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5. Zero-dimensional systems 1

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Contents 2 Types of QDs Metal clusters. Electronic properties Fullerenes Synthetic nanocrystals. Self-assembled QDs. QDs produced from heterostructures and lithografic etching Optical properties Coulomb blockade and single electron devices Summary

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Types of QDs Synthetic nanocrystals Self-assembled QDs QDs produced from heterostructures and lithographic etching Clusters of atoms grown from vapour-phase condensation increasing size Fullerenes 3

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Metal clusters. Electronic properties Clusters of atoms grown from vapour-phase condensation Metal cluster superatoms Nucleus Metal Cluster Atom Hydrogen (Z=1)Lead (Z=82) 4

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Shell-structures Metal clusters. Electronic properties closed shells of 2, 8, 20, 40, 58, 92, 138... electrons, sizes specially stables magic numbers of atomic clusters valence electrons in a spherical box Parabolic (harmonic osc.) V r 2 square well shell structure of nuclei 5

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Metal clusters. Electronic properties Jellium (sphere) model Electron spill-out 6

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Magic numbers of alkali metal atom clusters ion-mass spectrometer Fig 8.11 Supersonic beams produced by mixing the metal vapour with an inert carrier gas and ejecting the mixture through a nozzle Metal clusters. Electronic properties 7

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Spherical shell closing N=1430 atoms complete icosahedral clusters (e.g. Al 13 ) N=25000 atoms..... bulk material Metal clusters. Electronic properties 8

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Photoabsorption cross section of two potasim cluster ions Plasmon resonances Measured and calculated plasma resonances of potasim cluster ions Metal clusters. Electronic properties 9

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Fullerenes C 60 (next chapter) 10

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Synthetic nanocrystals. Obtained by chemical methods. Precipitates in colloids. Nanocrystals in insulating matrix, e. g., CdS, CdSe in glassy matrix, CuCl in NaCl crystals, Si, Ge... Size control (~1nm-> ~200nm) 11

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From the chemical point of view, QDs are molecular agregates. Cd 32 S 55 is a piece of CdS lattice (zincblende) Synthetic nanoparticles interesting because of optical properties. Reducing the size the gap changes, Higher fusion temperatures, estructural changes... e.g. the gap of CdSe can be tuned from red (1.7eV) to green (2.4 eV) when the particle diameter is reduced from 200 nm to 2 nm Aplications: lasers, LEDS, biosensors.... TEM image of wurtzite (hexagonal) Cd Se nanocrystals - Semiconductor Clusters, Nanocrystals and Quantum Dots, A. Paul Alisavatos, Science 271, 933 (1996) - Perspectives of the Physical Chemistry of Semiconductor Nanocrystals, A. Paul Alisavatos, J. Phys. Chem. 100, 13226 (1996) Synthetic nanocrystals. 12

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Synthetic nanocrystals. Higher fusion temperatures Fluorescence of semiconductor nanocrystals 13

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Self-assembled QDs. Obtained by deposition with MBE on semiconductor material of wider gap (e.g. In on Ga As). Nucleation islands (In x Ga 1-x As) form spontaneously. QDs are obtained when these islands are covered with an epitaxial layer of a second semiconductor of wider gap. 14

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- Self-assembling of nanocrystals fcc superlattice of CdSe nanocrystals of 48 in diameter - Opal: naturally occurring colloidal crystal of silica particles Interaction between QDs in supperlattice red-shift in optical absorption with respect to isolated nanocrystal Self-assembled QDs. 15

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Self-assembled QDs. 16

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Applications Optical and optoelectronical devices M. Bayer et al., Nature 405, 923 (2000) It is possible to create on individual exciton in a QD, by optical excitation. The study of the light emitted after the e-hole recombination provides information about the structure of the QD. Application: new lasers. Quantum rings Optical and optoelectronical devices Computation, quantum information P. Michler et al., Science 290, 2282, 2000 Combinig QDs with light cavities it is possible to perform quantum optics experiments, where emission is due to oneindividual photon. Applications : information and quantum computing A. Lorke et al., Phys Rev. Lett. 84, 2223 (2001) The energies of a quantum ring depend on the external magnetic field due to flux.crossing the ring Self-assembled QDs. 17

