . ⎦. E. Eα = E Eβ = E Eγ = E |α. ee ee − (Xsp − Xpp ), ee ee + (2Xsp + Xpp ), ee ee + (Xsp − Xpp ),. |α = (−2, 1, 1)t , |β = (1, 1, 1)t, |γ = (0, 1, −1)t .. |β |γ |α. Λz , Sze 4X → 3X. Szh. (S Se =. 1 2. )2.
(112) Eβ − Eα. |β ee 3Xpp. 3 2. Se =. hh 3Xpp. ee 3Xpp eh deh sp /dps. Sz = 0. 5. 4. 5X → 4X 5X. 5 4. 5X → 4X. 3 2 1 0 1.150. 1.200. 1.250. 1.300. 1.350. 1.400. 1.450. 1.500. ω/. Sztot = 0. 5X lztot = 0 Sze = 0. Sztot = 0 lztot = −1 Szh = 0. 4X. 16 × 16. ±1.
(113) −. Sztot = 0. 6×6 ps 2es 2ep 1hs 3hp 1es 3ep 2hs 2hp 5X → 4X. p+. p−. p− ss. |ψ1,...,4 5X → 4X lz. =0. |ψ5,6 Sz. =0.
(114) ψ1. ψ4. ψ5. ψ6. st. H {|ψi }6i=1 ⎡. ee −Xsphh −Xpp. 0. H=E. ⎤. eh Xpp. 0. 0. ⎢ ⎢ ee eh ⎢ −Xsphh 0 0 −Xpp 0 Xpp ⎢ ⎢ ⎢ −X ee eh 0 0 −Xsphh −Xpp 0 ⎢ pp ⎢ 1+⎢ ee eh ⎢ 0 −Xpp −Xsphh 0 0 −Xpp ⎢ ⎢ ⎢ X eh eh 0 −Xpp 0 0 −Xsphh ⎢ pp ⎣ eh eh 0 Xpp 0 −Xpp −Xsphh 0. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦. E. E. ee ee ee ee = 2(εes + εep ) + εhs + 3εhp + (Dss + Dpp + 4Dsp − 2Xsp ) hh hh hh hh eh eh eh eh + (3Dsp + 3Dpp − Xsp − Xpp ) − (2Dss + 6Dsp + 2Dps + 6Dpp ) .. {|ψi }6i=1 U ⎡. − 12. − 12. − 12. 0 ⎢ ⎢ 1 1 ⎢ 0 − 12 2 ⎢ 2 ⎢ ⎢ −1 1 1 0 ⎢ 2 2 2 ⎢ U=⎢ 1 ⎢ 12 0 − 12 2 ⎢ ⎢ ⎢ 0 − √1 0 0 ⎢ 2 ⎣ √1 0 0 0 2. 0. 1 2. ⎤. ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 12 ⎥ ⎥ ⎥ . 1 ⎥ 0 2 ⎥ ⎥ ⎥ 1 ⎥ √ 0 ⎥ 2 ⎦ √1 0 2 1 2.
(115) U† HU ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣. H ⎤. ee Xsphh − Xpp. 0. 0. Xsphh. √. 0. √. 0 eh 2Xpp. eh ee 2Xpp Xsphh + Xpp. 0. 0. 0. 0. 0. 0. 0. 0. 0. √ eh ee t = 2 2Xpp /Xpp ee Xpp E. 0. 0. 0. 0. 0. 0. 0. 0. 0. √ eh ee −Xsphh + Xpp − 2Xpp √ eh −Xsphh − 2Xpp 0. 0. 0 0 ee −Xsphh − Xpp. hh ee t˜ = Xsp /Xpp + t˜ + 12. H H. 1 H = 2. = − 32 ,. . −1 t. , t 1 3 = −2t˜1 − H ss , H = −2t˜ − , 2 √ √ E = − 32 E = ± 12 1 + t2 E −2t˜∓ 12 1 + t2. −2t˜ t˜ √ t=2 2 4X, i|P i {|ψi }6i=1 P. |5X, |5X,. 1 2. eh ∝ deh ps |ψ1 + dps |ψ5 +. ∝. deh ps |ψ4 . −. deh ps |ψ6 . E −2t˜− 32 t˜ ≈ 0.47. t. P. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ . ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦. +. t. eh hh Xpp = Xpp. |5X, j |5X, j 5X. |4X, i. ×. ,. ×. ..
(116) 3 2. ss−dim. 1. st−dim. 0. ss−bri g. ht. −1. ts−bright. −2. tt−bright. st−brig. ht. −3 −4 0. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. |t|. √ eh ee t = 2 2Xpp /Xpp. ee Xpp. + t˜+ 12. E t˜ ≈ 0.47. t. H. =. βi. j. |ψj Uj,3. 2 |deh ps |. (αi , βi )t. !α βi !!2 ! i √ + ! . ! 2 2 |4X,. {|ψi }6i=1. = αi |ψ˜2 + βi |ψ˜3 |4X, i = αi j |ψj Uj,2 +. i. t st ! βi !!2 2 ! αi √ − = |deh | ! ! ps 2 2 H H v = (−β, α)t. (α, β)t. v =. e↔h t ≈ −2.83. t ≈ −1.23.
