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Electronic Properties of Coupled Quantum Dots
M.Reimer, H. J. Krenner,
M. Sabathil, J. J. Finley.
Walter Schottky Institut, TU München
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Outline
Motivation
Project Objectives
Introduction to quantum dots
Electronic properties of Nanostructures
Quantum Wells
Self Assembled quantum dots
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Outline Cont’d
How to model a quantum dot
Electronic properties of coupled quantum dots
Photocurrent Spectroscopy of single and coupled quantum dots
Summary
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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
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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
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Introduction to Quantum Dots
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Electronic Subbands
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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
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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
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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
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How to Model a Quantum Dot
A step by step introduction
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Choose the Shape of Dot
Pyramide Lens Semiellipsoid
Dot shape has influence on strain and electronic structure
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• 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
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Define structure including substrate, wetting layer and QD on a finite differences grid.
Define the Structure
Resolution below 1nmsubstrate (GaAs)
wettinglayer QD
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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
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Calculate the Potential
,4 rrr
Solve Poisson equation.
(Piezo, Pyro, electrons and holes)
Conduction band profile including potential and shifts due to strain:
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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
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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
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Electronic Properties of Coupled Quantum Dots
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Coupled Quantum Dots
InGaAs-GaAs self assembled QD-molecules Self alignment via strain field
10nm
7nmWL d=6nm
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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
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• 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
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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)
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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)
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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
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6 nm
xx
Strain has Long Range Effect
WL
WL
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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
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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
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Bonding and Anti-Bonding State
anti-bondingbonding
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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
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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
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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
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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
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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
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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
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Photocurrent Spectroscopy
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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
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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
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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
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Photocurrent – Quantitative Measure of Absorption
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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
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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
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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
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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
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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
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Photocurrent vs. E and TSingle Layer
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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
![Page 57: Electronic Properties of Coupled Quantum Dots M.Reimer, H. J. Krenner, M. Sabathil, J. J. Finley. Walter Schottky Institut, TU München](https://reader035.vdocuments.mx/reader035/viewer/2022062421/56649dab5503460f94a99a23/html5/thumbnails/57.jpg)
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)
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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!
![Page 59: Electronic Properties of Coupled Quantum Dots M.Reimer, H. J. Krenner, M. Sabathil, J. J. Finley. Walter Schottky Institut, TU München](https://reader035.vdocuments.mx/reader035/viewer/2022062421/56649dab5503460f94a99a23/html5/thumbnails/59.jpg)
-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