1. Modeling of Energy States of Carriers in Quantum Dots
Michael Yu. Petrov,
St. Petersburg State University, Faculty of Physics

2. 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
Conclusion

3. 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

4. INTRODUCTION INTO THE QUANTUM DOT HETEROSTRUCTURES
WHAT IS A QUANTUM DOT?
D. Bimberg, M. Grundmann, N.N. Ledentsov,
Quantum Dot Heterostructures (Wiley, New York, 1999)

5. SELF-ORGANIZED QUANTUM DOTS
TEM of InAs/GaAs QDs (plan-view)
V.G. Dubrovskii, G.E. Cirlin, et al.,
Journal of Crystal Growth (2004).
HRTEM of InP/InGaP QDs
(front-view)
Y. Masumoto, T. Takagahara,
Semiconductor Quantum Dots: Physics, Spectroscopy and Applications,
(Springer, Berlin, 2002).

6. ENERGY SPECTRA (FROM BULK TO HETEROSTRUCTURES)
Typical PL spectrum
of InGaAs/GaAs QDs
Experimentalle Physik II,
Universitaet Dortmund, Germany
D.Bimberg, M.Grundmann, N.N.Ledentsov,
Quantum Dot Heterostructures (Wiley, New York, 1999)

7. SIMPLEST MODELS OF ENERGY STRUCTURE
Cube-like QD with infinite barriers
Sphere-like QD with infinite barriers
For InAs QD (me=0.023m0): cube: a=10 nm E111=0.49 eV
sphere: R0=6.2 nm E10 =0.42 eV
(cube volume = sphere volume)

8. MODELING 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 approximation
multi-band calculations
Coulomb interaction of carriers

9. SHAPE AND VOLUME OF QUANTUM DOTS
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

10. STRAIN PROFILES IN QUANTUM DOTS
Harmonic Continuum Elasticity Theory (CE)
Atomistic Valence-Force- Field Model (VFF)
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

11. STRAIN PROFILES IN QUANTUM DOTS (CONTINUE)
C. Pryor et al., J. Appl. Phys. 83, (1998)

12. INFLUENCE OF STRAIN ON BAND PROFILES
C. Pryor, Phys. Rev. B 57, (1998)

13. COMPARISON OF DIFFERENT METHODS OF CALCULATION OF ENERGY STATES OF CARRIERS
C. Pryor, Phys. Rev. B 57, (1998)

14. ELECTRON AND HOLE DENSITIES
O. Stier, M. Grundmann, D. Bimberg, Phys. Rev. B 59, (1999)

15. OPTICAL EXCITONIC TRANSITIONS
Strong Confinement Regime (simple consideration)
Hartree Approximation Ee Eh
E= Ee + Eh -EX
O. Stier, M. Grundmann, D. Bimberg,
Phys. Rev. B 59, (1999)

16. EXCITONIC SPECTRUM OF INGAAS QUANTUM DOTS
1e-1h
2e-2h
3e-3h

17. MODIFICATIONS OF THE ELECTRONIC STATES OF InGaAs QUANTUM DOTS EMBEDDED IN BOWED AIRBRIDGE STRUCTURES
left-up: SEM of structure;
right: PL spectrum;
left-down: Energy Shift
T. Nakaoka, T. Kakitsuka, et al.,
Journ. Appl. Phys. 94, 6812 (2003).

18. INFLUENCE OF ULTRA-HIGH MAGNETIC FIELD ON ENERGY STRUCTURE OF InGaAs/GaAs QUANTUM DOTS
Fock-Darwin spectrum
(left (c) – experiment,
right – 8-band k·p-model)
S. Raymond, S. Studenikin, et al.,
Phys. Rev. Lett. 92, (2004).

19. MODELING OF ENERGY SPECTRA OF ANNEALED INAS/GAAS QUANTUM DOTS
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: v4

20. INTERDIFFUSION OF INDIUM AND GALLIUM DUE TO THERMAL ANNEALING OF QUANTUM DOTS

21. MODIFICATION OF CARRIER DENSITIES DUE TO THERMAL ANNEALING
Indium concentration distribution
Electron density distribution
Hole density distribution

22. EXCITONIC SPECTRA OF ANNEALED QUANTUM DOTS

23. 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.)

24. Thank You For Your Attention!

25. 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, (1998).
C. Pryor, Phys. Rev. B 57, (1998).
O. Stier, M. Grundmann, D. Bimberg, Phys. Rev. B 59, (1999).
T. Nakaoka, T. Kakitsuka, et al., J. Appl. Phys. 94, (2003).
S. Raymond, S. Studenikin, et al., Phys. Rev. Lett. 92, (2004).
M.Yu. Petrov, I.V. Ignatiev, et al., Phys. Rev. B (submitted); also available in arXiv: v4