(117) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −3. −2. −1. 0. 1. 2. 3. ss t t<0. t < −2. 5X → 4X. eh ee Xpp ≈ Xpp. 6X → 5X sp 0X → 1X. 0X → 1X. 0X → 1X. 6X → 5X. ps.
(118) 11 10 9 8 7. 6X → 5X. 6 5 4 3 2 1 0 1.150. 1.200. 1.250. 1.300. ω/. 5X → 4X. 1.350. 1.400. 1.450. 1.500.
(119) 6X → 5X. 5X → 4X 4X → 3X 3X → 2X 2X → 1X 1X → 0X. 1 0 1.050. 1.100. 1.150. 1.200. 1.250. 1.300. 1.350. 1.400. 1.450. 1.500. ω/. Sztot = 0. eh ee Xpp ≈ Xpp. d = 5.7. h = 2.3. s. p. p k·p deh sp.
(120) p. s. p s. εep + εhs. εes + εhp. s. |a. 2X Ea = Eb < Ec < Ed. |a |b |c. |d.
(121) 6X → 5X 5X → 4X 4X → 3X 3X → 2X 2X → 1X 0.5. 1X → 0X. 0 0.650. 0.750. 0.850. 0.950. 1.050. 1.150. 1.250. 1.350. ω/. d = 5.7. h = 2.3 Sztot = 0. |a . |c. s. s. p s. |d s. s. | Ec < Ed. | . | . | |c. Ea = Eb <.
(122) |c. s. EA < EB < EC. | . | . |C. | |C. EC. s. s s. EA < EB <.
(123)
(124)
(125)
(126) β.
(127)
(128)
(129) H = H0. +H. + H0 + H .. H0 + H. H0. =. . εci c†i ci +. i. H. i. εvi vi† vi. ,. 1 1 = Vij,kl c†i c†j ck cl + V v † v † vk vl 2 ijkl 2 ijkl ij,kl i j + Vij,kl c†i vj† vk cl . ijkl.
(130) i, j, k, l i = (ν, σ). ci c†i |i vi vi†. i = (k , σ) εci. (c,v). εvi. εi. . λλ Vij,kl. H. 0 Hph. =. . ωξ. b†ξ bξ. ξ. ωξ ξ. 1 + 2. .. ξ e (q). q. b†ξ (bξ ) =±. ωξ = c|q| Uξ (r) = e (q) ei qr. H = −i. . gijξ c†i vj bξ + gijξ vi† cj bξ −. ijξ. H ˆ+ E. ˜ E i ξ Eξ Uξ (r)bξ. = −eEr ˆ =E ˆ+ +E ˆ− E.
(131) ˜ξ = E. ωξ /2 r 0 V. gijξ. = E˜ξ. V. d3 r ψic∗ (r)erUξ (r)ψjv (r) . ξ |i. |j. fie = c†i ci . b†ξ bξ . i . . . =. i. ∂ !! ! ∂t H. ∂ !! ! A = [A, H] . ∂t H . . .. {ρ0 . . .}. fie fie = i. =. ρ0. H. c†i ci . ∂ !! ! ∂t H. ξ. c†i ci =. ∂ !! i ! ∂t H. ci =. ∂ !! i ! ∂t H. c†i = −. ". . i. ∂ !! ! ∂t H. ∂ !! c†i ci + c†i i ! ∂t H. Vijkl (c†j ck + vj† vk )c†l. jkl. jkl. ∗ Vijkl cl (c†k cj + vk† vj ) ,. ci. #. ..
(132) ∂ !! i ! ∂t H. fie = −. . ∗ { Vijkl c†l (c†k cj + vk† vj )ci + Vijkl c†i (c†j ck + vj† vk )c†l } .. jkl. 1. 2. fie 3. H. H. H. H. 1 − → 2 − → 3 − → 4 − → ... N. N H. H. − →. ij. a†i a†j ak al ≈ a†i a†j ak al . hij a†i aj. = a†i al a†j ak − a†j al a†i ak ..
(133) H = H1 + H2 H1. i. H2. 2. 1. ∂ 1 = F1 {1} + F2 {2} ≈ F1 {1} + F2 {2 ∂t. } = F1 {1} + F2 {11} .. ∂ i ∂t 1. H1 F1 {1}. H2 F2. F2. 2. 2. 2. δ2 2 = 2. + δ2 = 11 + δ2 .. δ3 = 3 − = 3 − 1δ2 − 111 . 1δ2. 111. 3 N. 1 = δ1. N N.
(134) N N N >1. i. i. ∂ 1 = F1 {1} + F2 {2} = F1 {1} + F2 {11, δ2} ∂t. ∂ ∂ δ2 = i 2 − 11 = G1 {2, 11} + G2 {3, 21} ∂t ∂t ≈ G1 {δ2, 11} + G2 {1δ2, 111} . F1. F2. δ2 G1 {2, 11} G2 H2 2 −→ 3. 2 = 11 + δ2 H1 H1 H1 2 −→ 2 11 −→ 11 H2 3 21 H. 2 11 −→ 21. 3 = 1δ2 + 111 + δ3 δ3. 3SD = δ21 + 111.
