Electronic Properties of Coupled Quantum Dots M.Reimer, H. J. Krenner, M. Sabathil, J. J. Finley. Walter Schottky Institut, TU München

Outline Motivation Project Objectives Introduction to quantum dots Electronic properties of Nanostructures Quantum Wells Self Assembled quantum dots

Outline Cont’d Electronic properties of coupled quantum dots How to model a quantum dot Electronic properties of coupled quantum dots Photocurrent Spectroscopy of single and coupled quantum dots Summary

Quantum Information Processing Motivation wX Exciton Ground State a Rabi Oscillations have been observed for single quantum dots - Zrenner et al. Nature 418 (2002) Obtain coherent control of the two-level system via ps laser pulses State can be read by measuring a deterministic photocurrent Quantum Information Processing Target CNOT Initial State Final State Control For conditional quantum logic, two qubits are required Coupled Quantum Dots are needed Demonstrated by E.Biolatti et al. APS 85, 5647 (2000)

Project Objectives ? Project Objectives How? Study and understand the electronic properties of coupled quantum dots Determine the coupling between these dots using vertical electric fields How? Optical techniques Photocurrent Measurements Photoluminescence Experimental Setup Single Quantum Dot Ensemble Coupled Quantum Dots PC technique successfully applied to single layer of quantum dots Stark Shifts Oscillator Strengths ?

Introduction to Quantum Dots

Quantum Wells, Wires and Dots g0D(E) E E3D E0 E1 g2D(E) g1D(E) E00 Enm “Enlm“

Bulk ® Quantum Wells E2D E3D k=(kx,ky,kz) kxy=(kx,ky) tz tz~nm x=1 x=0 Band - j Subband - i x=0 x=1 tz~nm tz E3D k=(kx,ky,kz) Band - j

Quantum Wires and Dots Free motion Quantised Motion Dot Wire ty tz g0D(E) E “Exyz“ Wire g1D(E) E E00 Eyz tz ty Fully Quantised

Interest in Quantum Dots Lasers (Jth<6Acm-2) in visible and near infrared Optical data storage Optical detectors Quantum Information Processing and Cryptography “Atom-optics“ type experiemtns on man-made atoms

Requirements for Dot-Based Devices Size DEc and DEv >> 3kBT High crystal quality Low defect density Uniformity Homogeneous electronic structure Density High areal density Bipolar-confinement Bound electron and hole states should exist for optical applications Electrically active matrix Enables electrical excitation

Self-Assembled Quantum Dots Formed during epitaxial growth of lattice mismatched materials e.g. InAs on GaAs (7% lattice mismatch) Form due to kinetic and thermodynamic driving forces – energetically more favourable to form nanoscale clusters of InAs 1 x 1µm 10nm Some general properties Perfect crystalline structures High areal density (10-500µm-2) Strong confinement energies (100meV) Already many applications Lasers (Jth<6Acm-2) in visible and near infrared Optical data storage Optical detectors Quantum Information and Cryptography 10nm

SAQDs - Electronic Structure z For SAQDs - z-axis confinement is generally much stronger than transverse quantisation x,y (Ez>>Exy) QD states are often approximated as a 2D Harmonic oscillator potential – Fock-Darwin states y x z x,y Orbital character of QD states similar to atomic systems The shells n=1,2,3 - often termed s,p,d,.. in comparison with atomic systems DEe0-e1~50-70meV, DEh0-h1~20-30meV, Exciton BE ~30meV Dipole allowed optical transitions Dn=0 Single X transitions observable in absorption experiment PL requires state filling spectrosopy – excitons interact QW like potential ~ HO like potential 2D state (6) n=3 (4) n=2 0D states (2) n=1 Eg+Ez Eg Eg+Ez+Exy (2) (4) (6)

Properties of Excitons in QDs Aperture of a near field shadow mask Diffraction limited resolution of µ-PL 1 µm 100 - 500 nm Probe the optical properties of a QD Isolation of a single Quantum Dot Emission spectroscopy Power-dependence reveals the different configurations X0 2X0

Calculation of Eigenstates - QW 2D structures – V varies only in z-direction z Separate motion  and || to QW 1D Schrödinger equation along – z HH2 LH1 HH1 E1 E2 Envelope functions

Electronic Subbands

Materials Discontinuities Materials properties (e.g. m*) change accross interface Continuity equations for envelope functions z=0 mA* mB* Wavefunction continuous Probability flux=0 (Bound states) 1 2 Both conditions satisfield by BenDaniel-Duke form of Schrödinger equation. BenDaniel Duke SE

Contributions to Total Potential The total potential (VT(z)) in BenDaniel-Duke Schrödinger equation may have several contributions 1) Bandedge modulation 2) Electrostatic Potential 3) Coulomb Interactions (e-e, e-h) 4) Image Charges (e-varies) ewell ebarrier Additional contributions can exist in special cases e.g. due to piezo-electric charges etc

