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
Exciton
wX
1
Ground State
0
Rabi Oscillations have been observed
for single quantum dots
- Zrenner et al. Nature 418 (2002)
  0   1


Obtain coherent control of the two-level system via ps laser pulses
State can be read by measuring a deterministic photocurrent
Target
Control
00
CNOT
00
01
01
10
11
11
10
Initial State
Final State
• 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  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
How?
Experimental Setup
Single Quantum Dot Ensemble
Coupled Quantum Dots
100
T=200K
• PC technique
Photocurrent (pA)
80
successfully applied
to single layer of
quantum dots
60
40 -7.0V
• Stark Shifts
-6
20 -5
-4
-3
-2
-1
0
1.0
• Oscillator Strengths
0V
1.1
1.2
Photon Energy (eV)
1.3
Introduction to Quantum Dots
Quantum Wells, Wires and Dots
3
N (k ) 
4 3  L 
dN
1  2m 
k  .   g 3 D ( E ) 
 2 2 
3
dE 2   
 2 
Wire
Quantum Well
E3D
E1
g0D(E)
g1D(E)
“Enlm“
E00
E0
g2D (E) 
E
Dot
Enm
g2D(E)
3
2
m
 2
E
E
g1D ( E ) 
2m 1
2 E
E
g 0 D ( E )  discrete
Bulk  Quantum Wells
Band - j
Subband - i
E2D
3D
tz
Band - j
x=1
x=0
k=(k
kxy=(k
x,kx,k
y,kyz))
22
22 2
p k xy k
EE
  *  *E z
3 D2 D
2m2m* 2m
tz~nm
Quantum Wires and Dots
Quantised
Motion
Free motion
Wire
ty
2
tz
E1D
Dot
2
 kx

 E yz
*
2m
g0D(E)
Eyz
g1D(E)
E00
E0 D  Exyz
E
E
g1D ( E ) 
“Exyz“
2m 1
2 E
g 0 D ( E )  discrete
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
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
1 x 1µm
• 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
z
x,y
n=3
(6)
(4)
0D states
(2)
Eg
Eg+Ez
(2)
(4)
(6)
x
• Orbital character of QD states similar
to atomic systems
~ HO like potential
QW like potential
2D
state
y
n=2
n=1
Eg+Ez+Exy
• The shells n=1,2,3 - often termed
s,p,d,.. in comparison with atomic
systems
• DEe0-e1~50-70meV, DEh0-h1~2030meV, Exciton BE ~30meV
• 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
T=2 K
 L=632.8 nm
In0.4Ga0.6As/GaAs
s-Shell
p-Shell
PL(µW)
100 - 500 nm
1 µm
0.24
0.20
Probe the optical properties of a QD
 Isolation of a single Quantum Dot
 Emission spectroscopy
 Power-dependence reveals the
different configurations
PL intensity (arb. units)
0.15
0.11
0.08
s-p
p-s
0.06
0.05
2X
0.04
0.03
1X
X0
0.02
2X0
0.015
1280
1290
1300
Energy (meV)
1310
Calculation of Eigenstates - QW
2D structures – V varies only in z-direction
2
2


2 2
2 2
  







V
z

n x, y, z   Enn x, y, z 
*
2
*
2
*
2
2m y y
2mz z


 2mx x

z
• Separate motion  and || to QW
E2
E1
n x, y, z   n z e
i ( k x . xk y . y )
• 1D Schrödinger equation along – z
 2 2








V
z

z

E
 n z 


n
*
2
n
 2mz z

HH1
LH1
HH2
Envelope functions
Electronic Subbands
Materials Discontinuities
• Materials properties (e.g. m*) change accross interface
– Continuity equations for envelope functions
z=0
1
Wavefunction
continuous
 n z 
2
Probability flux=0
(Bound states)
 n z 
m*z z  z
1
mA*
• Both conditions satisfield by BenDanielDuke form of Schrödinger equation.
BenDaniel   2  1   V z  z   E  z 

 n
n
Duke SE  2 z m*z z  z

n
m B*
Contributions to Total Potential
• The total potential (VT(z)) in BenDaniel-Duke Schrödinger
equation may have several contributions
VT z   VSC z   e z   VXC z   Vim (z)
1) Bandedge
modulation
2) Electrostatic
Potential
 2 z 
e z 


