IFUAP-BUAP Mini-curso en Nanoestructuras

Transcription

1 IFUAP-BUAP Mini-curso en Nanoestructuras Parte II. Sergio E. Ulloa Ohio University Department of Physics and Astronomy, CMSS, and Nanoscale and Quantum Phenomena Institute Ohio University, Athens, OH Supported by US DOE & NSF NIRT

2 Preguntas? Más info? nano.gov

3 Resumen/Outline Quantum dots confinement vs interactions How to make / study them Coulomb blockade & assorted IV characteristics Optical effects excitons: selection rules, field effects Transport in complex molecules: the case of DNA

4 Quantum Dots: L ~ λ good things come in small packages Confinement: KE~ k 2 ~ L -2 Interactions: PE ~ q 2 /L For L~100nm, PE ~ 1meV while KE ~ 0.5meV (in GaAs) Total energy E = KE + PE For L~5nm, PE ~ 20meV ; KE ~ 200meV

5 Lithographically Quantum dot fabrication Kouwenhoven et al, U Delft Weis et al, MPI Stuttgart Ensslin et al, ETH Zurich

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7 Quantum dot fabrication Self-assembly Stranski Krastanow islands MBE in-plane densities ~ cm -2 size variations < 10% sharp photoluminescence features, frequency size

8 AlAs and GaAs antidots in InAs Tenne et al, PRB May 2000

9 Colloidal dots Quantum dot fabrication Chemical synthesis Quantum Dot Corp. CdSe, CdS, InP, etc Size ~ 5nm diameter Uses for biotags

10 soluble metal salt (AgNO 3 ) excess NaBH 4 colloidal METAL dots in dendrimers 0.6 -H 2 O Absorbance nm 481 nm Wavelength (nm) 800 Au or Ag ~3-5nm dots ph or hydration changes interdot separation aggregation changes interactions & color G. Van Patten, Ohio University

11 QD w/tunable size and e - number ~ artificial atoms I Electronic transport - Low bias: ground state Coulomb blockade High bias: excited states selection rules(!) Rings, phases & resonances ω ω Incident/outgoing photons - Visible/optical: exciton Far infrared: internal multi-electron excitations Raman: excitons/confined phonons B Capacitance - Ground state vs B-field Combination w/optics & in-plane transport

12 Resumen/Outline Quantum dots confinement vs interactions How to make / study them Coulomb blockade & assorted IV characteristics Optical effects excitons: selection rules, field effects Transport in complex molecules: the case of DNA

13 Coulomb blockade I

14 B field effects

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17 Ramirez et al, PRB 1999 QD molecules Blick et al Science 2002 in series in parallel future? multidots for quantum computing

18 Kondo effect Goldhaber-Gordon et al Nature 1998

19 Kondo primer zero-bias feature Kouwenhoven & Glazman, Phys World 2001

20 Aharonov-Bohm effect B 2 p 1 e H = p A 2m 2m c 2 Φ ρ B= A; A= Bρϕˆ

21 Phases, dots, rings and resonances 1. Phase in experiments AB interferometer 2. Phase of a resonance (single particle) 3. CB in QD ~ SP resonance? 4. What s the Fano effect / lineshape? 5. Fano as probe of coherence in QD? 6. QD in Kondo regime phases in expts? 7. Fano+Kondo to probe coherence in QDK?

22 Phase in experiments AB interferometer Schuster et al, Nature 1997 ~ CB peaks AB oscill s t QD e iθ Yacoby/Heiblum PRL 1995

23 Nature 1998 weaker AB signal when QPC is on

24 AB interferometer: phase measurer QD G~ t = t + e t 2 i ϕ 2 CE QD dir ϕ = 2π Φ/ Φ 0 G~ t + t + 2 t t cos( θ θ 2 2 QD dir QD dir QD dir e - dir ϕ) If θ dir is indep of V gate θ QD can be measured A pure resonance has the Breit-Wigner form: i Γ /2 tqd = C = t QD e E E + i Γ/2 θ QD = θ + C n tan 2(E E ) Γ 1 n iθ QD θ QD /π

25 BW SP resonances OK. coherent propagation through QD CB charging ~ not important but. sequence of BW resonances would be expected to accumulate while phase resets to 0 as t QD ~ 0!!? overlapping resonances BW?? e-e interactions? Why? N theory papers nice disc: Aharony et al cond-mat/ G(B)=G(-B) not valid here (open device θ 0,π)

