Selected Topics on Phonons and Electron-Phonon Interaction in

Semiconductors

朱邦芬清华大学物理系

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

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。

绝热近似：

)R(H)R,r(H

))R(VT())R,r(V)r(VT(

|RR|)Ze(

M|Rr|Ze

|rr|e

mH

Le

nmLeLeee

m,n mnn

N

n ni n nij,i jii

NZ

i

22

1

2222

1

2

21

221

21

2

)()()()(ˆ

')(),()()(ˆ),(

)(),()](ˆ[

RERRRH

EHRRrRRHRr

ERRrRH

inininiinL

iniin

iniiniiinLi

iniininiiL

LL HHH '忽略的非绝热算符：

),()(),(),(ˆ RrRRrRrH iiie

)(),( RRr ini 总波函数：

Using the static approximation we separate out the part describing motion of ions:

)(ˆ)(ˆ),(ˆ)],(),([))(()),()((ˆ

0

00

RHRHRrH

RrVRrVRVTRrVrVTH

pheLe

eLeLnmLeLeee

),()(),(),(ˆ0000 RrRRrRrH iiie

)()()]([

)(]),()(ˆ),()(ˆ[ 00*

0

RRRVT

RdrRrRHRrRH

inininiL

inipheiiL

)(),( 0 RRr ini Total wave-function ：

)()()]([)(ˆ RRRVTRH nnnLnion

We started with this equation:

•V represents the interaction between ions including

screening by the valence electrons

• 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。

)(ˆ RH phe

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

 optical vibrations dipole polar motion of a continuum obeying phenomenological Huang’s Equations:

 - 2w = b11 w + b12 E

P = b21 w + b22 E, (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

The dielectric continuum model

Applied to a superlattice:(i) bulklike modes: modes with frequencies TO(A), LO(A), TO(B), and LO(B) confi

ned to respective layers.

Derivation LO (A) bulklike modes:

For vibration modes w exp(it) ,

1st Eqn.

w = b12E/(2TO-2),

2nd Eqn.

(2LO-2) /(2

TO-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.

=LO (A)  in A-layers

A0, DA = 0 and DA = 0 in B-layers,

B0,

EB = 0 2B(z)/z2= k2B(z)

B(z) = C+ exp(kz) +C- exp(-kz)

 At two interfaces,

continuity with Dz=0 in A C+ = C-= 0

B(z) = EB = wB =0

LO (A) mode confined to the A-layers.

(ii) Interface modes:

frequencies within LO-TO gap of either material A or B.

A0, B0

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.

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？

  Model

   Basis 

  Boundary Condition

   Parity of potential m (z)

  Inter-face modes

Dielec-tricContinuum

Electrostatic Eqn.+Huang’s Eqn.

 (d/2)=0;wz(d/2) antinode

m=odd, m even; m=even, m odd

 yes

GuidedMode

Linear chain phonon modelcalculation? 

/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

真实超晶格样品的 Raman 谱Dips or Peaks?

“DOS enhanced by Fröhlich-interaction” Model

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

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)

Optical phononmodes

Scatteringconfiguration

Electron-phononinteraction

n=even Polarized Fröhlich

n=odd Depolarized Deformationpotential

(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

与之相关的低维纳米结构特点

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

A. Shilds, M. Cardona et al PRL 1994

Dips rather than peaks

1. The predominance of the IF modes with non-zero wavenumbers over other scattering channels

2. The spectrum dips, the minimum of DOS, and the anticrossing regions happen to be of identical frequencies

3. To map various optical-phonon Raman peaks into the bulk dispersion so accurately

4. Impossible to observe the resonant Raman profile related to the LO4 mode as observed in experiments by Sood et al.

Critique to the Dip-assignment

Symmetry for IF-Modes

DOS enhanced by Fröhlich-interaction” Model

if

SiC 纳米棒的 Raman 谱“DOS enhanced by Fröhlich-interac

tion” Model

30-40cm-1 shift? 表面模 ?

