modeling of energy states of carriers in quantum dots
DESCRIPTION
Modeling of Energy States of Carriers in Quantum Dots. Michael Yu. Petrov, St. Petersburg State University, Faculty of Physics e-mail: [email protected]. OUTLOOK. Motivation Introduction into the Quantum Dot Heterostructures What is a quantum dot? - PowerPoint PPT PresentationTRANSCRIPT
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]
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
zyxEzyxm
,,,,2
2
,2 2
22222
,, aNNN
mE zyx
NNN zyx
,3,2,1,, zyx NNN
azN
ayN
axN
azyx
NNN zyx
sinsinsin2 23
,,
,,,,2
2
rErm
,2 2
0
22
Rk
mE nl
nl
,2,1,0,3,2,1
ln
,2
0,, lm
nllmln r
Rkj
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
dudxdu
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
ijjkijij
ijij
ij
AE RRRr
RRR
E20
31
20
220220 8
383
0
440
120
114 ; ;3
aC
aC
aC
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
heshheeX rr
errE 14 0
222
2/
~~
eheheh
heheX
VV
EEEEE
eSe
hhhe
Ve
EV
02
~ hSh
eeeh
Ve
EV
02
~
Ee
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,,
trxDt
trx
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