26 Fano effect / lineshape interference of discrete autoionized state with a continuum asymmetric peaks in atomic excitation spectra Ugo Fano, PR 1961 In QD; expt: Göres PRB 2000 th: Clerk PRL 2001 In AB interf + QD; expt: Kobayashi PRL 2002

27 Fano ϕ & {ψ E } mix V E Ψ E = aϕ + E' b E' ψ E' 1 a = sin ( E) π V E 2 1 E (E) = tan E E ϕ F(E E' ; be' = cos (E) δ (E E') π VE E E' π V V sin (E) ; F(E ) 2 E' ) = = shift of resonance due to continuum E' V E E' excitation of Ψ E via an operator T from an initial state I yields: sin Ψ E T I = Φ T I ψ E T I cos π V E' E' Φ= + ϕ E' * E V ψ E E' modified state due to mix

28 Fano lineshape Φ T I E Eϕ F 2 q = ; ε = cot = ; Γ= 2 π V * E πv ψ T I Γ /2 E E E ΨE T I (q + ε ) = 2 2 ψ T I 1+ ε 2 2 = transition prob via "autoionized" state Is G ~ t QD + t dir 2 a Fano line? iβ z tdir = e G d ; tqd = ε + i with G = G ε + q ε + 1 d 2 z q = i+ e G d iβ 2 Notice prob can vanish due to interference Clerk PRL 2001

29 Fano as probe (proof?) of coherence in QD (CB) Kobayashi PRL 2002 direct paths through QD? how is QD intrinsic width affected by ring? decoherence? 1 vs N passes?

30 QD in Kondo regime phases in expts? peaks deform AB osc s clear Ji, Heiblum, Science 2000 phase lapses in CB but plateaus in K valley ~ π

31 phases in Kondo regime phase evolves from plateaus ~ π to lapses to 0 as Γ decreases th: Gerland PRL 2000 K res phase shift ~ π/2 (NRG)

32 phases in Kondo regime w/temperature w/dc bias

33 Fano+Kondo to probe coherence in QDK? direct ~ W Hofstetter PRL 2001 (~Bulka PRL 2001) r 1 dot and for e= 2[ ε +RΣ(0)]/ Γ 2 2 (e + q) sin b α 2 2 e + 1 e 2 4 R L/ ; q (1 G( ω) = [ ω ε Σ( ω)] b dot b AB phase/flux ϕ g = T + "generalized Fano form" + 1 α = Γ Γ Γ = α T)/T cosϕ

34 references on phases/kondo/resonances Yacoby, PRL 74, 4047 (1995) Schuster, Nature 385, 417 (1997) Aharony, cond-mat/ Fano, PR 124, 1866 (1961) Clerk, PRL 86, 4636 (2001) Kobayashi, PRL 88, (2002) Ji, Science 290, 779 (2000) Gerland, PRL 84, 3710 (2000) Bulka, PRL 86, 5128 (2001) Hofstetter, PRL 87, (2001)

35 IFUAP BUAP Minicurso en Nanoestructuras Tarea # 1 Entrega: 22 Julio [40pts] (a) Calcule el espectro para una partícula de masa m confinada en una caja rectangular en 3D y 2D de radio R. Suponga que la caja tiene paredes infinitamente duras. (b) Describa como varía la energía de confinamiento con respecto a R. (c) Escriba la forma completa de los eigenstados en 3D/2D. (d) Cuál es el valor de la energía (en ev) para los dos primeros estados en 2/3D si la masa de la partícula es 0.067m 0 (con m 0 la masa del electrón libre), y el radio es R = 5, 50, 100nm? Hint: funciones de Bessel. 2. [60pts] (a) Estime el efecto de la interacción de Coulomb entre electrones en un punto cuántico de radio R, utilizando U = e 2 /Rɛ, donde ɛ es la constante dieléctrica del material ( 12 para materials típicos), si el radio del punto es 5 o 100nm. Compare esta estimación con la diferencia entre los dos primeros niveles en un punto como se modeló en 1 arriba, = E 2 E 1. Para qué radio R son estas dos cantidades iguales (U = )? Haga la estimación tanto en 2D como en 3D. (b) Use teoría de perturbaciones para hacer la estimación más confiable: Ψ n (1)Ψ m (2) V (1 2) Ψ k (1)Ψ l (2), en donde V (1 2) = e 2 /ɛ r 1 r 2 es la interacción de Coulomb entre dos partículas, y las varias Ψ j son las funciones de onda (modeladas/escritas en 1(c) arriba). Use cualquier método de integración, numérica incluso, para evaluar (o al menos estimar) la integral en el caso que los dos electrónes están en el estado base (y diferente spin, por supuesto) en un punto en 3D. Compare con (a) y comente. Hint (aunque no esencial): 1 r = 4π q 2 e iq r d 3 q

