Quantum Dots in Photonic Structures Lecture 7: Low dimensional structures Jan Suffczyński Wednesdays, 17.00, SDT Projekt Fizyka Plus nr POKL.04.01.02-00-034/11 współfinansowany przez Unię Europejską ze środków Europejskiego Funduszu Społecznego w ramach Programu Operacyjnego Kapitał Ludzki

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

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

Bloch waves Bloch’s theorem: Solutions of the Schrodinger equation Felix Bloch 1905, Zürich - 1983, 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

Nearly free electron model Origin of a band gap! Kittel

Isolated Atoms

Diatomic Molecule

Four Closely Spaced Atoms conduction band valence band

Band formation

Electronic energy bands allowed energy bands

Brilluoin zones e (k): single parabola folded parabola

Electron velocity and effective mass in the k-space

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.

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).

Doping of semiconductors

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

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

Low-dimensional structures

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

Low-dimensional structures B A z B A z z A quantum dot quantum wire quantum well 21

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

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

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

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

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

Quantum Dots

QD as an artificial atom

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.

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…)

QD size Should be small enough to see quantum effect kBT at 4.2 K ~0.36 meV --> for electron maximum dimension in 1D ~100-200 nm Small size  larger energy level separation (Energy levels must be sufficiently separated to remain distinguishable under broadening, e.g. thermal)

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

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: http://www.evidenttech.com/products/core_shell_evidots/overview.php Sample papers: Steigerwald et al. Surface derivation and isolation of semiconductor cluster molecules. J. Am. Chem. Soc., 1988.

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

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. 4182. 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!)

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

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.

Flucutation type QDs Flucutation of QW thickness Flucutation of QW composition

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