2 z
e r z e o
3) Coulomb
Interactions
(e-e, e-h)
ebarrier
ewell
4) Image
Charges
(e-varies)
• Additional contributions can exist in special cases
• e.g. due to piezo-electric charges etc
Example
Undoped GaAs-Al0.3Ga0.7As Quantum Well
VT z   VSC z 
DEc~60%
E0
1940meV
1500meV
HH0
m*e~0.067mo , m*hh~0.34mo
n=1
n=0
•
  2 2 
2
En   * 2 n  1
 2m d 
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
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 B65 (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 (GaAs)
Resolution below 1nm
Calculate the Strain
Minimization of elastic energy in continuum model.
EEL 
e xx
1
Cijkle ij r e kl r dV

2V
e yz
Lead to Piezo electric polarization
GaAs
tensile
InGaAs
compressive
GaAs
-2
-1
0
1 [%] -2
-1
1
2
Calculate the Potential
Solve Poisson equation.
e r r   4 r, 
(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

1
mc r 
*
r   Ec r   er   Er 
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
H KS i  (T  KS (r ))i  e iKS i
KS (r )  x (r )  c (r )  ext (r )  H (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
10nm
WL
d=6nm
 InGaAs-GaAs self assembled QD-molecules
 Self alignment via strain field
7nm
Vertically Correlated QDs
• Upper layers of dots tend to nucleate in strain field
generated by lower layers
100
150
Strain field extends outside buried QD
200
-150
-100
-50
0
50
100
150
200
x (A)
100
150
200
-150
-100
-50
0
50
100
150
9.000 -- 10.00
8.000 -- 9.000
7.000 -- 8.000
6.000 -- 7.000
5.000 -- 6.000
4.000 -- 5.000
3.000 -- 4.000
2.000 -- 3.000
1.000 -- 2.000
0 -- 1.000
-1.000 -- 0
-2.000 -- -1.000
-3.000 -- -2.000
-4.000 -- -3.000
-5.000 -- -4.000
-6.000 -- -5.000
-7.000 -- -6.000
-8.000 -- -7.000
-9.000 -- -8.000
-10.00 -- -9.000
200
x (A)
10nm
Transmission Electron
Micrograph of single
coupled QD molecule
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
Energy [eV]
20
bonding
anti-bonding
0
width
[nm]
Indium content
Well 1
Well 2
5.0
5.0
0.305
0.300
potential [meV] 137
135
0
2
4
6
8
10
Well separation [nm]
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
Energy [meV]
100
anti-bonding
bonding
0
0
width
[nm]
Indium content
Well 1
Well 2
5.0
5.0
0.305
0.300
potential [meV] - 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 meV
Strain has Long Range Effect
exx
WL
6 nm
WL
Strain Deforms Valence Band
strain
HH-valence band
1.65
2nm
Slice through
center of QD
Energy [eV]
1.60
6nm
10nm
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
Heavy hole
-0.436
lower dot
lower dot
~ 22 meV
~ 3 meV
815
4
6
upper
dot
?
upper dot
bonding state
2
Energy (meV)
Energy (meV)
anti-bonding state
8
QD separation (nm)
10
strain
-0.444
2
4
6
8
10
QD separation (nm)
Quantum coupling
Quantum coupling
Strain
Strain
Bonding and Anti-Bonding State
bonding
anti-bonding
Excitonic Structure
quantum coupling + strain + Coulomb interaction
Exciton Energy [eV]
1.29
antibonding
1.28
1.27
1.26
indirect Ex
bonding
Coulomb
interaction [~20 meV]
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
Field
+
Energy
Linear
Stark shift
Energy
Dipole:
Quadratic
Stark shift
-
-+
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.(s0+F)
s0
DE=p.F=es0F+e F2
E = E0 + s0eF + eF2
 First order term provides a direct determination of s0
   Effective height of dot
Anomalous Stark Effect
QD separation 6nm
QD separation 2nm
indirect
