Selected Topics on Phonons and Electron-Phonon Interaction in Semiconductors

1. Selected Topics on Phonons and Electron-Phonon Interaction in Semiconductors 朱邦芬 清华大学物理系

2. Outline • Adiabatic Approx. Static Approx. Lattice Relaxation • Phonon and Electron-Phonon Interactions in Low-Dimensional Systems • Phonon Raman scattering in realistic QW • Effect of Electron-Phonon Interaction on transport through Quantum Dot

3. Adiabatic Approx. Static Approx. & Lattice Relaxation 绝热近似（Adiabatic Approx.)——由于原子实的质量远大于电子，可以认为，当原子振动的每个瞬间，电子将始终处于基态（能量不同） 静态近似（Static Approx.)——原子实位于平衡状态下研究电子态，晶格振动平衡位置依赖于电声子作用的对角项，依赖于电子态。

4. 绝热近似： 总波函数： 忽略的非绝热算符：

5. Using the static approximation we separate out the part describing motion of ions: Total wave-function：

6. We started with this equation: • Vrepresents the interaction between ions including • screening by the valence electrons

7. 只保留电子-声子互作用算符 对角项，把非对角项作为微扰处理； 原则上，不同的电子态对应于不同的晶格平衡位置；通常计算固体的振动性质时，假定电子处于基态。 激发态电子的晶格平衡位置与基态晶格平衡位置的差——晶格弛豫 由于晶格弛豫，激发态电子所对应的声子简正坐标与基态电子所对应的声子简正坐标不正交——多声子跃迁的理论基础 晶格弛豫和多声子跃迁在缺陷电子态和量子点中显著。

9. Phonons in Semiconductor SL • In general, the vibrational modes in SL can be classified as: • Zone-Folded Acoustic modes • Confined Optical modes • Interface modes • Macroscopic: Coulomb mode • Microscopic: AB/CD SL

10. The dielectric continuum model optical vibrations dipole polar motion of a continuum obeying phenomenological Huang’s Equations:  - 2w = b11w + b12E P = b21w + b22E, (b12 =b21) w reduced optical displacement, b-coefficients expressible in terms of 0, and TO. In bulk materials leading to Transverse waves: TO; Longitudinal waves: LO=TO(0/)1/2

11. Applied to a superlattice: (i) bulklike modes: modes with frequencies TO(A), LO(A), TO(B), and LO(B) confined to respective layers. Derivation LO (A) bulklike modes: For vibration modes w  exp(it) , 1st Eqn. w = b12E/(2TO-2), 2nd Eqn.  (2LO-2) /(2TO-2), Superlattice: electrostatics solution of alternating layers with Aand B . Electrostatic potential : V(r) =(z)exp(i k·x), E = -V(r) Only equation to satisfy:  ·D = 0.

12. =LO (A)  in A-layers A0,DA = 0 and DA = 0  in B-layers, B0, EB = 0 2B(z)/z2= k2B(z)  B(z) = C+ exp(kz) +C- exp(-kz) At two interfaces, continuity with Dz=0in A C+ = C-= 0 B(z) = EB = wB =0 LO (A) mode confined to the A-layers.

13. (ii) Interface modes: frequencies within LO-TO gap of either material A or B. A0, B0 A(z) =A+ exp(kz) +A- exp(-kz),in A-layer or B(z) =B+ exp(kz) +B- exp(-kz),in B-layer to be joined in accordance with the periodicity, subject to the electrostatic connection rules at the interfaces.

14. 疑问： 为什么连续介电模型得到的界面模符合实验，而类体模不符合实验？

15. Model Basis Boundary Condition Parity of potential m (z) Inter-face modes Dielec-tric Continuum Electrostatic Eqn.+ Huang’s Eqn. (d/2)=0; wz(d/2) antinode m=odd, m even; m=even, m odd  yes Guided Mode Linear chain phonon model calculation? /z(d/2) =0; (d/2) antinode m = odd, m odd; m = even, m even no Huang -Zhu Dipole lattice models (d/2)=0; /z(d/2) =0 m=odd, m odd; m=even,m even yes

17. 真实超晶格样品的Raman谱 Dips or Peaks? “DOS enhanced by Fröhlich-interaction” Model

18. Raman scattering • Raman scattering: a third order process with phonon scattering between two intermediate states • Two types of electron-optical phonon interaction: Fröhlich scattering and Deformation potential scattering • Two types of Optical phonon modes: Confined modes and Interface(Surface) modes