(135) a†i a†j a†k al am an SD = + a†i al δa†j a†k am an − a†i am δa†j a†k al an + a†i an δa†j a†k al am − a†j al δa†i a†k am an + a†j am δa†i a†k al an − a†j an δa†i a†k al am + a†k al δa†i a†j am an − a†k am δa†i a†j al an + a†k an δa†i a†j al am − a†i al a†j am a†k an + a†i al a†j an a†k am − a†i am a†k al a†j an + a†i an a†j am a†k al − a†i an a†j al a†k am + a†i am a†j al a†k an . 1δ2. 111. i. b†ξ (t). =. ∂ !! † † gijξ c†i vj . ! bξ = −ωξ bξ + i ∂t H ij. b†ξ (0)e+iωξ t. 1 + gijξ ij. . t 0. . dt e+iωξ (t−t ) c†i (t )vj (t ) .. b† v † c. b†q = 0 b†ξ a†i a†j ak al SD = + δb†ξ a†i al a†j ak + δb†ξ a†j ak a†i al − δb†ξ a†j al a†i ak − δb†ξ a†i ak a†j al ..
(136) δb†ξ a†i aj = b†ξ a†i aj . b†ξ b†ξ a†i aj SD = b†ξ b†ξ a†i aj .. δ.
(137)
(138) c†i cj i. ∂ † ci cj = − (εci − εcj )c†i cj ∂t $ ∗ † † ∗ % ∗ b†ξ v1† cj + gj1ξ bξ v1 ci −i gi1ξ 1ξ. +. 234. −. . & ' Vi234 c†j c†2 c3 c4 − c†j v2† c4 v3 ' & † † ∗ Vj234 c4 c3 c2 ci − c†4 v3† ci v2 .. 234. 1, 2, 3. 4 H0. H. c =: Cj234. ( )* + † † cj c2 c3 c4 = δc†j c†2 c3 c4 −c†j c3 c†2 c4 + c†2 c3 c†j c4 , c†j v2† c3 v4 = δc†j v2† c3 v4 −c†j c3 v2† v4 . * +( ) X =: Cj234. v2† c3 c†j v4 b† v † c (i = j) i. ∂ † c ci = − 2i ∂t i + 2i. $ 1ξ. $. ∗ gi1ξ b†ξ v1† ci . %. c X Vi234 C2i43 − Ci243 − c†i c3 c†2 c4 . 234. + c†2 c3 c†i c4 + c†i c4 v2† v3 . %. ..
(139) i. i. $. fiλ. λ. c % † † † † † † X Vi234 C2i43 − Ci243 − ci c3 c2 c4 + c2 c3 ci c4 + ci c4 v2 v3 . 234. 2↔3 ! ∂! i ∂t !. H. i. fic = 0. b†ξ bξ . ξ d b† b dt ξ ξ. b†ξ bξ i. i↔4. ∂ † bξ bξ = − (ωξ − ωξ )b†ξ bξ ∂t ∗ † † ∗ ∗ +i b†ξ v2† c1 + i g12ξ g12ξ bξ v2 c1 . 12. i. ∂ † b bξ = 2i ∂t ξ. 12. $ 12. ∗ g12ξ b†ξ v2† c1 . %. ξ = ξ † % ∂ $ † b bξ + ci ci = 0 . ∂t ξ ξ i.
(140) b† v † c. i. ∂ † † bξ vi cj = (εcj − εvi − ωξ )b†ξ vi† cj ∂t & ' X +i g21ξ c†2 cj (δi1 − vi† v1 ) + C2ij1 12. +i. . & ' b†ξ bξ g1iξ c†1 cj − gj1ξ vi† v1 . 1ξ . −. . & V123i b†ξ v1† cj c†2 c3 − b†ξ v1† c3 c†2 cj . 123. +. 234. + b†ξ v1† cj v2† v3 − b†ξ v2† cj v1† v3 . '. & Vj234 b†ξ vi† c4 c†2 c3 − b†ξ vi† c3 c†2 c4 . + b†ξ vi† c4 v2† v3 − b†ξ v2† c4 vi† v3 H0 H0. b†ξ bξ b†ξ bξ . '. ..
(141) CX , Cc. Cv. CX Cc. c†i vj† ck vl . C. c†i vj† ck vl HF. ∂ !! i ! ∂t H. c†i vj† ck vl = −. & 123. Cv. X. V123i c†1 (c†2 c3 + v2† v3 )vj† ck vl ) + V123j c†i v1† (c†2 c3 + v2† v3 )ck vl ) ∗ − V123k c†i vj† (c†3 c2 + v3† v2 )c1 vl ) ∗ c†i vj† ck (c†3 c2 + v3† v2 )v1 ) − V123l. 8 × 6 = 48. vi† cj 8 × 9 = 72. '. .. 1δ2. 1, 2, 3 1δ2. δ2 c†i vj† ck vl HF =. δ2 † −ci ck vj† vl 11 c† c. v†v 12. (c† c + v † v). 4×2 = 8 2. 2 = δ2 + 11. c†i vj† ck vl . 111 c†i vj† ck vl CX. CX.
