MODELING OF ENERGY STATES MODELING OF ENERGY STATES OF CARRIERSOF CARRIERSIN QUANTUM DOTSIN QUANTUM DOTS

Michael Yu. Petrov,

St. Petersburg State University, Faculty of Physics

e-mail: M.Yu.Petrov@gmail.com

OUTLOOK Motivation Introduction into the Quantum Dot Heterostructures

What is a quantum dot? Self-organized semiconductor quantum dots Energy Spectra

Modeling of Real Quantum Dots Shape of real dots Band profiles (including its modifications via strain effects) Calculation models (effective mass approximation and multi-band k·p-

method) Optical transitions in real quantum dots (Coulomb interaction in

excitons) Applications of Modeling

Air-bridge detector device Fock-Darwin spectra in ultra-high magnetic field Optical transition of annealed quantum dots

Conclusion2

MOTIVATION

Quantum dot is a model object of fundamental research in modern semiconductor physics

Quantum dot is an object for applications and technology including: Laser Technology Optoelectronic Devices Spintronics and Quantum Information Processing

Modeling because of a model object

3

INTRODUCTIONINTRODUCTIONINTO THE QUANTUM DOT HETEROSTRUCTURESINTO THE QUANTUM DOT HETEROSTRUCTURES

4D. Bimberg, M. Grundmann, N.N. Ledentsov,Quantum Dot Heterostructures (Wiley, New York, 1999)

WHAT IS A QUANTUM DOT?

SELF-ORGANIZED QUANTUM DOTS

5

TEM of InAs/GaAs QDs (plan-view)

V.G. Dubrovskii, G.E. Cirlin, et al.,Journal of Crystal Growth 267 47-59 (2004).

HRTEM of InP/InGaP QDs(front-view)Y. Masumoto, T. Takagahara,Semiconductor Quantum Dots: Physics, Spectroscopy and Applications,(Springer, Berlin, 2002).

ENERGY SPECTRA(FROM BULK TO HETEROSTRUCTURES)

6D.Bimberg, M.Grundmann, N.N.Ledentsov,Quantum Dot Heterostructures (Wiley, New York, 1999)

Typical PL spectrumof InGaAs/GaAs QDs

Experimentalle Physik II,Universitaet Dortmund, Germany

SIMPLEST MODELS OF ENERGY STRUCTURE

7

Cube-like QD with infinite barriers

Sphere-like QD with infinite barriers

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,,,,2

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,2 2

22222

,, a

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azN

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xN

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2 23

,,

,,,,2

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Rk

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R

kj

For InAs QD (me=0.023m0): cube: a=10 nm E111=0.49 eVsphere: R0=6.2 nm E10 =0.42 eV(cube volume = sphere volume)

MODELING OF REAL QUANTUM DOTSMODELING OF REAL QUANTUM DOTS

Important parameters for real QDs: shape and volume of QDs in sample band profiles (including its modification via strain)

Different methods of calculation of energy structure: one-band effective mass approximationmulti-band calculations

Coulomb interaction of carriers

8

SHAPE AND VOLUME OF QUANTUM DOTS

9

A “regularly shaped” QDs are available only at excellent growth conditions

Size spread is approximately 10% for self-organized QDs

It is not possible to describe the QD ensemble by microscopy of single QD

Two most popular models of QD shape: pyramid, lens

STRAIN PROFILES IN QUANTUM DOTS

10

Harmonic Continuum Elasticity Theory (CE)

Atomistic Valence-Force-Field Model (VFF)

222

44

12

222112

1

2

zxyzxy

xxzzzzyyyyxx

zzyyxxCE

C

C

CE

i

j

j

iij dx

du

dxdu

21

The solution for strain tensor, εij, can be obtain by minimizing the elastic energy, ECE, by modifying the displacement vectors, ui

The solution for strain tensor, εij, can be obtain by minimizing the elastic energy, ECE, by modifying the atomic positions

ijk

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ijij

ij

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8

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11

4 ; ;

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Ca

Ca

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STRAIN PROFILES IN QUANTUM DOTS(CONTINUE)