Example E0 HH0 Undoped GaAs-Al0.3Ga0.7As Quantum Well DEc~60% 1940meV m*e~0.067mo , m*hh~0.34mo 1940meV 1500meV n=0 n=1 Infinite-well approximation reasonable for estimating E0, HH0 Better for wider wells (d>75Å) Approximation poor for excited states (n>0)

How to Model a Quantum Dot A step by step introduction

Choose the Shape of Dot Dot shape has influence on strain and electronic structure Pyramide Lens Semiellipsoid

Enhanced lateral confinement Choose the Alloy Profile of Dot Linear Trumpet Inverted pyramidal P.W. Fry et al., PRL 84, (2000) T. Walther et al. PRL 86 (2001) M. Migliorato et al. PR B65 (2002) Enhanced lateral confinement N. Liu et al. PRL. 84, (2001)

Define the Structure Define structure including substrate, wetting layer and QD on a finite differences grid. wetting layer QD substrate (GaAs) Resolution below 1nm

Calculate the Strain Minimization of elastic energy in continuum model. Lead to Piezo electric polarization GaAs tensile InGaAs compressive GaAs -2 -1 1 [%] -2 -1 1 2

Calculate the Potential Solve Poisson equation. (Piezo, Pyro, electrons and holes) Conduction band profile including potential and shifts due to strain:

Calculate the Quantum States Solve single- or multi-band (k.p) Schrödinger equation Electron wavefunctions s p p d d Hole wavefunctions

Calculation of Few-Particle States Possible methods: Quantum Monte Carlo (QMC) Configuration interaction (CI) Density functional theory (DFT) Kohn- Sham Equations DFT in local density approximation (LDA): Exchange and correlation depends on local density r(r) Binding energy for exciton in typical QD ~ 20 meV

Electronic Properties of Coupled Quantum Dots

Coupled Quantum Dots InGaAs-GaAs self assembled QD-molecules 10nm WL d=6nm 7nm Materials system, perhaps TEM / strain profile of coupled QD molecule What can be varied ? - Coupling strength (7-13nm) Growth conditions – MBE, density, high pairing probability ? InGaAs-GaAs self assembled QD-molecules Self alignment via strain field

Vertically Correlated QDs Upper layers of dots tend to nucleate in strain field generated by lower layers Strain field extends outside buried QD 10nm Transmission Electron Micrograph of single coupled QD molecule

Potentially useful as coupled QBITs for Quantum Logic Operations Stacking Probability 5 – vertically aligned InAs QDs d 10nm STM-image For InAs QDs in GaAs - Pairing probability ~ 1 for d<25nm Enables fabrication of coupled layers of dots and QD superlattices Potentially useful as coupled QBITs for Quantum Logic Operations

1D Model of Coupled Wells: Holes Holes in a double well as a function of well separation 2 4 6 8 10 bonding anti-bonding Energy [eV] Well separation [nm] 20 Well 1 Well 2 width [nm] 5.0 Indium content 0.305 0.300 potential [meV] 137 135 Weak splitting due to large effective mass (mh~ 10 × me)

1D Model of Coupled Wells: Electrons Electrons in a double well as a function of well separation 2 4 6 8 10 Energy [meV] Well separation [nm] bonding anti-bonding 100 Well 1 Well 2 width [nm] 5.0 Indium content 0.305 0.300 potential [meV] - 215 -212 Strong splitting due to small effective mass (mh~ 10 × me)

What happens in a Real Structure? Quantum mechanical coupling Splits electron states into bonding and anti-bonding Leaves hole states almost unaffected Strain effects - Increased hydrostatic strain increases gap which leads to higher transition energies - Complicated effect on holes Coulomb interaction of electron and hole in exciton - Binding energy between direct and indirect excitons differs by ~ 20 meV

Strain has Long Range Effect xx WL 6 nm WL

Strain Deforms Valence Band HH-valence band 1.65 2nm 6nm 10nm 1.60 Slice through center of QD 1.55 Energy [eV] 1.50 1.45 1.40 1.35 10 20 30 40 50 60 Growth axis [nm]

Single Particle States Electron 2 4 6 8 10 815 840 Energy (meV) QD separation (nm) upper dot lower dot anti-bonding state bonding state ~ 3 meV ~ 22 meV Quantum coupling Strain 2 4 6 8 10 -0.444 -0.436 Energy (meV) QD separation (nm) upper dot lower dot ? strain Heavy hole Quantum coupling Strain

Bonding and Anti-Bonding State

Excitonic Structure quantum coupling + strain + Coulomb interaction anti- bonding 1.29 1.28 1.27 indirect Ex Exciton Energy [eV] bonding 1.26 Coulomb interaction [~20 meV] 1.25 direct Ex 1.24 2 4 6 8 10 Dot separation [nm]

