Electronic Properties of Coupled Quantum Dots M Reimer

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

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

Project Objectives Ø Study and understand the electronic properties of How? Experimental Setup coupled quantum dots Ø Determine the coupling between these dots using vertical electric fields Ø Optical techniques Photocurrent Measurements Photoluminescence 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 Quantum Well Wire Dot Enm g 2 D(E) E 3 D E 1 g 0 D(E) g 1 D(E) “Enlm“ E 00 E E E

Bulk ® Quantum Wells Band - j Subband - i E 2 D 3 D tz Band - j x=1 x=0 k=(k kxy=(k x, k y, k z) x, k y) tz~nm

Quantum Wires and Dots Free motion Wire ty Quantised Motion Dot tz g 0 D(E) Eyz g 1 D(E) “Exyz“ E 00 E E Fully Quantised

Interest in Quantum Dots • Lasers (Jth<6 Acm-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 >> 3 k. BT • 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. In. As on Ga. As (7% lattice mismatch) • Form due to kinetic and thermodynamic driving forces – energetically more favourable to form nanoscale clusters of In. As 10 nm • Some general properties • Perfect crystalline structures • High areal density (10 -500µm-2) • Strong confinement energies (100 me. V) • Already many applications • Lasers (Jth<6 Acm-2) in visible and near infrared • Optical data storage 1 x 1µm • Optical detectors • Quantum Information and Cryptography 10 nm

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 2 D Harmonic oscillator potential – Fock-Darwin states z x, y 2 D state n=3 (6) (4) 0 D states (2) Eg Eg+Ez (2) (4) (6) x • Orbital character of QD states similar to atomic systems ~ HO like potential QW like potential y n=2 n=1 Eg+Ez+Exy • The shells n=1, 2, 3 - often termed s, p, d, . . in comparison with atomic systems • DEe 0 -e 1~50 -70 me. V, DEh 0 -h 1~2030 me. V, Exciton BE ~30 me. V • Dipole allowed optical transitions Dn=0 • Single X transitions observable in absorption experiment • PL requires state filling spectrosopy – excitons interact

Properties of Excitons in QDs Aperture of a near field shadow mask Diffraction limited resolution of µ-PL 100 - 500 nm 1 µm Probe the optical properties of a QD Ø Isolation of a single Quantum Dot Ø Emission spectroscopy Ø Power-dependence reveals the different configurations X 0 2 X 0

Calculation of Eigenstates - QW 2 D structures – V varies only in z-direction z • Separate motion and || to QW E 2 E 1 HH 1 LH 1 HH 2 • 1 D Schrödinger equation along – z Envelope functions

Electronic Subbands

Materials Discontinuities • Materials properties (e. g. m*) change accross interface – Continuity equations for envelope functions z=0 1 Wavefunction continuous 2 Probability flux=0 (Bound states) m. A * • Both conditions satisfield by Ben. Daniel. Duke form of Schrödinger equation. Ben. Daniel Duke SE m. B *

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

Example Undoped Ga. As-Al 0. 3 Ga 0. 7 As Quantum Well DEc~60% E 0 1940 me. V 1500 me. V HH 0 m*e~0. 067 mo , m*hh~0. 34 mo n=1 n=0 • Infinite-well approximation reasonable for estimating E 0, HH 0 • 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 shape has influence on strain and electronic structure Pyramide Lens Semiellipsoid

Choose the Alloy Profile of Dot • Linear P. W. Fry et al. , PRL 84, (2000) • Trumpet T. Walther et al. PRL 86 (2001) M. Migliorato et al. PR B 65 (2002) • Inverted pyramidal 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 QD layer substrate (Ga. As) Resolution below 1 nm

Calculate the Strain Minimization of elastic energy in continuum model. Lead to Piezo electric polarization Ga. As tensile In. Ga. As -2 compressive -1 0 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 Hole wavefunctions p d d

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 me. V

Electronic Properties of Coupled Quantum Dots

Coupled Quantum Dots 10 nm WL d=6 nm Ø In. Ga. As-Ga. As self assembled QD-molecules Ø Self alignment via strain field 7 nm