36 IFUAP-BUAP Mini-curso en Nanoestructuras Parte II. Toma 2 Sergio E. Ulloa Ohio University Department of Physics and Astronomy, CMSS, and Nanoscale and Quantum Phenomena Institute Ohio University, Athens, OH Supported by US DOE & NSF NIRT

37 Resumen/Outline Quantum dots confinement vs interactions How to make / study them Coulomb blockade & assorted IV characteristics Optical effects excitons: selection rules, field effects Transport in complex molecules: the case of DNA

38 Quantum dot fabrication Self-assembly Stranski Krastanow islands MBE in-plane densities ~ cm -2 size variations < 10% sharp photoluminescence features, frequency size

39 Photoluminescence in SINGLE QD Excitons Excited excitons (s- and p-shell) Biexcitons, X -, X --, etc. laser power Findeis et al, Sol. State Comm, 2000

40 Excitons and dot shapes Dot asymmetries reflected in exciton properties: Binding energy vs dot size Oscillator strength / optical response Influence of magnetic field Raman differential cross section and intensity --- experiments and phonon mode confinement: Selection rules Carrier masses Scanning experiments see part I Trallero & Ulloa PRB 1999 Quantum rings: excitonic Aharonov-Bohm effect for neutral/polarizable entity

41 Simple geometry: Parabolic confinement V(x,y) w y x B L y w x L x y V ez L z z E g H = HCOM + Hrel + Hc V hz H = H + H + H e h eh 2 1 e H j = p j ± A j + mj( ωxxj + ωyy j) + V( zj) 2m c 2 H eh j 2 e = ε r r Harmonic Oscillator in 2D e h Song& SU; Pereyra & SU PRB 2000 effective confinement increases with B field

42 off-diagonal Coulomb interaction exact expression: Song & SU PRB 1995 off-diagonal one-particle elements η = ω ω η = ω ω B= 0 x / y x / y B>> 1 η 1 (circular limit)

43 Coulomb interactions in harmonic oscillator basis

44 binding energy binding energy = E exc E 0 = E b E b ~ 1/L L x /L y = 1:1 2:1 + 3:1 saturates to 2D value for L>>a B & all shapes exciton size exciton shrinks as L 0, r S L but r s a B as L>> a B

45 χ = linear optical susceptibility ~ PLE signal ~ optical response COM gnd state & replicas excited states of internal degrees of freedom area of dots = 5x5 nm 2

46 Magnetic field effects diamagnetic shift orbital Zeeman splitting

47 binding energy exciton size η=1 η=1.25 suppression of Zeeman orbital splitting asymmetry of structure

48 SAQD RINGS!! smallest COHERENT ring potential for electrons and/or holes 80nm Lorke et al PRL 1999

49 J Garcia 1999

50 Is there AB effect for excitons? flux lines e e φ = Ad l = Φ c c v v 0 AB is the nonadiabatic version of Berry phase in a flux exciton charge = ZERO no ABE? M. Berry, Proc. R. Soc. Lond. A 392, 45 (1984) BUT... Ground state of 1D excitons (Romer & Raikh 2000):

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52 Results for microscopic calculation Song & SU PRB 2001 soft/parab walls hard wall binding energy NO AB oscillations!!

53 AB oscillations seen if ring is narrower ~ 1D like width = 10nm heavy hole light hole Hu et al PRB 2001

54 BUT excitons are polarizable! flux lines 80 nm flux lines Φ e v 0 e- v e- e- h+ R Φ h+ e φe φh = ( Φ e Φ h ) c net flux is nonzero!

