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

1

0

X

10

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

00

01

10

11

00

01

11

10

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)

Motivation

Project Objectives

Study and understand the electronic properties of coupled quantum dots

Determine the coupling between these dots using vertical electric fields

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

How?

Project Objectives

Introduction to Quantum Dots

Quantum Wells, Wires and Dots

Em

dE

dNEg

LkkN D

23

223

33 2

2

1)(

2.

3

4)(

Quantum Well Wire Dot

22 )(m

Eg D E

mEg D

1

2

2)(1

g0D(E)

E

E3D

E0

E1g2D(E)

E

g1D(E)

E

E00

Enm “Enlm“

discreteEg D )(0

Bulk Quantum Wells

E3D

k=(kx,ky,kz)

Band - j

*

22

*

2

3 22 m

k

m

pE D

E2D

kxy=(kx,ky)

Band - jSubband - i

zxy

D Em

kE

*

22

2 2

x=0

x=1

tz~nmtz~nm

tz

Quantum Wires and Dots

Wire

E

mEg D

1

2

2)(1

g1D(E)

E

E00

Eyz

tz

ty

yzx

D Em

kE

*

22

1 2

Dot

g0D(E)

E

“Exyz“

discreteEg D )(0

xyzD EE 0

Free motion QuantisedMotion

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– Ec and Ev >> 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

1 x 1µm

10nm

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

• 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

xy z

x,y

• 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

2D state

0D states

~ HO like potentialQW like potential

(2)

(4)

(6)

n=1

n=2

n=3

(2)(4)

(6)

Eg+EzEgEg+Ez+Exy

• 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

• Ee0-e1~50-70meV, Eh0-h1~20-30meV, Exciton BE ~30meV

• Dipole allowed optical transitions n=0

• Single X transitions observable in absorption experiment

• PL requires state filling spectrosopy – excitons interact

Properties of Excitons in QDs

Probe the optical properties of a QD Isolation of a single Quantum Dot Emission spectroscopy Power-dependence reveals the

different configurations

1280 1290 1300 1310

0 .24

P (µW )L

0 .20

0 .15

0 .11

0 .08

0 .06

0 .05

0 .04

0 .03

0 .02

0 .015

2X

s-p p-s

s-Shell p-She ll

1X

E nergy (m eV )

PL

inte

nsity

(ar

b. u

nits

)

T=2 K=632.8 nm L

In G a A s/G aAs0.4 0 .6

Aperture of a nearfield shadow mask

Diffraction limited

resolution of µ-PL

1 µm 100 - 500 nm

X0 2X0

Calculation of Eigenstates - QW

HH2LH1HH1

E1

E2

zyxEzyxzVzmymxm nnn

zyx

,,,,222 2

2

*

2

2

2

*

2

2

2

*

2

2D structures – V varies only in z-direction

z• Separate motion and || to QW

• 1D Schrödinger equation along – z

)..(,, ykxkinn

yxezzyx

zEzzVzm nn

zn

2

2

*

2

2

Envelope functions

Electronic Subbands

Materials Discontinuities• Materials properties (e.g. m*) change accross interface

– Continuity equations for envelope functions

mA*

mB*

z=0

• Both conditions satisfield by BenDaniel-Duke form of Schrödinger equation.

zEzzVzzmz nn

zn

*

2 1

2

BenDanielDuke SE

zn z

z

zmn

z

*

1

Wavefunction continuous

Probability flux=0(Bound states)1 2

Contributions to Total Potential

)(zVzVzezVzV imXCSCT

• The total potential (VT(z)) in BenDaniel-Duke Schrödinger equation may have several contributions

• Additional contributions can exist in special cases • e.g. due to piezo-electric charges etc

1) Bandedge modulation

3) CoulombInteractions

(e-e, e-h)4) ImageCharges(-varies)

well

barrier

2) ElectrostaticPotential

or z

ze

z

z

2

2

ExampleUndoped GaAs-Al0.3Ga0.7As Quantum Well

HH0

E0

Ec~60%

m*e~0.067mo , m*

hh~0.34mo

1940meV1500meV

2

2*

22

12

ndm

En

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)

zVzV SCT

How to Model a Quantum Dot

A step by step introduction

Choose the Shape of Dot

Pyramide Lens Semiellipsoid

Dot shape has influence on strain and electronic structure

• Linear

• Trumpet

• Inverted pyramidal

T. Walther et al. PRL 86 (2001)

M. Migliorato et al. PR B65 (2002)

N. Liu et al. PRL. 84, (2001)

Enhanced lateral confinement

P.W. Fry et al., PRL 84, (2000)

Choose the Alloy Profile of Dot

Define structure including substrate, wetting layer and QD on a finite differences grid.

