1. Quantum Dots in Photonic Structures
Lecture 7: Low dimensional structures
Jan Suffczyński
Wednesdays, 17.00, SDT
Projekt Fizyka Plus nr POKL /11 współfinansowany przez Unię Europejską ze środków Europejskiego
Funduszu Społecznego w ramach Programu Operacyjnego Kapitał Ludzki

2. Plan for today Reminder 2. Doping and holes 3.
Low dimensional structures

3. Wigner-Seitz Cell construction
Form connection to all neighbors and span a plane normal
to the connecting line at half distance

4. Bloch waves Bloch’s theorem: Solutions of the Schrodinger equation
Felix Bloch
1905, Zürich , Zürich
for the wave in periodic potential U(r) = U(r+R) are:
Periodic (unit cell) part
Envelope part
Bloch function:
modified slide from Rob Engelen

5. Nearly free electron model
Origin of a band gap!
Kittel

6. Isolated Atoms

7. Diatomic Molecule

8. Four Closely Spaced Atoms
conduction band
valence band

9. Band formation

10. Electronic energy bands
allowed energy bands

11. Brilluoin zones
e (k): single parabola
folded parabola

12. Electron velocity and effective mass in the k-space

13. Electron velocity and effective mass in the k-space
On crossing the zone boundary, the phase velocity changes direction: the electron is reflected
Velocity is zero at the top and bottom of energy band.

14. Electron velocity and effective mass in the k-space
Velocity is zero at the top and bottom of energy band, the.
Efective mass: m*>0 at the band bottom, m*<0 at the band top, in the middle: m*→±∞ (effective mass description fails here).

16. Doping of semiconductors

17. Holes
Consider an insulator (or semiconductor) with a few electrons excited from the valence band into the conduction band
Apply an electric field
Now electrons in the valence band have some energy states into which they can move
The movement is complicated since it involves ~ 1023 electrons

18. Holes
We can “replace” electrons at the top of the band which have “negative” mass (and travel in opposite to the “normal” direction) by positively charged particles with a positive mass, and consider all phenomena using such particles
Such particles are called Holes
Holes are usually heavier than electrons since they depict collective behavior of many electrons

19. Low-dimensional structures

20. Dimensionality 2 1 2 21 2 22 2 23 Increase of the dimension
in one direction
Increase of
the volume 2 1 2 21 2 22
Jednym z ulubionych fraktali Mandelbrota jest wybrzeże Wielkiej Brytanii. Ktoś mógłby naiwnie sądzić, że brzeg ma dobrze określoną długość. Mandelbrot wykazał jednak, że to nieprawda: wynik zależy od skali, w jakiej wykonujemy pomiary. Gdybyśmy mierzyli długość angielskiego wybrzeża stukilometrowym prętem, pominęlibyśmy takie jego cechy jak ujście Tamizy, Wash, Firth of Forth czy Kanał Bristolski. Pręt o długości 10 km pozwoliłby uwzględnić takie duże struktury, ale pominąłby ujścia mniejszych rzek i niewielkie zatoki. Można byłoby je uwzględnić, używając pręta o długości 1 km lub 1 m, ale wtedy nadal ignorowalibyśmy wiele drobnych elementów, takich jak głazy. Im mniejsza skala pomiaru, tym większa długość linii brzegowej. Wynik zależy od skali ponieważ wybrzeże ma cechy fraktalne: wykazuje przybliżone samopodobieństwo na wszystkich poziomach, od wielkich kanałów do małych kamyków. Wymiar wybrzeża leży zatem pomiędzy 1 i 2 - nie jest to ani regularna krzywa, ani też pełny, dwuwymiarowy obiekt geometryczny. 2 23

21. Low-dimensional structuresB A z B A z z A
quantum dot
quantum wire
quantum well 21

22. Discrete States Quantum confinement  discrete states
Energy levels from solutions to Schrodinger Equation
Schrodinger equation:
For 1D infinite potential well
If confinement in only 1D (x), in the other 2 directions  energy continuum
x=0
x=L V

23. Quantum Wells
Energy of the first confined level
Decrease of the level energy when width of the Quantum Well decreased
W. Tsang, E. Schubert, APL’1986

24. Quantum Wells Energy of confined levels GaAs/AlGaAs Quantum Well
R. Dingle,
Festkorperprobleme’1975

25. In 3D… For 3D infinite potential boxes Simple treatment considered
Potential barrier is not an infinite box
Spherical confinement, harmonic oscillator (quadratic) potential
Only a single electron
Multi-particle treatment
Electrons and holes
Effective mass mismatch at boundary

