Far from equilibrium and time-dependent phenomena for electron - PowerPoint PPT Presentation

QIMP11, May 29 th – June 10 th , 2011, Dresden PRÉSENTATION Far from equilibrium and time-dependent phenomena for electron transport in quantum dots Renaud Leturcq IEMN – CNRS, Department ISEN, Villeneuve d'Ascq, France

Outline Introduction Part I: Single electron transport in quantum dots Electron and spin in quantum dots Time-resolved single electron detection Single electron manipulation Interaction with photons and phonons Part II: Kondo effect in quantum dots

Introduction

Challenges for quantum electronic transport • Low-frequency linear transport in non interacting systems is well understood G = e 2 h ∑ T n – Landauer-Büttiker theory n • Understanding the experiments requires to go beyond! – non-equilibrium effects (large bias voltage, current noise) – role of electron-electron interactions – interaction with the environment ⇒ finite coherence time – high-frequency response (adiabatic or non-adiabatic regime) – role of the electron spin • Quantum dots as an “ideal” playground to provide answers

Transport in quantum dots quantum source drain dot  S gate  D  S  D QD source drain k B T E C  tunnel barriers • Small island k B T ≪ E C – large capacitance C charging energy E C = e 2 / C transport through a single atomic level – quantum confinement level spacing Δ ~ ħ 2 /( m * r 2 ) trapped electron = quantum impurity connected to Fermi leads

Fabrication of semiconductor quantum dots • Most successful up to now: GaAs heterostructures R. Hanson et al. , Rev. Mod. Phys. 79 , 1217 (2007) Most of the demonstrative experiments on quantum dots were performed on this system.

Fabrication of semiconductor quantum dots • Most successful up to now: GaAs heterostructures – extensive tuning of parameters • number of electrons N quantum source drain dot • confinement potential Δ  S  D • coupling to the leads Γ S , Γ D • bias voltage V SD eV SD • What is difficult to achieve with GaAs heterostructures? k B T E C – change the intrinsic electronic properties ( m* , g -factor,...) – coupling with other materials (superconductors, ferromagnetic)  – new geometries (interaction with motion) – optically active quantum dots

Fabrication of semiconductor quantum dots • Necessity of tuning the material properties – change the intrinsic electronic properties 2 Δ≈ ℏ • effective mass  broader range of level spacing ∗ r 2 m • spin-orbit interaction (InAs, InSb: strong SOI) • zero nuclear spin (Si, C)  long spin coherence time – coupling with other materials • superconductors • ferromagnetic materials – new geometries • suspended nanostructures (nanowire, nanotubes) • heterogeneous integration – optically active quantum dots (see lecture A. Imamoglu)

Fabrication of semiconductor quantum dots • Carbon-based nanostructures – carbon nanotubes, fullerene M. Bockrath et al. , Science 275 , 1922 (1997) H. Park et al. , Nature 407 , 57 (2000) – graphene L. A. Ponomarenko et al. , Science 320 , 356 (2008) • Semiconductor nanowire – InP, Si, InAs, Ge, InSb S. De Franceschi et al. , Appl. Phys Lett. 83 , 244 (2003) Z. Zhong et al. , Nano Lett. 5 , 1143 (2003) M. T. Björk et al. , Nano Lett. 4 , 1621 (2004) Y. Hu et al. , Nature Nanotechnol. 2 , 622 (2007) H. A. Nilsson et al. , Nano Lett. 9 , 3151 (2009)

Take-away message (1) High tunability of semiconductor quantum dots for transport through quantum impurities electronic properties material properties interaction with the environment next: what can we probe in transport experiments?

Part I Single electron transport in quantum dots 1. Transport mechanisms in quantum dots 2. Time-resolved single electron detection 3. Single electron manipulation 4. Interaction with photons 5. Interaction with phonons

Single electron transistor (SET) review: Single Charge Tunneling , ed. Graber & Devoret, Plenum Press (1992) gate  S  D − Ne = Q g  Q L  Q R SET source drain C = C L  C R  C g V g = Q L − Q g = Q R − Q g tunnel barriers C L C g C R C g Ne ≡ 2 2 2 E cl  N ,V g = Q L  Q R  Q g I 2 C L 2 C R 2 C g C G + + -V /2 + = = 2 +V /2 ≈ N e − C g V g  = V G 2 C

Single electron transistor (SET) gate review: Single Charge Tunneling , ed. Graber & Devoret, Plenum Press (1992)  S  D E cl ( N , V G ) SET source drain tunnel barriers N -2 N -1 N N +2 N +3 N +1 Ne ≡ V G  E C = e 2 /C I C G + + -V /2 + = = +V /2 = current at low bias voltage I V G energy conservation  E cl ( N , V g ) = E cl ( N +1, V g ) 2 E cl  N ,V g ≈ N e − C g V g  2 C V G