11

C. Pryor et al., J. Appl. Phys. 83, 2548-2554 (1998)

INFLUENCE OF STRAIN ON BAND PROFILES

12

C. Pryor, Phys. Rev. B 57, 7190-7195 (1998)

COMPARISON OF DIFFERENT METHODS OF CALCULATION OF ENERGY STATES OF CARRIERS

13

C. Pryor, Phys. Rev. B 57, 7190-7195 (1998)

ELECTRON AND HOLE DENSITIES

14O. Stier, M. Grundmann, D. Bimberg, Phys. Rev. B 59, 5688-5701 (1999)

OPTICAL EXCITONIC TRANSITIONS

15

Strong Confinement Regime (simple consideration)

Hartree Approximation

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EV

0

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Eh

E= Ee + Eh -EX

O. Stier, M. Grundmann, D. Bimberg,Phys. Rev. B 59, 5688-5701 (1999)

EXCITONIC SPECTRUM OF INGAAS QUANTUM DOTS

16

1e-1h

2e-2h

3e-3h

MODIFICATIONS OF THE ELECTRONIC STATES OF InGaAs QUANTUM DOTS EMBEDDED IN BOWED AIRBRIDGE STRUCTURES

17

left-up: SEM of structure;right: PL spectrum;left-down: Energy ShiftT. Nakaoka, T. Kakitsuka, et al.,Journ. Appl. Phys. 94, 6812 (2003).

INFLUENCE OF ULTRA-HIGH MAGNETIC FIELD ON ENERGY STRUCTUREOF InGaAs/GaAs QUANTUM DOTS

18

Fock-Darwin spectrum(left (c) – experiment,right – 8-band k·p-model)S. Raymond, S. Studenikin, et al.,Phys. Rev. Lett. 92, 187402 (2004).

MODELING OF ENERGY SPECTRA OF ANNEALEDINAS/GAAS QUANTUM DOTS

19

Bell-like shaped QD for describing the average in ensemble

Diffusion Law for describing thermal annealing

Model of Constant Potentials for carriers

One-band Effective Mass Approximation for energy states calculations

z

M.Yu. Petrov, I.V. Ignatiev et al., Phys. Rev. B (submitted);also available in arXiv: 0710.5091v4

INTERDIFFUSION OF INDIUM AND GALLIUMDUE TO THERMAL ANNEALING OF QUANTUM DOTS

20

0,,

trxD

ttrx

A

AA kT

EDTD exp0

EA

a

MODIFICATION OF CARRIER DENSITIES DUE TO THERMAL ANNEALING

21

Electron density distribution

Indium concentration distribution

Hole density distribution

EXCITONIC SPECTRA OF ANNEALED QUANTUM DOTS

22

CONCLUSION

The basic principles of calculations of energy structure of quantum dots were demonstrated The main important parameter is a built-in strain For approximation of lowest state the simplest constant

potential models of QD can be used Describing of excited states requires more complex models

(band mixing, coulomb interaction etc.)

23

24

Thank You For Your Attention!Thank You For Your Attention!

REFERENCES D. Bimberg, M. Grundmann, N.N. Ledentsov, Quantum Dot

Heterostructures (Wiley, New York, 1999). Y. Masumoto, T. Takagahara, Semiconductor Quantum Dots:

Physics, Spectroscopy and Applications, (Springer, Berlin, 2002).

C. Pryor et al., J. Appl. Phys. 83, 2548-2554 (1998). C. Pryor, Phys. Rev. B 57, 7190-7195 (1998). O. Stier, M. Grundmann, D. Bimberg, Phys. Rev. B 59, 5688-

5701 (1999). T. Nakaoka, T. Kakitsuka, et al., J. Appl. Phys. 94, 6812

(2003). S. Raymond, S. Studenikin, et al., Phys. Rev. Lett. 92, 187402

(2004). M.Yu. Petrov, I.V. Ignatiev, et al., Phys. Rev. B (submitted);

also available in arXiv: 0710.5091v4 25