55 strong Coulomb interaction limit: * 0 = ( e + h)/2 0 R R R a 2 2 Φ , Φ = π e h Φ 0 0 E exc = L + + E 2 MR ( R R ) B. weak interaction limit: R a * 0 0 correlated motion Φ e Φ h exc = 2 e h 2meRe Φ 0 2mhR Φ h 0 E L L, 2 e( h) π Re( h) B, Φ = L = Le + Lh independent motion

56 weak interaction limit optical emission strongly suppressed in B field windows dark window L tot non zero bright window L tot = 0 Govorov et al PRB 2002

57 strong interaction limit optical emission strongly suppressed after critical B field effective persistent current associated with Berry s phase and angular momentum in ground state

58 Optical emission from a charge tunable quantum ring Warburton et al Nature 2000

59 Optical detection of the AB effect in a charged particle ~30nm Bayer et al PRL 2003 ~90nm no oscillations in X 0 only diamagnetic shift lithographic ring PL

60 experiments calculations reasonable agreement w/calculations interactions do not change period and only slightly the amplitude

61 Science 2000

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63 IFUAP BUAP Minicurso en Nanoestructuras Tarea # 2 Efecto Aharonov-Bohm en un anillo conductor. (a) En la figura debajo se muestra la conductancia experimental de un anillo construido en un gas de electrones de dos dimensiones en un semiconductor. Estime de la gráfica el período de éstas oscilaciones de Aharonov-Bohm. El eje horizontal es campo magnético en milésimas de Tesla. Por cierto, porqué se esperaría que la conductancia G fuera simétrica al invertir B, tal y como aparece en la figura? (b) Suponiendo que éstas oscilaciones son producidas por el efecto AB, la conductancia G se puede aproximar por: G G 0 1+ exp(iφ/φ 0 ) 2, en done Φ=BA es el flujo a través del anillo de area A, y Φ 0 =h/e es el cuanto de flujo magnético. Deduzca entonces el diámetro del anillo. (c) Suponga que el anillo fuera ancho y que entonces tuviera mas de un canal transversal. Si el anillo es de 1 micra de radio interno y 2 micras de radio externo, estime el numero de canales permitidos para una longitud de Fermi de 50nm. De este número, estime que rango de períodos de oscilaciones AB se podrían ver en el experimento. Se esperaría que las oscilaciones sobrevivieran? Explique brevemente su respuesta.

64 IFUAP-BUAP Mini-curso en Nanoestructuras Parte II. Toma 3 Sergio E. Ulloa Ohio University Department of Physics and Astronomy, CMSS, and Nanoscale and Quantum Phenomena Institute Ohio University, Athens, OH Supported by US DOE & NSF NIRT

65 Photons, excitons, spins, energy transfer & entaglement all in QDs Gammon/Steel optical control of quantum dot state Imamoglou single photon source from quantum dots Klimov Förster coupling between quantum dots Zrenner coherent control of quantum dot photodiode

66 Gammon/Steel optical control of quantum dot state

67 Gammon/Steel optical control of quantum dot state Bonadeo et al PRL 1998 Naturally Formed GaAs QDs Al mask with apertures 500 Å GaAs cap layer 500 Å Ga 0.3 Al 0.7 As 42 Å GaAs layer 250 Å Ga 0.3 Al 0.7 As 42 Å 40nm STM image showing the island formation and elongation. Gammon et al., Phys. Rev. Lett.76, 3005 (1996) Samples are provided by D.S. Katzer, D. Park, and E. S. Snow, NRL

68 B QD Magneto-excitons * - g factor Photoluminescence of a Single Exciton splitting shift T T T Magnetic Field (Tesla) Weak field limit: H int = g * exµ B B 0 S z + αb 2 0 g * ex = g * e + g * hh g * hh = 6κ + 27q/2 Zeeman splitting T T T T T * First reported by S. W. Brown et al., Phys. Rev. B 54, R (1996) Energy (mev)

69 Photoluminescence in SINGLE QD Excitons Excited excitons (s- and p-shell) Biexcitons, X -, X --, etc. laser power Findeis et al, Sol. State Comm, 2000

70 Experimental Techniques Linear Spectroscopy: Photoluminescence Indirect probe of exciton resonances Requires spectral diffusion of excited carriers CW Frequency Stabilized Dye Lasers Coherent Nonlinear Spectroscopy: CW differential transmission Resonant excitation Probe coherent interaction in the system E NL (ε) ~ χ (3) (ε) E pump E* pump E probe I signal = Re ( E NL E* probe ) D Steel et al U Michigan RF Electronics Reference Lock-in Amplifier E pump E NL Detector E probe AOM ƒ~100 MHz E probe 5 K Imaging