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

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

1 D Model of Coupled Wells: Holes in a double well as a function of well separation Energy [e. V] 20 bonding anti-bonding 0 width [nm] Indium content Well 1 Well 2 5. 0 0. 305 0. 300 potential [me. V] 137 135 0 2 4 6 8 10 Well separation [nm] Weak splitting due to large effective mass (mh~ 10 × me)

1 D Model of Coupled Wells: Electrons in a double well as a function of well separation Energy [me. V] 100 anti-bonding 0 0 width [nm] Indium content Well 1 Well 2 5. 0 0. 305 0. 300 potential [me. V] - 215 -212 2 4 6 8 10 Well separation [nm] 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 me. V

Strain has Long Range Effect xx WL 6 nm WL

Strain Deforms Valence Band strain HH-valence band 1. 65 2 nm Slice through center of QD Energy [e. V] 1. 60 6 nm 10 nm 1. 55 1. 50 1. 45 1. 40 1. 35 0 10 20 30 40 Growth axis [nm] 50 60

Single Particle States Electron 840 -0. 436 lower dot ~ 22 me. V ~ 3 me. V upper dot 2 4 6 8 QD separation (nm) 10 Energy (me. V) anti-bonding state 815 Heavy hole -0. 444 upper dot ? strain 2 4 6 8 10 QD separation (nm) Quantum coupling Strain

Bonding and Anti-Bonding State bonding anti-bonding

Excitonic Structure quantum coupling + strain + Coulomb interaction Exciton Energy [e. V] 1. 29 antibonding 1. 28 1. 27 1. 26 indirect Ex bonding Coulomb interaction [~20 me. V] 1. 25 direct Ex 1. 24 2 4 6 8 Dot separation [nm] 10

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

Analysis of Stark Shift § Origins of quadratic and linear components of Stark Shift ? § Anisotropic QD shape – e-h separation at F=0 Zero Field e-h separation Field Induced e-h separation p(F)=e. (s 0+b. F) s 0 DE=p. F=es 0 F+e b. F 2 E = E 0 + s 0 e. F + eb. F 2 § First order term provides a direct determination of s 0 § b Effective height of dot

Anomalous Stark Effect QD separation 6 nm QD separation 2 nm indirect 1. 28 1. 26 single QD direct 1. 24 1. 22 1. 20 -60 Exciton energy [e. V] 1. 32 1. 30 1. 28 1. 26 1. 24 bonding 1. 22 1. 20 -40 -20 0 20 40 Applied Field [k. V/cm] -60 60 1. 0 0. 8 0. 6 0. 4 0. 2 -40 -20 0 20 40 60 Applied Field [k. V/cm] 1. 0 0. 0 -60 anti-bonding 1. 30 Overlapp Exciton energy [e. V] 1. 32 0. 6 0. 4 0. 2 -40 -20 0 20 Applied Field (k. V/cm) 40 0. 0 -60 -40 -20 0 20 40 Applied Field [k. V/cm] 60

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

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 Ø Excitation source - monochromated 150 W Halogen Lamp Ø Photocurrent measured using lock-in amplifier Ø Low noise screened setup (<50 f. A) Ø Low incident optical power density (~3 m. W/cm 2) <<1 e-h pair per dot

How Does it Work? Thermal activation e. V EMISSION Tunnelling ABSORPTION Ø QD-molecules embedded in n-i Schottky photodiodes Ø Electric field dependent optical spectroscopy

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

Photocurrent – Quantitative Measure of Absorption

QDM Photocurrent T=300 K E 0 Ø Strong Stark shift Ø Oscillator strength Ø Observations differ strongly from single QD layer samples

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 ü Stark Shift Qualitatively Similar, but off by a factor 3 ü Enery splittings similar ~ 30 -40 me. V

Comparison with Theory: Oscillator Strength üGround State quenches at higher electric fields üMore 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~5 K - Tunneling escape dominates • T>200 K - Thermal activation dominates • All absorbed carriers contribute to measured signal – PC=Absorption

Temperature Dependence: Coupled Layer

Summary Ø PC technique provides a direct measurement of the absorption Ø Ensemble of single dot layer exhibits quadratic stark shift in electric field • Maximum transition energy occurs for non-zero field Ø Behavior of coupled quantum dots strongly different ü Stark Shift: Qualitatively similar ü Energy splittings of same order ~ 30 -40 me. V ü Oscillator Strength: Ground state quenches at higher electric fields ü 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