(142) ! ∂! i ∂t !. X Cijkl = $ . − 1234 V1234 c†i c4 vj† v3 δ1k − c†1 ck δ2l − v2† vl . − δi4 − c†i c4 δj3 − vj† v3 c†1 ck v2†vl X+v † X+c + v1† vl δ4j − vj† v4 δ1l Ci2k3 − ci c4 δ1k + c†1 ck δ4i C2j3l X X − v2† vl δ4i − c†i c4 δl2 C1jk3 + vj† v4 δ2k − c†2 ck δ4j Ci13l X X − c†2 ck δ4i − c†i c4 δk2 C1j3l + vj† v4 δ2l − v2† vl δ4j Ci1k3 X + v3† vl δ4k + δ3l (c†4 ck − δk4 ) Cij12 X − c†i c3 δ4j + δ3i (v4† vj − δj4 ) C21kl H. X X X X −δi4 c†1 c3 C2jkl − δj4 v1† v3 Ci2kl + δk1 c†2 c4 Cij3l − δl1 v2† v4 Cijk3 % X X X X (c†2 c3 + v2† v3 )[δi4 C1jkl + δj4 Ci1kl − δk1 Cij4l − δl1 Cijk4 ] . X+v X c Cijkl = Cijkl + Cijkl. i. X+c X v Cijkl = Cijkl + Cijkl. ∂ !! X C X = −(εci + εvj − εck − εvl ) Cijkl ! ∂t H 0 ijkl. $ ∗ † ∂ !! X ∗ i ! Cijkl = i ci c1 − gi1ξ v1† vl b†ξ vj† ck g1ξ ∂t H 1ξ % + b†ξ c†i vl g1jξ c†1 ck − gk1ξ vj† v1 . Cc. Cv. CX. c†i vj Cc. Cv.
(143) ωξ H. gijξ gijξ = giξ δij. c†i cj . i = j. gijξ gijξ = E˜ξ E˜ξ =. . ωξ /2. 0 V. ζic (r). d3 r ψic∗ (r)erUξ (r)ψjv (r) ,. V. ζjv (r) c,v uk≈0 (r). ψic (r) . ˜ξ d gijξ ≈ E. ψjv (r). d3 r ζic∗(r)Uξ (r)ζjv (r). d. |i. gνν ξ ≈ E˜ξ dcv Uξ (r0 )δνν .. |j.
(144) r0. gk k ξ. 1 ≈ E˜ξ dcv A. z0. δνν . . d2 ρ e−i(k −k. )ρ. ˜ ξ (k − k , z0 ) . Uξ (ρ, z0 ) = E˜ξ dcv U ˜ ξ (q , z0 ) U. z xy q = 0. k 1 A. . |gk k ξ |2 f (k ). d2 ρ |h(ρ)|2. f (k ). k. k ˜ 2 k |hk | = ˜ 2 1 d2 ρ |dcv Uξ (ρ ρ , z = 0)|2 |gk k ξ |2 f (k ) ≈ f (k )E ξA ˜ ξ (q , z0 ) U q =0 ,. gk k ξ ≈ δk ,k . E˜ξ. 1 A. . . ρ , z = 0)|2 . d2 ρ |dcv Uξ (ρ. Uξ (r). iqr. e (q) e. δk ,k gk k ξ ∝ δk ,k +q|| q||. gk k ξ ∝ δk ,k . gijξ = giξ δij i. j s. p. |dcv e (q)|.
(145) z lz liz,v vi† vi z. =. . z,(c,v). liz,c c†i ci +. li. |i. z Vijkl ∝ δliz +ljz ,lkz +llz. [c†i cj , lz ] = −(liz,c − ljz,c )c†i cj. −(liz,c − ljz,c )c†i cj =. {ρ0 [c†i cj , lz ]} =. {c†i cj [lz , ρ0 ]} = 0 . {A[B, C]} =. ρ0 z. [l , ρ0 ] = 0. ρ0 ∝ e−βH. [lz , H] = 0. {|ψi }i {|ψi }i. c†i cj = 0. {B[C, A]}. ρ0. liz,c = ljz,c ,. l. z. i. pi |ψi ψi |.
(146) |i. c†i cj . |j. liz. s c†i cj . p. i=j c†i c†j ck cl . c†i c†j ck cl = δliz +ljz ,lkz +llz c†i c†j ck cl e Cijkl. a†i aj . a†i aj = fia δij. i = j. b†ξ vi† cj = δij b†ξ vi† ci z,c † lz = li ci ci + liz,v vi† vi. bξ. i. ∂ † c ci = − 2i ∂t i + 2i. $ ξ. $. ∗ giξ b†ξ vi† ci . %. e % X Vi234 C2i43 − Ci243 ,. 234. i. ∂ † b bξ = +2i ∂t ξ. $ 1. ∗ g1ξ b†ξ v1† c1 . %.
(147) i. ∂ † † εci − ε˜vi − ωξ − iΓ)b†ξ vi† ci bξ vi ci = (˜ ∂t X + igiiξ fic (1 − fiv ) + i g1ξ C1ii1 + i(fic − fiv ). . 1. g1ξ b†ξ bξ . ξ. + (fic − fiv ). . Vi1i1 b†ξ v1† c1 .. 1. ε˜ci = εci −. . Vi1i1 f1c +. 1. ε˜vi. =. εvi. −. . . V1ii1 (f1c + f1v − 1). 1. Vi1i1 (f1v. 1. − 1) −. . V1ii1 (f1c + f1v − 1). 1. ωξ ξ. Γ −1. f1v − 1. ε˜v,c = εv,c i i b†ξ vi† ci . (fic − fiv ) = −(1 − fie − fih ).