71 Non-degenerate Nonlinear Spectroscopy: Advantages Probe single excitonic state decay dynamics Measure both T 1 and T 2 * Probe coupling between different excitonic states Probe inter-dot energy transfer and dot-dot coupling Study excited states of excitons Probe multi-exciton correlation effects ; Study the coherent interaction between exciton doublet. * Nicolas H. Bonadeo et al., PRL, 81, 2759 (1998)

72 hh Excitonic Level Diagram c v B 0 = 0 B 0 0 σ σ σ- state Scattering states Two-exciton state E Bound state of multi-excitons σ+ state Ground state: 0-exciton state Non-degenerate experiments can excite both σ+ and σ- excitons by varying the frequency and the polarization of the excitation beams (pump and probe beam).

73 Exciting Two Electrons 3 2 c v σ- state Pump σ σ σ+ state σ level diagram in 2-electron basis + Probe σ+ 3 2 Ground state Resonant and coherent excitation of two electrons Two contributions: 1. Incoherent: Ground state depletion 2. Coherent: Zeeman coherence between σ- and σ+ state

74 Optically Entangling Two Systems: the Importance of Coupling coupling H total = H 0 + H H 0 + H Without coupling Product state of the two subsystems. A strong coupling allows one system to see the excitation of the other. Coulomb interaction between charged particles: trapped ions Magnetic dipole interaction: NMR systems Exciton-exciton Coulomb coupling: excitons Mutual coherence between E and E is essential.

75 Coulomb Correlation*: Coulomb interaction between electron-hole pairs within single QD 1 2 σ- state Two-exciton state 2 E Bound state of multi-excitons Pump σ Probe σ+ 3 2 Scattering states σ+ state Ground state: 0-exciton state Two-exciton state cancels the signal due to the symmetry in the level diagram. σ- σ But Two electrons are involved within a single QD. Coulomb interaction between the two excitons is important * Kner et al Phys. Rev. Lett. 78, 1319 (1997) Ostreich et al Phys. Rev. Lett. 74, 4698 (1995)

76 Turn on the Coulomb Correlation - Pump σ g σ σ+ + Probe σ+ Turn off the Coulomb Correlation - + Pump σ g Probe σ+ Pump: σ Groundstate Depletion Zeeman Coherence γ + = + γ γ = γ = =1 No Signal!! Total Signal Probe ( γ ) Probe ( γ )

77 Experiment : Coulomb Correlation and Zeeman Coherence σ+ σ Pump: σ + γ = γ = + γ = γ=1 Ground state depletion Zeeman Coherence Total Signal Degenerate and Non-degenerate Coherent Nonliear Response Evidence for Zeeman Coherence in Single QDs Degenerate Non-degenerate pump σ σ σ+ 1.3 Tesla 5K probe σ Probe ( γ ) Creation of two-electron entanglement Energy (mev)

78 Ultimate Goal Combine: Optical control of individual QDs Long spin lifetimes QD nanostructure engineering To produce: Qubit register of QD spins Coherence of a single QD can be controlled Gammon et al have demonstrated this for excitons [Stievator et al., PRL (2001)] Next: QNOT and other quantum gates Spins have long coherence times: dephasing times T2* = 5ns-300ns [Dzhioev et al., PRL (2002)] Next: explore in single QDs; optical read-out and initialization schemes

79 Science 2001

80 Imamoglou single photon source from QDs

81 Imamoglou single photon source from QDs Michler et al., Science 290, 2282 (2000) It is difficult to isolate a single photon, or fix the number of photons in a pulse Fluctuations in photon number mask the quantum features of light A stable train of laser pulses have Poissonian (photon number) statistics It is desirable to have single photon sources: single photon turnstile Complete regulation of photon generation

