Report

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 1. 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 2 d2 ˆ V(r ) Ψ k (r ) k Ψ k (r ) 2 2m dr for the wave in periodic potential U(r) = U(r+R) are: Bloch function: k (r ) e i k r uk (r ) Envelope part Periodic (unit cell) part uk (r ) uk (r R ) Felix Bloch 1905, Zürich 1983, Zürich Nearly free electron model Origin of a band gap! Kittel Isolated Atoms Diatomic Molecule Four Closely Spaced Atoms conduction band valence band Band formation allowed energy bands Electronic energy bands Brilluoin zones 2 k k 2m 2 (k): single parabola folded parabola Electron velocity and effective mass in the k-space 2 2 k 2 m* Electron velocity and effective mass in the k-space 2 2 k 2 m* Velocity is zero at the top and bottom of energy band. Electron velocity and effective mass in the k-space 2 2 k 2 m* 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 Increase of the dimension in one direction Increase of the volume 2 1 2 21 2 22 2 23 Low-dimensional structures B A B B A B A z z quantum well z quantum wire quantum dot Discrete States • Quantum confinement discrete states • Energy levels from solutions to Schrodinger Equation • Schrodinger equation: V 2 2 V (r ) E 2m • For 1D infinite potential well ( x) ~ sin( nLx ), n integer x=0 • If confinement in only 1D (x), in the other 2 directions energy continuum T otalEnergy n2h2 8 mL2 p 2y 2m p z2 2m x=L 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 ( x, y, z ) ~ sin( nLxx ) sin( mLyy ) sin( qLzz ), n, m, q integer Energylevels n2h2 8 mL x 2 m2h2 8 mL y 2 q 2h2 8mL 2 z • 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 DoS dN dN dk dE dk dE N (k ) k space vol vol per state 4 3k 3 (2 )3 V Structure Degree of Confinement dN dE Bulk Material 0D E Quantum Well 1D 1 Quantum Wire 2D 1/ E Quantum Dot 3D d(E) Quantum Dots QD as an artificial atom QD as an artificial atom 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 QD as an artificial atom - differences Atom 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 (Energy levels must be sufficiently separated to remain distinguishable under broadening, e.g. thermal) • Small size larger energy level separation 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”) CdSe core with ZnS shell QDs Si nanocrystal, NREL 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 Verma/NIST • Mismatch of bandgaps potential energy well Overgrowth • 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