(148) 1234. . X V1234 δi4 c†1 c3 C2jkl =. . 1234. X V323i f3c C2jkl =. . 23. X V1234 δi4 c†1 c3 C2jkl. V3i3i f3c. X . Cijkl. 3. c†1 c3 = δ13 f3c V323i ∝ V3i3i δi2. Vijij. i. 1. 4. |i. ∂ X X εci + ε˜vj − ε˜ck − ε˜vl ) Cijkl C = − (˜ ∂t ijkl − Vklji fkc fjv (1 − fic )(1 − flv ) − (1 − fkc )(1 − fjv )fic flv X c X v V1lj2 (Ci1k2 + Ci1k2 ) + (fic − fkc ) Vk12i (C1j2l + C1j2l ) + (fjv − flv ) 12. + (1 −. flv. −. fkc ). . 12. X Vkl12 Cij21. − (1 −. fic. −. fjv ). 12. +. (fic + flv ). +. (fkc. . −. . X V12ji C12kl. 12. X V1l2i C1jk2 −. (fkc + fjv ). X Vk1i2 C1j2l. (flv. 12. fic ). . X Vk1j2 Ci12l. 12. +. −. fjv ). . X V1l2j Ci1k2. 12 12 $ ∗ % † † ∗ ∗ + i δil δjk (fic − fiv ) giξ bξ vj cj + (fjc − fjv ) gjξ b†ξ vi† ci . ξ. ξ. CX.
(149) X Cijkl. i. ∂ c c C = − (˜ εci + ε˜cj − ε˜ck − ε˜cl ) Cijkl ∂t ijkl ∗ ∗ − Vijkl − Vijlk (1 − fic ) (1 − fjc ) fkc flc − fic fjc (1 − fkc ) (1 − flc ) c X c X + (fkc − fjc ) Vk12j (Ci1l2 + Ci1l2 ) − (fic − flc ) Vl12i (Cj1k2 + Cj1k2 ) 12. −. (fkc. −. fic ). 12. + (1 −. flc. −. fkc ). −. (fic. −. flc ). +. (fkc. −. fjc ). 12. c Vk12i (Cj1l2. . +. X Cj1l2 ). +. (fjc. −. − (1 −. fic. −. fjc ). −. fic ). −. flc ). 12. . c V12ji C12kl. 12. c Vl1i2 C1jk2. +. (fkc. c V1k2j Ci12l. −. (fjc. 12. . c X Vl12j (Ci1k2 + Ci1k2 ). 12. c Vkl21 Cij12. 12. . . flc ). . c V1k2i C1j2l. 12. 12. Cv CX. c Vl1j2 Ci1k2.
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(151) I(ω) . I(ω) dω =. ξ ωξ ∈[ω,(ω+dω)]. ∂ † b bξ . ∂t ξ. [ω, (ω + dω)] ξ. q. =±. e (q). 1 V. q. →. 1 (2π)3. . 3. dq. q.
(152) I. =. ω I(ω) =. ∂ † 2 bξ bξ = ∂t ξ ξ. . ∗ giξ b†ξ vi† ci . i. =. . 2 . i. † † |giξ |2 b ξ vi ci .. ξ. † † † † giξ b ξ vi ci = bξ vi ci . ω = c|q|. . ξ. ξ. † † † † b ξ vi ci = bω vi ci q. ω = cq. f . |giξ |2 f (q) →. ξ. 1 1 2 0 r (2π)3. q . |dcv |2 = 2 6π 0 r c3. . q. 3. q cq. . |dcv ep (q )|2 f (q). p=± 3. ωω f. giξ |d e (q )|2 = |dcv |2 sin2 θ cv p p=± dcv. . |q |. ω. .. c. θ q. ω 2 . f (q) I(ω) = |˜ giω |2 =. 2 . . † † |˜ giω |2 b ω vi ci .. i. |dcv |2 ω 3 6π 2 0 r c3. q. ∂ ∂t. . |giξ |2. † =± bq. bq . sin2 θ. † † i bξ vi ci .
(153) b†ξ vi† ci . I(ω) = = δΓ (E) =. =−. i giξ fic (1 − fiv ) . εci − εvi − ωξ − iΓ. −i |˜ giω |2 f c (1 − f v ). 2 . i. 2π . εci. −. i. εvi. i. − ω − iΓ. |˜ giω |2 fic (1 − fiv ) δΓ (εci − εvi − ω). i. 1 Γ π E 2 +Γ2. δ εci − εvi fic (1 − fiv ) s p. eh. $ 1. (˜ εci. −. ε˜vi. − ω − iΓ)δ1i +. (fic. −. fiv )Vi1i1. %. p. b†ξ v1† c1 = −igiξ fic (1 − fiv ) ,. fie fih.