Define the Structure

Resolution below 1nmsubstrate (GaAs)

wettinglayer QD

Calculate the Strain

-2 -1 0 1 [%] -2 -1 1 2

xx yz V

klijijklEL dVCE rr 2

1

Minimization of elastic energy in continuum model.

compressive

tensile

InGaAs

GaAs

GaAs

Lead to Piezo electric polarization

Calculate the Potential

,4 rrr

Solve Poisson equation.

(Piezo, Pyro, electrons and holes)

Conduction band profile including potential and shifts due to strain:

Calculate the Quantum States

Electron wavefunctions

Hole wavefunctions

rrrr

r EeE

mc

c*

1

Solve single- or multi-band (k.p) Schrödinger equation

s p p d d

Calculation of Few-Particle StatesPossible methods:• Quantum Monte Carlo (QMC)• Configuration interaction (CI)• Density functional theory (DFT)

( ( )) KSKS i KS i i iH T r

Kohn- Sham Equations

( ) ( ) ( ) ( ) ( )KS x c ext Hr r r r r

DFT in local density approximation (LDA): Exchange and correlation depends on local density (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 Self alignment via strain field

10nm

7nmWL d=6nm

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

• For InAs QDs in GaAs - Pairing probability ~ 1 for d<25nm• Enables fabrication of coupled layers of dots and QD superlattices

Stacking Probability

Potentially useful as coupled QBITs for Quantum Logic Operations

d

10nm

5 – vertically aligned InAs QDs

STM-image

Well 1 Well 2

width [nm] 5.0 5.0

Indium content 0.305 0.300

potential [meV] 137 135

Holes in a double well as a function of well separation

0 2 4 6 8 10

bonding

anti-bonding

En

erg

y [e

V]

Well separation [nm]

0

20

1D Model of Coupled Wells: Holes

Weak splitting due to large effective mass (mh~ 10 × me)

Electrons in a double well as a function of well separation

Well 1 Well 2

width [nm] 5.0 5.0

Indium content 0.305 0.300

potential [meV] - 215 -212

1D Model of Coupled Wells: Electrons

0 2 4 6 8 10

En

erg

y [m

eV]

Well separation [nm]

bonding

anti-bonding

0

100

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

6 nm

xx

Strain has Long Range Effect

WL

WL

Strain Deforms Valence Band

0 10 20 30 40 50 601.35

1.40

1.45

1.50

1.55

1.60

1.65

En

erg

y [

eV

]

Growth axis [nm]

2nm 6nm

10nmSlice through center of QD

strain HH-valence band

Single Particle States

Electron

2 4 6 8 10815

840

En

erg

y (

me

V)

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

En

erg

y (m

eV)

QD separation (nm)

upper dot

lower dot

?

strain

Heavy hole

Quantum coupling

Strain

Bonding and Anti-Bonding State

anti-bondingbonding

2 4 6 8 10

1.24

1.25

1.26

1.27

1.28

1.29

Ex

cito

n E

ne

rgy

[e

V]

Dot separation [nm]

indirect Ex

direct Ex

anti-bonding

bonding

Coulomb interaction [~20 meV]

Excitonic Structure

quantum coupling + strain + Coulomb interaction

Coupled Dots in an Electric Field

What do we Expect?Direct exciton Indirect exciton

+- +-

EL

HL

EL

HL

Dipole:

Field

QuadraticStark shift

Linear Stark shift

Field

En

erg

y

En

erg

y

Analysis of Stark Shift

First order term provides a direct determination of s0

Effective height of dot

E = E0 + s0eF + eF2E = E0 + s0eF + eF2

p(F)=e.(s0+F)

E=p.F=es0F+e F2

Zero Field e-hseparation

Field Induced e-hseparation

s0

Origins of quadratic and linear components of Stark Shift ?

Anisotropic QD shape – e-h separation at F=0

Anomalous Stark Effect

-60 -40 -20 0 20 40 60

Exc

ito

n e

ner

gy

[eV

]

Applied Field [kV/cm]-60 -40 -20 0 20 40 60

1.20

1.22

1.24

1.26

1.28

1.30

1.32

Exc

ito

n e

ner

gy

[eV

]

Applied Field [kV/cm]

QD separation 6nm QD separation 2nm

1.20

1.22

1.24

1.26

1.28

1.30

1.32

-60 -40 -20 0 20 400.0

0.2

0.4

0.6

0.8

1.0

Ove

rlap

p

Applied Field (kV/cm)-60 -40 -20 0 20 40 60

0.0

0.2

0.4

0.6

0.8

1.0

Ove

rlap

p

Applied Field [kV/cm]

single QD

indirect

direct

anti-bonding

bonding

Influence of Coupling

-40 -20 0 20 40 601.20

1.21

1.22

1.23

1.24

1.25

1.26

1.27

2nm 4nm 6nm

Exc

ito

n e

ner

gy

(eV

)

Applied Field (kV/cm)

Ground state energy

Weak coupling Kink Strong coupling Smooth

-40 -20 0 20 40 60

0,0

0,2

0,4

0,6

2nm 4nm 6nm

e-h

Ove

rlap

Applied Field (kV/cm)

e-h overlap

Progressive quenching 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

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?

