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

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

3. 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

4. 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)

5. 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 ?

6. Introduction to Quantum Dots

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

8. 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

9. 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

10. 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

11. 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

12. 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

13. SAQDs - Electronic Structurez
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)

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

15. 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

16. Electronic Subbands

17. 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

18. 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

19. 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)

20. How to Model a Quantum Dot
A step by step introduction

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

22. 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)

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

24. 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

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

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

27. 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

28. Electronic Properties of Coupled Quantum Dots

29. 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

30. 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

31. 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

32. 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)

33. 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)

34. 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

35. Strain has Long Range Effect
xx WL
6 nm WL

36. 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]

37. 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

38. Bonding and Anti-Bonding State

39. 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]

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

41. 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)

42. 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

43. 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]

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

45. 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

46. Photocurrent Spectroscopy

47. 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

48. 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

49. 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

50. Photocurrent – Quantitative Measure of Absorption

51. 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

52. Single Layer vs. Coupled Layer
Reverse Bias

53. 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 ~ meV

54. 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

55. Spacing Layer Dependence
Expect dipole to increase with increased separation

56. Photocurrent vs. E and T Single Layer

57. 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

58. Temperature Dependence: Coupled Layer

59. 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 ~ 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!

60. 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