(154) M(ω) ψ = −iF . † † b ξ vi ci . ψ M(ω) F. fic (1 − fiv ) M(ω) ω M(ω). ˜ M. 1. ˜ M (fic − fiv ) fic = 0 D=. ˜ M. fiv = 1 S. −1. ˜S=D S M F˜ = S−1 F. (λ1 , λ2 , . . . , λn ). I(ω) =. 2 . . v˜it. i. ˜ M F˜i = fic (1−fiv ). b† v † c. v˜ = St 1. 1. −i|˜ giω |2 ˜ Fi . λi − ω − iΓ. S S = 1 ⇒ v˜i = 1. ˜ − (ω + iΓ)1 M. b† v † c. λi. λi ˜ M λi = εci − εvi λi. X Cijkl. λi.
(155) 3 · 1010 −2 9.13 · 10−29. CX , Ce C. v. 0.03. X {Cijkl }. 0.025 0.02 0.015 0.01 0.005 0. 0. 2. 4. 6. t[. 8. 10. ] 34. X Cijkl. ne =. 1.5 · 1010. 5. −2. T = 30K.
(156) 34 = 81. i, j, k, l ∈ {0, +1, −1}. X Cijkl. 1. 1. t. 10. s 8. 6. 4. p 2. 0 -150. -100. (ω − E. ne = 3.0 · 1010. −2. T = 200K. -50. )[. ].
(157) 3.0 · 1010. T = 200. −2. s −137 −111 p. −80. p. εcp + εvp = s. εcs + εvs = −55.5 s s. p. 10. 8 8 6 4. 6. 2 0. 4. 2 -138. -136 1. s. p. 2 0. 0. -140. -120. (ω − E. ne = 3.0 · 1010. k=0. -80. -100. )[. −2. -80. ]. T = 200K.
(158) 1.5 · 1010. −2. T = 30 s. p. 8. s 8 6. 6. 4 2. 4. 0 -134 -132. 2. p 0. -2. -140. -120. (ω − E. -100. )[. -80. ]. ne = 1.5 · 1010. T = 30K. v˜i. C X , C e, C v. F˜i. −2.
(159) -60. -80. -100. -120. -140 0. 0.2. 0.4. 0.8. 0.6. 1. T = 30, 70. s p. p s. p. M −133.75, −73.62. s s. p. p. −67.60.
(160) 1. g1ξ c†1 v1 vi† ci . giξ fic (1 − fiv ) +. 1. X g1ξ C1ii1. fic (1 − fiv ). ∂ c 2 fi = − ∂t =−. i |giξ |2 f c (1 − f v ) i i c v εi − εi − ωξ − iΓ ξ. fic (1 − fiv ) . τi ξ 1/τi Γ → 0. fic. 1/τi fiv.
(161) c† vv † c. fc. ∂ c fc f =− . ∂t τ. n ˆen ˆh = n ˆe. c† vv † c = c† c n ˆe n ˆh. n ˆen ˆh. ˆ ne n ˆ h = ˆ ne ˆ nh . n ˆe.
(162) nE. nH. nE nH. p. nE. nE = 1/2 nE = 1/2 nH = 1/2 nE nH = 1/4. ⎫ ⎪ ⎬ ⎪ ⎭. nH = 1/2 ⇒ nE nH = nE nH . nE nH = 1/2. nE nH nE nH = nE nH nE nH = ne . ne nh = ne nh ne nh = ne . fc. nH. ⎫ ⎪ ⎬ ⎪ ⎭. nE nH. p. ⇒ nE nH = nE .
(163) s ∂ Ce ∂t. i. ∂ X C = 2i ∂t. ∝ Ce. C e (t) = 0. % $ gξ∗ b†ξ v † c . (f c − f v ) ξ ∂ c f ∂t. CX † †. b v c i. ∂ CX ∂t. ∂ c f = −2i ∂t. $. gξ∗b†ξ v † c. % .. ξ. ∂ v = − ∂t f ∂ = ∂t (f c f v ) C X = f cf v + f c = 0, f v = 1, C X = 0 C X = f cf v c v X gξ {f (1 − f ) + C } gξ f c ∂ c f ∂t. 1.5 · 1010. X −iγCijkl. −2.
(164) I 0. 1000. 500. 1500. 2000. t [ps] γ = 0.001. ∂ c,v !! f ! ∂t i Fic,v (T ). =−. τ. = 1ps. fic,v − Fic,v (T ) , τ c,v T 1. γ = 0.001.
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(167) gνξ ≈ E˜ξ dcv Uξ (r0 ) . Uξ dcv.
(168) κ. ω ¯ ξ = ωξ − i 2ξ ξ ω ¯ξ. ω ¯ ξ∗. bξ. b†ξ. b†ξ bξ (t) = b†ξ bξ (0)e−κξ t. Δωξ = κξ. Q = ω0 /Δω. Q F ξ F = λξ. V |Uξ |2. |gνξ |2 = τ. 3 Qξ λ3ξ , 4π 2 V ,ξ ,ξ. |U (r)|2 = d3 r |Uξ |2 ξ. |Uξ (r)|2. 2 ωξ 1 F |UQ (r0 )|2 cos(φ ) , τ (ωξ ) 4 Q |UQ |2. (ωξ ) ωξ. n. φ.
