Modeling of Energy States of Carriers in Quantum Dots
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MODELING OF ENERGY STATES MODELING OF ENERGY STATES OF CARRIERSOF CARRIERSIN QUANTUM DOTSIN QUANTUM DOTS
Michael Yu. Petrov,St. Petersburg State University, Faculty of Physicse-mail: [email protected]
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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
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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
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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?
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SELF-ORGANIZED QUANTUM DOTS
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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).
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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
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SIMPLEST MODELS OF ENERGY STRUCTURE
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Cube-like QD with infinite barriers
Sphere-like QD with infinite barriers
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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
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SHAPE AND VOLUME OF QUANTUM DOTS
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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
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STRAIN PROFILES IN QUANTUM DOTS
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Harmonic Continuum Elasticity Theory (CE)
Atomistic Valence-Force-Field Model (VFF)
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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
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STRAIN PROFILES IN QUANTUM DOTS(CONTINUE)
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C. Pryor et al., J. Appl. Phys. 83, 2548-2554 (1998)
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INFLUENCE OF STRAIN ON BAND PROFILES
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C. Pryor, Phys. Rev. B 57, 7190-7195 (1998)
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COMPARISON OF DIFFERENT METHODS OF CALCULATION OF ENERGY STATES OF CARRIERS
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C. Pryor, Phys. Rev. B 57, 7190-7195 (1998)
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ELECTRON AND HOLE DENSITIES
14O. Stier, M. Grundmann, D. Bimberg, Phys. Rev. B 59, 5688-5701 (1999)
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OPTICAL EXCITONIC TRANSITIONS
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Strong Confinement Regime (simple consideration)
Hartree Approximation
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O. Stier, M. Grundmann, D. Bimberg,Phys. Rev. B 59, 5688-5701 (1999)
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EXCITONIC SPECTRUM OF INGAAS QUANTUM DOTS
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1e-1h
2e-2h
3e-3h
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MODIFICATIONS OF THE ELECTRONIC STATES OF InGaAs QUANTUM DOTS EMBEDDED IN BOWED AIRBRIDGE STRUCTURES
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left-up: SEM of structure;right: PL spectrum;left-down: Energy ShiftT. Nakaoka, T. Kakitsuka, et al.,Journ. Appl. Phys. 94, 6812 (2003).
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INFLUENCE OF ULTRA-HIGH MAGNETIC FIELD ON ENERGY STRUCTUREOF InGaAs/GaAs QUANTUM DOTS
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Fock-Darwin spectrum(left (c) – experiment,right – 8-band k·p-model)S. Raymond, S. Studenikin, et al.,Phys. Rev. Lett. 92, 187402 (2004).
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MODELING OF ENERGY SPECTRA OF ANNEALEDINAS/GAAS QUANTUM DOTS
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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
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M.Yu. Petrov, I.V. Ignatiev et al., Phys. Rev. B (submitted);also available in arXiv: 0710.5091v4
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INTERDIFFUSION OF INDIUM AND GALLIUMDUE TO THERMAL ANNEALING OF QUANTUM DOTS
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MODIFICATION OF CARRIER DENSITIES DUE TO THERMAL ANNEALING
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Electron density distribution
Indium concentration distribution
Hole density distribution
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EXCITONIC SPECTRA OF ANNEALED QUANTUM DOTS
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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.)
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Thank You For Your Attention!Thank You For Your Attention!
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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