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QDs produced from heterostructures and lithografic etching Lithographic etching Reed et al., PRL 60, 535 (1988) Obtained from GaAs/AlGAAs heterostructures, whre a 2D electron gas has been formed. Lateral (in the plane of the 2D electron gas) QDs are produced by lithographic etching. Also vertical QDs are produced bt MBE growth in the direction perpendicular to the electron confinement plane. Size: 10-20 nm to ms 18

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Vertical quantum dots QDs produced from heterostructures and lithografic etching 19

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Vertical QDs from lithographic etching: different shapes QDs produced from heterostructures and lithografic etching 20

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QDs produced from heterostructures and lithografic etching 21

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QDs obtained by lithography Double QDs QDs produced from heterostructures and lithografic etching 22

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Optical properties Assume spherical QDs... 1)Weak- confinement regime R Mott-Wannier exciton radius Reduced mass for e - -hole bound state Exciton binding energy Exciton mass for center-of-mass motion dielectric constant of semiconductor 23

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Weak confinement Strong confinement Optical properties 24

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2)Strong- confinement regime weak e - -hole correlation e - and hole levels particle-in-a-box model: e.g. s-sate Optical properties 25

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blue shift in absorption e - -hole pair 1s-state (n=1) e - -hole interaction inside sphere remanent of exciton interactions due to quantum confinement effects Optical properties 26

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CdS in glass strong confinement CuCl in NaCl weak confinement theory R 30 29 77 Blue shift of energy gap Optical properties 27

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Optical absorption intensity, CdS in glass blue shift R= 5nm 1.7 nm Enhanced exciton effects (subbands) Optical properties 28

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Radiative interband transitions e.g. bulk Si in indirect- band-gap semiconductors low intensity porous Si: - etching isolated Si-columns substrate surface k not a good quantum number direct transitions without phonon allowed Photoluminiscence, electroluminiscence direct- band-gap semiconductor Optical properties 29

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indirect- band-gap direct transitions phonon Optical properties 30

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Coulomb blockade Single electron transistors (SETs) Diferencial conductance vs. V gate Peaks corresponding to one electron addition Coulomb blockade model Electron non-coherent tunnelling Coulomb blockade and single electron devices 31

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Coulomb blockade and single electron devices Different modes to confine electrons in QDs Coplanar metal QD Etched vertical QD (mesa) Potential Lateral GaAs QD insulator QD GaAs Al x Ga 1-x As Source GaAs Gate Electrons at interface

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Differencial conductance vs. gate voltage Coulomb blockade and single electron devices

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Coulomb blockade model Adding 1 e- E=e 2 /2C, C=capacity between dot and surrounding threshold energy above E F for current flow - If k B T < e 2 /2C no current, Coulomb blockade only evident at very low T

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But a current can be made to flow varying Vg voltage E=QV g + Q 2 /2C parabola with min. at Q 0 =-CV g, but charge is quantized Q=Ne exact for metallic QDs, 10 7 electrons, extra charge on surface Q 2 /2C exact Approximative for semiconductor QDs, N < 50 electrons, confinement of electron wave function espatial size quantization discrete energy spectrum Coulomb blockade and single electron devices 35

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Coulomb blockade and single electron devices V g =Ne/C E(N+1)-E(N)= e 2 /2C V g =(N+1/2)e/C E(N+1)=E(N) electron-hole tunneling gap e 2 /C E=QV g + Q 2 /2C 36

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Electron transport controlled e - by e - single electron transistor Coulomb blockade and single electron devices 37

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A single electron transistor made from a CdSe nanocrystal, D.L. Klein et al, Nature 389, 699 (1997) It is possible to add electrons one by one into the nanocrystal Distance between electrodes: 10 nm The particle (d=5.5 nm) is joined to the contacts by hexanoditiol molecules, which form tunel barriers of 1.2 nm between particle and leads. Coulomb blockade and single electron devices 38

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Coulomb blockade and single electron devices Double QD 39

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Coulomb blockade and single electron devices Coulomb blockade and Kondo effect 40

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Coulomb blockade and single electron devices What is the Kondo effect? The Kondo physics appears at very low T, when there is a quantum impurity (with charge and spin freedom) coupled to the Fermi sea itinerant electrons by tunnel effect. The virtual tunnel effect processes, together with the strong Coulomb repulsion, give rise to non-trivial physical effects, effective spin-flip originating the Kondo effect. + spin-flip The Kondo effect appears in the DOS of the impurity, as a sharp p resonance placed at the fermi level of the itinerant electrons at T