(169) 2 & ∗ † † ' d f =− g b v c , dt d † 2 & ∗ † † ' b b = −κb† b + g b v c , dt d † † κ b v c = −i(εc − εv − hωc − i )b† v † c + gf . dt 2. b† v † c = −igf /(εc −εv −hω −iκ/2) b† b =. 12 κ. & ∗ † † ' g b v c . 12 b† b(t) = f (t) κ. |g|2 εc − εv − hω − i κ 2. .. b† b(t) = Cf (t). C. (1 + C). d f = −κCf . dt. C 1. τ. 1/κC C 1 ω0 τ 104. C1. C 106. C ≈. κτ. 1. F Q. ≤. F κτ. =. F Q ω0 τ.
(170) 1 τ. =. F τ. ( κ )2 |UQ (r0 )|2 2 cos(φ ) . (εc − εv − hω)2 + ( κ )2 |UQ |2 2. τ F <1. UQ (r0 ) κ. φ ωc. (εc − εv ). d.
(171) I 0. 500. 1000. 1500. 2000. t [ps]. γ = 0.001. τ. = 1ps.
(172) I. 1. 0.1. 0. 1000. 500. t [ps] ne = nh = 2 · 109 0.52. −2. ne = nh = 3 · 109 −2 −2 nQD = 3 · 109. 1500.
(173) 1010. 1011. −2. μ. 3 · 109. −2. 1.5.
(174) ωξ |Uξ (r)|. ρ. Qξ. ξ ωξ = ωξ (1 − b†ξ bξ ≈ δξξ b†ξ bξ Q. k||. i ) 2Q.
(175) 1. Norm. Intensity. 0.1. 0.01. 0.001. 0.0001 0. 300. 900 600 Time [ps]. 1200. 1500 μ. 1× 109. fie fih. −2. 5× 109. −2.
(176) Intensity (arb. units). 0.1. 0.01. 6μm 4μm 2μm. 0.001. 1μm 0.0001. 0. 300. 900 600 Time [ps] 2 × 109. F ∝ Q/V. −2. 1200. 1500.
(177) r/R0. 0.5 0 −0.5. −131. −130. −129. −128. −127. ω − E. Qξ. ρ = 0. μ.
(178) Exc. Density 5.56 Wm-2. 0.56 Wm -2. 1.6 7W m -2 5.5 6W m -2. 16.67 Wm-2. Norm. Intensity [log. scale]. Norm. Intensity [log. scale]. d=5 μm. 6 μm. 4 μm. 3 μm 2 μm. -2. 25 Wm. 1 μm. -2. 30 Wm. 0 300 600 900 120015001800. 0. 300 600 900 1200 1500. Delay [ps]. Decay [ps]. 5μ −2.
(179) t=0.
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(183) p. C3v p.
(184) g 2(τ ).
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(187) φα |H|φβ |φi . |φ = P i. . (e†i )ni. e. i nei =Ne. P j. (h†j )nj |0 . h. j nh j =Nh. .. |φ. H0e H0e =. . εi e†i ei .. i. φα |H0e |φβ =. . εi nei (α)φα |φβ = δα,β. i. nei (α) = 1 |φα nei (α) = 0 H0h. i. |i. εi ni (α).
(188) Hee =. 1 V e† e† ek el . 2 ijkl ij,kl i j. Hee φα |Hee |φβ = φhα |φhβ φeα |Hee |φeβ , |φλ . λ ∈ {e, h}. . |φλ = P i. Hee. (a†i,λ )ni |0 . λ. i nλ i =Nλ. |i |φeα . |j. |φeβ . |k. |l. φeα |Hee |φeβ . Ne Ne = 0. |φeα = |φeβ φeα |Hee |φeα =. 1 e ni (φα )nej (φα ) Vij,ji − Vij,ij . 2 i,j. Ne = 1 |φeα . |φ¯e . |φeβ α ¯ |φeα = (−1)Pα eα+¯ |φ¯e , ¯ β. |φeβ = (−1)Pβ e+ |φ¯e . β¯ |φ¯e . α ¯. β¯. |φeβ . |φeα. 1 Vij,kl φ¯e |eα¯ e†i e†j ek el e+ |φ¯e β¯ 2 ijkl . = (−1)Pα +Pβ nei (φα ) Vαi,i . ¯ β¯ − Viα,i ¯ β¯. φeα |Hee |φeβ = (−1)Pα +Pβ. i. |φ¯e .
(189) α = β Ne = 2. α ¯ 1 ,α ¯2. |φeα = (−1)Pα. ¯ ,β ¯ β 1 2. |φeβ = (−1)Pβ. eα+¯1 eα+¯2 |φ¯e , e+ e+ |φ¯e . β¯1 β¯2. . φeα |Hee |φeβ = (−1)Pα +Pβ Vα¯1 α¯2 ,β¯2 ,β¯2 − Vα¯1 α¯2 ,β¯1 ,β¯2 . Ne > 2 φeα |Hee |φeα = 0 .. Heh =. . Vij,kl h†i e†j ek hl .. ijkl. φα |Heh |φβ =. . Vij,kl φeα |e†j ek |φeβ φhα|h†i hl |φhβ .. ijkl. φeα |e†j ek |φeβ . Ne = 0 φeα |e†j ek |φeβ = δj,k nej (φα ) . Ne = 1. |φeα . |φ¯e . |φeβ . φeα |e†j ek |φeβ = (−1)Pα +Pβ δj,α¯ δk,β¯ . Ne > 1. e†j ek φeα |e†j ek |φeβ = 0 ..