82 Single Photon Sources Quantum Cryptography: Quantum Computation: Available sources: Highly attenuated laser pulse Parametric down conversion secure key distribution by single photon pulses single photons + linear optical elements E. Knill, R. Laflamme, and G.J. Milburn, Nature 409, 46 (2001) Poisson fluctuations Random generation of single photons Possible solution: Deterministic (triggered) single photon emission: Single Photon Turnstile Device A. Imamoglu, Y. Yamamoto, Phys. Rev. Lett. 72, 210 (1994) Experiments: Coulomb blockade of electron/hole tunneling in a mesoscopic pn-junction: J. Kim et al. Nature 397, 500 (1999) Single Molecule at room temperature: B. Lounis and W.E. Moerner, Nature 407, 491 (2000) Single InAs Quantum Dot in a microcavity: P. Michler et al., Science 290, 2282 (2000)

83 Signature of a triggered single-photon source Intensity (photon) correlation function: g (2) ( τ ) = : I ( t) I ( t + τ ) : I ( t) 2 Single quantum emitter (I.e. an atom) driven by a cw laser field exhibits photon antibunching. g (2) (τ) 1 0 τ Triggered single photon source: absence of a peak at τ=0 indicates that none of the pulses contain more than 1 photon. g (2) (τ) 0 τ

84 Photon antibunching Intensity correlation (g (2) (τ)) of light generated by a single two-level (anharmonic) emitter. Assume that at τ=0 a photon is detected: - We know that the system is necessarily in the ground state g> - Emission of another photon at τ=0+ε is impossible. Photon antibunching: g (2) (0) = 0. pump (nonclassical light) photon at ω p g (2) (τ) recovers its steady-state value in a timescale given by the spontaneous emission time. If three are two or more 2-level emitters, detection of a photon at τ=0 can not ensure that the system is in the ground state (g (2) (0) >0.5).

85 Single InAs Quantum Dots InAs/GaAs Single QD P (W/cm 2 ) s-shell 650 p-shell T=4K Intensity (a.u.) X 1X 17 x5 in intensity Energy (ev) x2 in intensity Two main wavelengths emitted from each shell (For s-shell) 2X recombination while there is already an e-h pair in the s-shell (biexciton) 1X recombination while there is no other e-h pair in the s-shell (exciton) Due to carrier-carrier interaction Typically hν 1X = hν 2X + 2-3meV Unique wavelengths for 1X and 2X transitions

86 Exciton linewidth measured by a scanning Fabry-Perot Under non-resonant pulsed excitation Free Spectral Range: 62 µev 250 Intensity (a.u.) X2 X1 Intensity (a.u.) Energy (ev) Bins linewidth: 5.6 µev

87 Photon antibunching from a Single Quantum Dot (c) W/cm 2 g 2 (0)=0.1 τ = 750 ps PL spectrum Correlation Function g (2) (τ) (b) Coincidence Counts n(τ) 55 W/cm 2 g 2 (0)=0.0 τ = 1.4 ns Intensity (a.u.) P=55W/cm 2 T=4K 2X 1X (a) W/cm 2 g 2 (0)=0.0 τ = 3.6 ns Energy (ev) Delay Time τ (ns) proof of atom-like behavior

88 A single quantum dot excited with a short-pulse laser can provide single-photon pulses on demand BUT How about embedding quantum dots in a microcavity to increase collection efficiency and fast emission?

89 Q=13500 Q>17000 Microdisk Cavities No roughness on the sidewall up to 1nm! Q>18000 for 4.5µm diameter microdisk Q=9000 for 2µm diameter microdisk Michler et al., Appl. Phys. Lett. 77, 184 (2000) Fundamental whispering gallery modes cover a ring with width ~ λ/2n on the microdisk

90 A single quantum dot in a microdisk Larger width of the peaks due to larger lifetime of the quantum dot Intensity (a.u.) P=20W/cm 2 T=4K 2X WGM 1X Q = 6500 Pump power well above saturation level Energy (ev)

91 Tuning the quantum dot into resonance with a Cavity Mode Energy (mev) 1323 WGM 1X 1322 Cavity coupling can provide better collection T=44K WGM Intensity (a.u.) Temperature (K) WGM E 0.0 1X -E 0.5 (mev) 1.0 Small peak appears at τ=0: Purcell effect: reduction of emission time

92 Klimov Förster coupling between QDs

93 Klimov Förster coupling between QDs Crooker et al PRL 2002 Non-Radiative Energy Transfer Mechanism D * + A D + A * Coulomb-driven interaction Dipole-dipole interaction (Förster 1946) Higher multipoles interaction (Förster Dexter) Exchange-driven interaction (Dexter)