Coupled Dots in an Electric Field What do we Expect? Direct exciton Indirect exciton + - EL HL Dipole: Field Quadratic Stark shift Linear Energy

Single QD Layer – Stark Shift Transitions exhibit quadratic dependence on electric field Maximum transition energy occurs for F0 Source: P W Fry et al Phys. Rev. Lett 84, 733 (2000)

Analysis of Stark Shift Origins of quadratic and linear components of Stark Shift ? Anisotropic QD shape – e-h separation at F=0 p(F)=e.(s0+bF) DE=p.F=es0F+e bF2 Zero Field e-h separation Field Induced e-h s0 E = E0 + s0eF + ebF2 First order term provides a direct determination of s0 b  Effective height of dot

Anomalous Stark Effect QD separation 6nm QD separation 2nm 1.32 1.32 indirect direct anti-bonding bonding 1.30 1.30 1.28 1.28 single QD Exciton energy [eV] 1.26 Exciton energy [eV] 1.26 1.24 1.24 1.22 1.22 1.20 1.20 -60 -40 -20 20 40 60 -60 -40 -20 20 40 60 Applied Field [kV/cm] Applied Field [kV/cm] -60 -40 -20 20 40 0.0 0.2 0.4 0.6 0.8 1.0 Overlapp Applied Field (kV/cm) -60 -40 -20 20 40 60 0.0 0.2 0.4 0.6 0.8 1.0 Overlapp Applied Field [kV/cm]

Influence of Coupling Ground state energy e-h overlap Progressive quenching Not observed for single layer Weak coupling  Kink Strong coupling  Smooth

Electronic Structure: Coupled QDMs The electronic structure of coupled quantum dots is determined by three main effects that are all of the same order: Strain effects Quantum coupling Coulomb coupling Comparison to recent experimental results shows qualitative agreement

Photocurrent Spectroscopy

Experimental Setup Excitation source - monochromated 150W Halogen Lamp Photocurrent measured using lock-in amplifier Low noise screened setup (<50fA) Low incident optical power density (~3mW/cm2) <<1 e-h pair per dot

How Does it Work? eV EMISSION ABSORPTION Thermal activation eV Tunnelling EMISSION ABSORPTION Sample structure Operation in PL and PC regimes Ensemble measurements to obtain general properties QD-molecules embedded in n-i Schottky photodiodes Electric field dependent optical spectroscopy

What Does it Tell Us? T>200K - thermal activation faster than excitonic spontaneous lifetime All photogenerated carriers contribute to measured photocurrent  PC  Absorption Electronic Structure Information about excited states Oscillator strengths of the transitions Advantages over Luminescence Provides a sensitive method for measuring low noise absorption spectra Provides a direct measure of the electronic states in the single exciton regime Excited state energies can be determined (Luminescence probed the ground state) Absorption techniques give the oscillator strengths of the transitions

Photocurrent – Quantitative Measure of Absorption

QDM Photocurrent E0 Strong Stark shift Strong Stark shift Oscillator strength  Observations differ strongly from single QD layer samples Strong Stark shift Oscillator strength  Observations differ strongly from single QD layer samples T=300K E0

Single Layer vs. Coupled Layer Reverse Bias

Comparison with Theory: Transition Energies Estimated dipole of ground state (black line): exp~ 2.1 nm theory~ 3.6 nm Stark Shift Qualitatively Similar, but off by a factor 3 Enery splittings similar ~ 30-40 meV

Comparison with Theory: Oscillator Strength Ground State quenches at higher electric fields More rapid quenching of the ground state is observed with increased distance between layers

Spacing Layer Dependence Expect dipole to increase with increased separation

Photocurrent vs. E and T Single Layer

Carrier Escape Mechanisms Carrier escape mechanisms – sensitive to Temperature and E-field T~5K - Tunneling escape dominates T>200K - Thermal activation dominates All absorbed carriers contribute to measured signal – PC=Absorption

Temperature Dependence: Coupled Layer

Summary In good agreement with predicted theoretical calculations! PC technique provides a direct measurement of the absorption Ensemble of single dot layer exhibits quadratic stark shift in electric field Maximum transition energy occurs for non-zero field Behavior of coupled quantum dots strongly different Stark Shift: Qualitatively similar Energy splittings of same order ~ 30-40 meV Oscillator Strength: Ground state quenches at higher electric fields More rapid quenching of the ground state is observed with increased distance between layers In good agreement with predicted theoretical calculations!

Discussion Both dots assumed to be identical – in reality, the upper dot is ~ 10% larger Further investion of theoretical modelling required Demonstrates an asymmetric curve about the crossing points