(190) z 1 † (a ai↑ − a†i↓ ai↓ ) , 2 i i↑ † ai↑ ai↓ , S+ = Sz =. i. S− =. . a†i↓ ai↑ .. i. . a†iσ ajσ ,. σ. . 1 Vijkl a†iσ a†jσ akσ alσ 2 ijkl iσ σσ † †. † = εi aiσ aiσ + Vijkl a†iσ alσ ajσ akσ − δlj aiσ akσ. H=. εi a†iσ aiσ +. σ. i. σ. ijkl. σ. σ. Vijkl. i. j σ. a†iσ akσ cv. H cv. =. 1 Viσλ,jσ λ ,kσ λ ,lσλ a†iλσ a†jλ σ akλ σ alλσ 2 ijkl σσ λλ. λ λ. σ σ.
(191) Viσλ,jσ λ ,kσ λ ,lσλ = Viσ,jσ ,kσ ,lσ. c. ↑. v. ↓. 1 † S˜z = (a ai,c − a†i,v ai,v ) =: Pz , 2 i i,c † S˜+ = ai,c ai,v = P † , i. S˜− =. . a†i,v ai,c = P. i. P [H cv , P] = 0 P. H0cv =. H0cv. Viσλ,jσ λ ,kσ λ ,lσλ = Viσ,jσ ,kσ ,lσ cv c † v † = i εi ai,c ai,c + i εi ai,v ai,v εci = εvi. εc − εv † εc + εv †. i i i i ai,c ai,c + a†i,v ai,v + ai,c ai,c − a†i,v ai,v . 2 2 i i εci. −. εvi. Pz , P † ,. P. (εci − εvi )Pz. [H cv , P] = [H cv + H0cv , P] = [H0cv , P] = (εci − εvi )[Pz , P] = −(εci − εvi )P .. P.
(192) . λλ Vijkl = Vijkl. λ ∈ c, v. εh. εe. s p {εhi }i. {εei }i. → {εci }i. {εvi }i. → p p. s. K. Tn = 1. T. n. z λ A A(α|ψ + β|φ) = αA|ψ + βA|φ α∗ A|ψ + β ∗ A|φ. n. A(α|ψ + β|φ) =.
(193) 2π. m = 0, 1, . . . , n − 1. λ = ei n m. n>2. K. Kψ(r) = ψ ∗ (r). [H, K] = 0. T ψ(r) = ψ(T −1 r). T. KT ψ(r) = Kψ(T −1 r) = ψ ∗ (T −1 r) = T ψ ∗ (r) = T Kψ(r) , K. K. H T. T. φ(r). H. T. E. λ n > 2 H(Kφ(r)) = K(Hφ(r)) = E(Kφ(r)) T (Kφ(r)) = K(T φ(r)) = λ∗ (Kφ(r)) ∗ λ = λ φ(r) Kφ(r) T Tn = 1 n>2 K T. 2π 3. R. z. n=3. IC4z C2z C2z IC4z. n = 2 n=4. Q. αi.
(194) k|| ω αs1 (ω) αs2 (ω) [αs1 , αs2]. α |k|||. k0 + Δk/2. / 2 k⊥ + k||2 = ω/c. k0 k0 − Δk/2. αs2. αs1 αi |k⊥ | |k⊥ | |k|| |. αi. |k⊥ | |k|| |. |k|| | αs1. αs2. ω0. αs1. αs2. ω 2 k⊥ + k||2 = ω02/c2. ω0 = ck0 (0, k0 ) αs1 = arcsin k0 −Δk/2 k0. αs2. π 2. αs1 > αi αi 0. αs1.
(195) π 2. − αs1. π 2. |˜ gωi |2. |˜ gωi |2 = ζ |˜ gωi |2 , ζ 1 − 34 Δk k Δk. 3 4. sin(αs1 ) + 14 sin3 (αs1 ). k0 Δk/2. Δλ λ0. ≈. ζ. 2 2 1 n1 −n2 π n1 n2. 90% κ. Lz → 0 Lz.
(196) b a. f (x)dx ≈. . f (xi )Δx. i. {wi }. i. {xi }. wi f (xi ). f (x) 2π 1 2π i − sin( i) (b − a) + a , 2π N + 1 N +1 b−a 2π 1 wi = 1 − cos( i) . 2π N +1 N +1 xi =. i. 1 a. I(k) = I(k) =. . . b. dk V (k, k )f (k ) . {xi }. N. . V (k, k ). k = k . . dk V (k, k )[f (k ) − f (k)α(k, k )] + f (k) α(k, k ). dk V (k, k )α(k, k ) ,. α(k, k) = 1. f (k) α(k, k ). QR. (NX − 1).
(197) d. d y(t) = f [y(t), t] dt. y(t = 0) = y0 ,. y(t) t Δy/Δt y(t + Δt) = y(t) + f y(t), t Δt + O[(Δt)2 ], Δt Δt. Δt = 1 106.
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(202) 1−x. x. 1−x. x.
(203) x. 1−x.
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(206) 2. In0.2 Ga0.8 N.
(207) x. 1−x.
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