Effects of Electron-Phonon Interaction on Nonequilibrium Transport Through Single-Molecule Tra

nsistor and Quantum Dots

• Phonon-Assisted Resonant Tunneling

• Phonon-Kondo effect• Phonon-Fano effect

Introduction

Nature 407, 57 (2000)

Single molecule transistor (SMT)

Exhibiting phonon characteristics due to the vibrational feature of SMT.

quantum, coherent, exhibiting strong correlation, …

transport:

Phys. Rev. Lett. 93, 266803 (2004)

Phonon satellites persist even into the Kondo regime.

Model and Technique

Anderson-Holstein Model:

Current formula: EPI

Canonical Transformation:

Local polaron condition:

Mean field approximation:

where

Decoupling of Green functions:

Improved scheme:

Previous scheme:

Derivation of dressed Green functions:

•Nonequilibrium Green function formulas•Equation of motion

techniques:

Decoupling approximation in truncation:

•usually:

Generalize Lacroix’s approximation to nonequilibrium cases with finite U and Zeeman splitting. (JPC )

•Lacroix’s: Works below Kondo temperature.(J. Phys. F: Metal Phys. 11, 2389-97 (1981))

Valid near or above Kondo temperature.

•Ours:

Phonon-Assisted Resonant Tunneling

Phys. Rev. B 71, 165324 (2005)

•The occupation is determined by Fermi surfaces in leads•The channel is enactive only when it lies between two Fermi surfaces.

Resonant tunneling: (T=0),

Sidebands can only be developed above the level for hole, while below for electron.

Isolate polaron: (T=0), Phonon sidebands are developed by emitting phonons

Joint effects of EPI and resonant tunneling:

•Sidebands developed

•Renormalized effect

Spectral function

Spectral weights

(equilibrium)

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

Tunneling current and differential conductance

Zero temperature

The difference between previous works arises from the difference of decoupling schemes.

There is no phonon peaks in G for

Finite temperature

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

Kondo effect in quantum dot

L. P Kouwenhoven group, Science 281,540(1998)

Phonon-Kondo Effect

Spectral functionspin degenerate case

J. Phys.: Condens. Matter 18 (2006) 5435–5446

Kondo satellites on different sides are associated with different types of spin singlet

hole spin singlet electron spin singlet

differential conductance

The differential conductance has phonon peaks at

The Kondo satellites will be split into two sets when

finite Zeeman splitting case

Kondo satellites only appear in one side in the spin resolved spectral function when wider than the

Kondo peak

Fano Resonance in parallel-coupled double Quantum Dots

Science, 297,70 (2002)

Lu, Lv, and Zhu Phys. Rev, B71,235320(2005)

There is a phase shift difference between channel 1 and channel 2, leading to the Fano interference .

Phonon-Fano Effect in Parallel-coupled double QD syst

em

The Fano effect between two resonate leve

ls is renormalized.

The phonon sidebands exhibit no Fano effect between the other reso

nant peak.

The phase shifts through the resonate levels are renormalized by the EPI.

Phonons act as the detector of “which way”.

Interference between dots dressed by the identical phonon

For different values of detuning Δ: (a) The interference between the same order phonon sidebands of two dots gives rise to the dips at integer times of the phonon frequency. (b) No interference between the sidebands with different orders, their contributions to the conductance are just direct summation.

An experimental proposal. Part of the tube 1 is suspended, allowing the phonon mode to be excited. The tube 2 is grown on the Pt surface, optical vibrations in this tube are suppressed, and no side peaks yet experimentally observed in this type of tight-lying nanotube.

Summary

By the improved canonical transformation scheme and the EOM method, we have studied the effects of the EPI in the SMT, especially focused on the behavior of the phonon satellites. These satellites are found to be sensitive to the lead Fermi surfaces, and the Kondo satellites on different sides are associated with different types of spin singlet. We have also shown that there is no Fano interference between the phonon satellite and the other main channel,

demonstrating the “which way” rule.

Thank you ！