26. Density of states Structure Degree of Confinement Bulk Material 0D
Quantum Well 1D 1
Quantum Wire 2D
Quantum Dot 3D
d(E)

27. Quantum Dots

28. QD as an artificial atom

29. QD as an artificial atom
Quantum Dot
3D confinement of electrons
Discrete density of electron states
Emission spectrum composed of individual emission lines
Non-classical radiation statistics
(e. g. single photon emission)
Creation of „molecules” possible
Większość ze współcześnie dyskutowanych zastosowań rzeczywiście wykorzystuje właściwości
pojedynczych kropek kwantowych wynikające z w pełni dyskretnego widma energetycznego
i podobieństw do atomów. Stąd teŜ kropki kwantowe są często nazywane sztucznymi atomami.
NaleŜy mieć jednak świadomość, Ŝe analogie te wcale nie są za daleko idące, i Ŝe system
otrzymywany sztucznie przez człowiek w ciele stałym, róŜni się wciąŜ istotnie pod wieloma
względami od naturalnych atomów. Na Rys. 2.3 pokazano porównanie obrazu atomu helu oraz
typowej samo rosnącej kropki półprzewodnikowej z arsenku indu w matrycy z arsenku galu.
Widoczna jest natychmiast róŜnica w skali wielkości, czy geometrii.

30. QD as an artificial atom - differences
Quantum Dot
Size
0.1 nm
10 nm
Confining potential
Coulombic (~1/r2)
Parabolic
Electron binding energy
10 eV
100 meV
Interaction of electron with environement
Weak
Strong (phonons, charges, nuclear spins…)
Anisotropy of confining potential No
Yes (shape, compoistion, strain…)

31. QD size Should be small enough to see quantum effect
kBT at 4.2 K ~0.36 meV --> for electron maximum dimension in 1D ~ nm
Small size  larger energy level separation
(Energy levels must be sufficiently separated to remain distinguishable under broadening, e.g. thermal)

32. QD types and fabrication methods
Goal: to engineer potential energy barriers to confine electrons in 3 dimensions
Basic types/methods
Colloidal chemistry
Electrostatic
Lithography
Epitaxy
Fluctuation
Self-organized
Patterned growth
- „Defect” QDs

33. Colloidal Particles
Engineer reactions to precipitate quantum dots from solutions or a host material (e.g. polymer)
In some cases, need to “cap” the surface so the dot remains chemically stable (i.e. bond other molecules on the surface)
Can form “core-shell” structures
Typically group II-VI materials (e.g. CdS, CdSe)
Size variations ( “size dispersion”)
Si nanocrystal, NREL
CdSe core with ZnS shell QDs
Red: bigger dots!
Blue: smaller dots!
Evident Technologies:
Sample papers: Steigerwald et al. Surface derivation and isolation of semiconductor cluster molecules. J. Am. Chem. Soc., 1988.

34. Electrostatically defined QDs
Only one type of particles (electron or holes) confined
--> (No spectroscopy)

35. Lithography defined QDs
QW etching and overgrowth QW
Etching
V.B. Verma, M.J. Stevens, K.L. Silverman, N.L. Dias, A. Garg, J.J. Coleman and R.P. Mirin. Photon antibunching from a single lithographically defined InGaAs/GaAs quantum dot. Optics Express. Vol. 19, No. 5, Feb. 28,  2011, p
Verma/NIST
Overgrowth
Mismatch of bandgaps  potential energy well
The advantage: QD shaping and positioning
The drawback: poor optical signal (dislocations due to the etching!)

36. Lithography defined QDs
V. B. Verma et al., Opt. Express’2011

37. Lithography Etch pillars in quantum well heterostructures
Quantum well heterostructures give 1D confinement
Pillars provide confinement in the other 2 dimensions
Disadvantages: Slow, contamination, low density, defect formation
A. Scherer and H.G. Craighead. Fabrication of small laterally patterned multiple quantum wells. Appl. Phys. Lett., Nov 1986.

38. Flucutation type QDs Flucutation of QW thickness
Flucutation of QW composition

39. Epitaxy: Self-Organized Growth
Lattice-mismatch induced
island growth
Self-organized QDs through epitaxial growth strains
Stranski-Krastanov growth mode (use MBE, MOCVD)
Islands formed on wetting layer due to lattice mismatch (size ~10s nm)
Disadvantage: size and shape fluctuations, strain,
Control island initiation
Induce local strain, grow on dislocation, vary growth conditions, combine with patterning