Transport in quantum dots k B T ≪ E C quantum quantum source drain source drain dot dot  S  D T = 50 mK N-1 G SD (10 -3 e 2 /h) E C E C (+ Δ ) N k B T k B T E C N+1   V PG (mV)

High bias spectroscopy E C • Charge stability diagram (SET) V SD N-1 ↔ N ↔ N+1 N-1 ↔ N N ↔ N+1 E C N-1 N N+1 0 V G + E C I 0 -

High bias spectroscopy • Charge stability diagram (SET) V SD N-1 N N+1 0 V G dI/dV

High bias spectroscopy • Charge stability diagram (QD): spin filling N even V SD E N-1 N N+1 0 V G DOS dI/dV

High bias spectroscopy • N ↔ N+1 excited states E C E C V SD N-1 N N+1 0 E C V G dI/dV

High bias spectroscopy • Constant interaction model: E C independent of N

High bias spectroscopy • Spectroscopy of an InAs nanowire QD GL GC S D -20 GR T = 100 mK -10 GL GC GR V SD (mV) E C Δ E C +Δ S D 0 N +1 N +2 E C ≈  ≈ 6 meV N even 10 gives a QD radius of 20 nm 20 -0.04 -0.02 0 V g ates

Spin spectroscopy • At high magnetic field: splitting of the degenerate spin states → can be used as a spin filter ∗  B B  E Z = g |g*| = 5.5, due to quantum confinement (bulk InAs, | g*| = 15) see also: R. Hanson et al. , Phys. Rev. Lett. 91 , 196802 (2003)

Signature of spin-orbit interaction • Spin-orbit Hamiltonien: coupling of the spin and orbital 2  =− B ⋅ B eff H SO =− B ⋅  degrees of freedom p × E 2 m c • Mixing of spin states in InAs quantum dots C. Fasth et al. , PRL 98 , 266801 (2007) A. Pfund et al. ,PRB 76 , 161308(R) (2007) T - (2,0) 2 electrons states T 0 (2,0) T - g*µ B B T T 0  ST T + (2,0) T + | g* |= 7 S  SO = 0.2 meV S(2,0)

Single electron transport mechanisms • Sequential tunneling model – master equation approach Beenakker, Phys. Rev. B 44 , 1646 (1991) d dt ∣ p ,t 〉 =−̂ L ∣ p ( t ) 〉 p n = probability to find the system in a state n L mn =δ n, m γ n −Γ m ← n γ n = ∑ Γ m ← n m ≠ n  n = transition rate from state n to state m

Single electron transport mechanisms • Higher order processes: – elastic and inelastic cotunneling S. de Franceschi et al. , PRL 86, 878 (2001)

Take-away message (2) Transport experiment can probe the quantum structure of the quantum dot electron and spin states... … assuming the constant interaction model ! transport mechanism via sequential co-tunneling next: can we access the transport time-scales?

Time scales for single electron transport time-resolved detection (I.2) pulsed gate experiments (I.3) energy time frequency 1 s 1 Hz 4 feV 0.5 nK • Inverse tunneling rates 1/ Γ S , 1/ Γ D = 10 ps – infinity source drain quantum dot  S 1 ms 1 kHz 4 peV 0.5 μK – time scale for a trapped microwave experiments (I.4)  D electron to escape ES  d GS • Charge or spin decay time k B T 1 μs 1 MHz 4 neV 0.5 mK 1/ Γ d = few ns – 1 second E C – coherent manipulation 1 ns 1 GHz 0.5 K 4 μeV  • h / E C , h / Δ = 1 – 100 ps – non-adiabatic transistion 1 ps 1 THz 4 meV 500 K

2. Time-resolved single electron detection shot noise: S I = 2 eI (Schottky, 1918) conductor A ? time time

Single charge detection with a quantum point contact gate source drain gate dot source drain gate I QPC working point V gates M. Field et al. , PRL 70, 1311 (1993)

Time-resolved single electron detection • Thermal fluctuations between leads and dot W. Lu et al. , Nature 423 , 422 (2003) R. Schleser et al. , APL 85, 2005 (2004) L. Vandersypen et al. , APL 85 , 4394 (2004) source quantum drain dot  S  D k B T R. Scheser et al. , APL 85 , 2005 (2004)

Time-resolved detection of single electron transport • Large bias voltage ⇒ directional flow S. Gustavsson, RL et al. , PRL 96, 076605 (2006) N quantum source drain dot  S N+1  D k B T current time

Histograms of current fluctuations S. Gustavsson, RL et al. , PRL 96, 076605 (2006) • Poisson distribution for • Sub-Poisson distribution asymmetric coupling for symmetric coupling Theory: Hershfield et al., PRB 47, 1967 (1993) Bagrets & Nazarov, PRB 67, 085316 (2003)

Histograms of current fluctuations • Asymmetric coupling • Symmetric coupling – statistics dominated by the – Coulomb blockade thicker barrier “orders” the electrons quantum quantum source drain source drain dot dot  S  S  D  D k B T k B T