QD-molecules embedded in n-i Schottky photodiodes Electric field dependent optical spectroscopy

E F

n -G aAs+ i-G aAs M eta l

F

EMISSION

h

esc rad<<

eV

ABSORPTION

Thermal activation

Tunnelling

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

Strong Stark shift Oscillator strength Observations differ strongly

from single QD layer samples

1120 1160 1200

0

50

100

150

200

250

Ph

oto

cu

rre

nt

(pA

)

Energy (meV)

-4V

-3V

-2V

-1V

0V

1100 1150 1200 12500

100

200

300

400

500

600

700

800

900

1000

-5V

Ph

oto

cu

rre

nt

(pA

)

Energy (meV)

0V

Strong Stark shift Oscillator strength Observations differ strongly

from single QD layer samples

QDM Photocurrent

T=300K

1100 1150 1200 12500

100

200

300

400

500

0V

Ph

oto

cu

rre

nt

(pA

)

Energy (meV)

E0

Single Layer vs. Coupled Layer

1050 1100 1150 1200 12500

200

400

600

800

1000

1V

2V

3V

4V

0V

5V

Ph

oto

curr

ent

(pA

)

Energy (meV)

Reverse Bias

Comparison with Theory:Transition Energies

20 40 60 80 1001201401601802002201200

1220

1240

1260

1280

13000 -1 -2 -3 -4

1120

1100

1080

1060

1040

1020

En

erg

y (m

eV

)

Electric Field (KV/cm)

Wav

elen

gth

(n

m)

Bias Voltage (Volts)

0 20 40 601.22

1.24

1.26

1.28

1.30

1.32

En

erg

y (e

V)

Electric Field (kV/cm)

Stark Shift Qualitatively Similar, but off by a factor 3 Enery splittings similar ~ 30-40 meV

Estimated dipole of ground state (black line): exp~ 2.1 nm theory~ 3.6 nm

Comparison with Theory:Oscillator Strength

20 40 60 80 100120140160180200220

0.0

0.2

0.4

0.6

0.8

1.0

0 -1 -2 -3 -4

7nm 13nm

e-h

Ov

erl

ap

Electric Field (KV/cm)

Bias Voltage (Volts)

-40 -20 0 20 40 60

0.0

0.2

0.4

0.6

2nm 4nm 6nm

Os

cil

lato

r S

tre

ng

thApplied Field [kV/cm]

Ground State quenches at higher electric fields

More rapid quenching of the ground state is observed with increased distance between layers

20 40 60 80 1001201401601802002201180

1200

1220

1240

0 -1 -2 -3 -4

1040

1020

1000 7nm 13nm

En

erg

y (

me

V)

Electric Field (KV/cm)

Wa

ve

len

gth

(n

m)

Bias Voltage (Volts)

Spacing Layer Dependence

-40 -20 0 20 40 601.20

1.21

1.22

1.23

1.24

1.25

1.26

1.27

2nm 4nm 6nm

Exc

ito

n e

ner

gy

(eV

)

Applied Field (kV/cm)

• Expect dipole to increase with increased separation

Photocurrent vs. E and TSingle 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

1200 1240 1280 13200

200

400

600

800

1000

230 kV/cm

15 kV/cm

Ph

oto

cu

rre

nt

(pA

)

Energy (meV)

E

20 40 60 80 1001201401601802002201200

1220

1240

1260

1280

13000 -1 -2 -3 -4

1120

1100

1080

1060

1040

1020

En

erg

y (m

eV)

Electric Field (KV/cm)

Wa

ve

len

gth

(n

m)

Bias Voltage (Volts)

40 80 120 160 200

0.0

0.2

0.4

0.6

0.8

1.0

0 -1 -2 -3 -4

e-h

Ove

rlap

Electric Field (KV/cm)

Bias Voltage (Volts)

Summary

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 differentStark Shift: Qualitatively similarEnergy splittings of same order ~ 30-40 meVOscillator Strength: Ground state quenches at higher

electric fieldsMore rapid quenching of the ground state is observed

with increased distance between layers

In good agreement with predicted theoretical calculations!

-60 -40 -20 0 20 40 601.20

1.22

1.24

1.26

1.28

1.30

1.32

Ex

cit

on

en

erg

y (

eV

)

Applied Field (kV/cm)

2nm

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