94 trioctylphosphine oxide (TOPO) ~11A A acceptor: larger dot D donor: smaller dot NQDs have better characteristics than biological light harvesting compounds, eg LH2

95 (a) PL decays from a dense film of monodisperse R=12.4A/9A CdSe/ZnS NQDs at the energies specified in the inset. Inset: cw PL spectra from film (solid) and original solution (dashed). (b) Dynamic redshift of the peak emission. Inset: PL spectra at the specified times. Crooker et al PRL 2002

96 (a) Schematic of NQD energy-gradient bilayer for light harvesting 13 A dots on 20.5 A dots. (b) Instantaneous PL spectra at 500 ps intervals (from 0 to 5 ns), showing rapid collapse of emission from 13 A dots.

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98 Our Goal Study the excitation energy transfer in quantum-dot arrays using an appropriate model Hamiltonian N N ( ) N N H = T c c T d d Uc c d d U c c d d e i j + i j + h i i i i + NN i i j j + Vc s i d i djcj ij, i ij, = NN i j

99 Probability of each basis state as a function of time Period of small oscillation ~ 1.3 ~ 1 2T V e c Period of large oscillation ~ 15 ~ 1 2V f V c > 1001> 0110> 0011> total probability of exciton state Calculations V c for two t/(2π) V c dots t/(2π ) Probability of each basis as a function Probability of each basis as a function of time without Förster Probability of each basis as a function of time T e /V c =-0.8 V f =0 V cnn /V c = > 1001> 0110> 0011> of time without tunneling T e /V c =0 V f /V c =-0.04 V cnn /V c = > 1001> 0110> 0011> V c /(2π) V t/(2π)

100 3.0 V c =10 mev 1 Psec V c =100 mev Oscillator strength T e /V c = V c,nn /V c =0.1 V f /V c =0.04 T h = V c t/2π Time evolution of the oscillator strength of an exciton initially localized at dot 12 in a 24 dot chain G. W. Bryant, Physica B 314, 15 (2002).

101 The movie of the 24 dots exciton probability at each dot (12) (11) (14) (10) (15) (9) (16) (8) (17) (7) (18) (6) (19) (5) (20) (4) (21) (3) (22) (2) (23) (1) (24) V c t/2π

102 1.0 probability of exciton state in each dot exciton probability at each dot From graph/calculation Förster rate for polymer h 2π V c 2π t = = 3.1 n n + 1 8U 8U V c V c c /(2π) V c t/2π 23 2π t 2.2 V c efficient interdot transfer rate 24

103 Zrenner coherent control of quantum dot photodiodes

104 Zrenner coherent control of quantum dot photodiodes Zrenner et al Nature 2002 Laser focus Shadow mask Top contact: 5nm Ti (semitransparent) QDs InGaAs Back contact Possibles measurements Photolumenescence (PL) spectrum (for τ rad < τ tunnel ) Photocurrent (PC) spectrum (for τ rad > τ tunnel )

105 Rabi Oscillation in a two level system ω 0 For ω = ω 0 and using RWA H ω0 Ω() t = σ z + cos( ω t ) σ x 2 2 Ω () t = µ ε () t P 2 = 0 X sin Θ Exciton Population τ Θ= Ω() tdt π 2π Pulsed Area (Θ) 3π 4π

106 rad Time scales for the device τ 1 ns τ tunnel 3 ps Tuned by Vb Photoluminescence τ rad < τ tunnel τ rad < τ tunnel τ rad > τ tunnel NeHe laser Photocurrent τ rad > τ tunnel Ti:sapphire laser τ pulse 1 ps f pulse 82 MHz F. Findeis et al. Appl. Phys. Lett. 78, 2958 (2001)

107 Artificial ion (charged exciton) τ rad < τ tunnel F. Findeis et al. Phys. Rev. B 63, (2001)

108 How does the Zrenner device work? τ rad > τ tunnel A. Zrenner et al. Nature 418, 612 (2002)

109 How does the Zrenner device work? τ pulse 1 ps f pulse 82 MHz I = eρ f XX pulse ρ XX Θ 2 2 = sin

110 Mesoscopic optical spectrum analyser QCSE

111 Rabi Oscillation in the photocurrent I = eρ f XX pulse