19. Selection Rule for Raman Spectra in Semicon-ductor Superlattices (Back-scattering): • B. Jusserand and M. Cardona “Light Scattering in Solids” • V, Springer(1989) • K. Huang, B. F. Zhu, and H. Tang, PRB 41,5825(1990)

20. 与之相关的低维纳米结构特点 (1) 低维微结构生长过程中不可避免的杂质缺陷——即使质量最好的量子阱也存在界面不平整性：准动量守恒定律松弛 (2)低维量子限制效应，特别是一维或零维结构的电子态密度奇异性 (3)体材料特殊的电子与声子色散关系：如碳纳米管 (4)样品的离散性

21. 缺陷和杂质对Raman谱的效应 晶体：严格的选择定则 非晶体：正比于态密度 低维纳米材料：准动量守恒定律松弛 ΔL Δq ~1 ——Raman峰移动与展宽 缺陷诱导的模式

22. Dips rather than peaks A. Shilds, M. Cardona et al PRL 1994

23. Critique to the Dip-assignment • The predominance of the IF modes with non-zero wavenumbers over other scattering channels • The spectrum dips, the minimum of DOS, and the anticrossing regions happen to be of identical frequencies • To map various optical-phonon Raman peaks into the bulk dispersion so accurately • Impossible to observe the resonant Raman profile related to the LO4 mode as observed in experiments by Sood et al.

24. Symmetry for IF-Modes

26. if DOS enhanced by Fröhlich-interaction” Model

31. SiC纳米棒的Raman谱 “DOS enhanced by Fröhlich-interaction” Model 30-40cm-1 shift? 表面模?

33. Effects of Electron-Phonon Interaction on Nonequilibrium Transport Through Single-Molecule Transistor and Quantum Dots

34. Phonon-Assisted Resonant Tunneling • Phonon-Kondo effect • Phonon-Fano effect

35. Introduction Single molecule transistor (SMT) transport: quantum, coherent, exhibiting strong correlation, … Exhibiting phonon characteristics due to the vibrational feature of SMT. Nature 407, 57 (2000)

36. Phonon satellites persist even into the Kondo regime. Phys. Rev. Lett. 93, 266803 (2004)

37. Model and Technique Anderson-Holstein Model: Current formula: EPI

38. Canonical Transformation: where Local polaron condition: Mean field approximation:

39. Decoupling of Green functions: Improved scheme: Previous scheme:

40. Derivation of dressed Green functions: • Nonequilibrium Green function formulas • Equation of motion techniques: Decoupling approximation in truncation: • usually: Valid near or above Kondo temperature. Works below Kondo temperature. (J. Phys. F: Metal Phys. 11, 2389-97 (1981)) • Lacroix’s: GeneralizeLacroix’s approximation to nonequilibrium cases with finite U and Zeeman splitting. (JPC) • Ours:

41. Phonon-Assisted Resonant Tunneling Phys. Rev. B 71, 165324 (2005) Resonant tunneling: (T=0), • The occupation is determinedby Fermi surfaces in leads • The channel is enactive only when it lies between two Fermi surfaces.

42. Isolate polaron: (T=0), Phonon sidebands are developed by emitting phonons Sidebands can only be developed above the level for hole, while below for electron.

43. Joint effects of EPI and resonant tunneling: (equilibrium) Spectral function Spectral weights • Sidebands developed • Renormalized effect

44. The phonon sidebands are manipulated by lead Fermi surfaces (T=0)

45. Tunneling current and differential conductance Zero temperature There is no phonon peaks in G for The difference between previous works arises from the difference of decoupling schemes.

46. Finite temperature The difference between previous work is negligible, and the phonon characteristics smear out for higher T .

47. Kondo effect in quantum dot • L. P Kouwenhoven group, Science 281,540(1998)

48. Phonon-Kondo Effect spin degenerate case Spectral function J. Phys.: Condens. Matter 18 (2006) 5435–5446

49. Kondo satellites on different sides are associated with different types of spin singlet hole spin singlet electron spin singlet

50. differential conductance The Kondo satellites will be split into two sets when The differential conductance has phonon peaks at