1.28
1.26
single QD
direct
1.24
1.22
1.20
-60
Exciton energy [eV]
1.32
1.30
anti-bonding
1.30
1.28
1.26
1.24
1.20
-40
-20
0
20
40
-60
60
0.8
0.8
Overlapp
1.0
0.6
0.4
0.2
-40
-20
0
20
40
60
Applied Field [kV/cm]
1.0
0.0
-60
bonding
1.22
Applied Field [kV/cm]
Overlapp
Exciton energy [eV]
1.32
0.6
0.4
0.2
-40
-20
0
20
Applied Field (kV/cm)
40
0.0
-60
-40
-20
0
20
40
Applied Field [kV/cm]
60
Influence of Coupling
Ground state energy
e-h overlap
0,6
1.26
1.25
e-h Overlap
Exciton energy (eV)
1.27
1.24
1.23
2nm
4nm
6nm
1.22
1.21
1.20
-40
-20
0,4
2nm
4nm
6nm
0,2
0,0
0
20
40
60
Applied Field (kV/cm)
 Weak coupling  Kink
 Strong coupling  Smooth
-40
-20
0
20
40
60
Applied Field (kV/cm)
 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?
Thermal
activation
F
eV
Tunnelling
EF
h
+
n -GaAs
i-GaAs
EMISSION
Metal
esc<<rad
ABSORPTION
 QD-molecules embedded in n-i Schottky photodiodes
 Electric field dependent optical spectroscopy
What Does it Tell Us?
Sample M1638
T = 300 K
EL2
EL Intensity (a.u.)
Photocurrent (pA)
10
E3
E2
EL1
5
E1
E0
0
0.95
1.00
1.05
1.10
1.15
1.20
1.25
 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
1.30
Energy (eV)
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
12
Current (nA)
120
Photocurrent (nA)
10
8
Bulk GaAs
80
Wetting
layer
40
0
1.3
6
1.4
1.5
Photon Energy (eV)
Single layer (x5)
4
Five layers
Single
layer
2
0
1
1.1
1.2
1.3
Photon Energy (eV)
1.4
QDM Photocurrent
1000
250
400
T=300K
900
0V
300
200
E0
100
0
1100
0V
200
800
1150
1200
1250
Energy (meV)
 Strong Stark shift
 Oscillator strength 
 Observations differ strongly
from single QD layer
samples
Photocurrent (pA)
Photocurrent (pA)
Photocurrent (pA)
500
-4V
700
150
600
-3V
500
100
400
300
50
200
-2V
-5V
-1V
0V
100
0
0
1120 1150 1160
1100
1200
Energy (meV)
1200
1250
Single Layer vs. Coupled Layer
1000
60
1.09
800 Reverse
1.08
Photocurrent (pA)
Photocurrent (pA)
80
Transition Energy (eV)
100
1.07
0 100 200 300
Electric Field (kV/cm)
Reverse
Bias
40
20
0
7V
6V
5V
4V
3V
2V
1V
0V
M1638
T = 200K
1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35
Energy (eV)
Bias
600
5V
400
4V
3V
200
2V
1V
0 0V
1050 1100 1150 1200 1250
Energy (meV)
Comparison with Theory:
Transition Energies
Bias Voltage (Volts)
0
-1
-2
-3
1.32
-4
1020
1040
1260
1060
1240
1080
1100
1220
1.30
Energy (eV)
1280
Wavelength (nm)
Energy (meV)
1300
1.28
1.26
1.24
1120
1200
20 40 60 80 100120140160180200220
Electric Field (KV/cm)
1.22
0
20
40
Electric Field (kV/cm)
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
60
Comparison with Theory:
Oscillator Strength
Bias Voltage (Volts)
1.0
e-h Overlap
0.8
-1
-2
-3
-4
7nm
13nm
0.6
0.4
0.2
0.0
20 40 60 80 100120140160180200220
Electric Field (KV/cm)
Oscillator Strength
0
0.6
2nm
4nm
6nm
0.4
0.2
0.0
-40
-20
0
20
40
Applied Field [kV/cm]
Ground State quenches at higher electric fields
More rapid quenching of the ground state is observed with
increased distance between layers
60
Spacing Layer Dependence
Bias Voltage (Volts)
1.26
1240
1.25
1.24
1.23
2nm
4nm
6nm
1.22
1.21
1.20
-40
-20
0
20
40
Applied Field (kV/cm)
60
-1
-2
-3
-4
7nm
13nm
1220
1000
1020
1200
1040
1180
20 40 60 80 100120140160180200220
Electric Field (KV/cm)
• Expect dipole to increase with increased separation
Wavelength (nm)
0
Energy (meV)
Exciton energy (eV)
1.27
Photocurrent vs. E and T
Single Layer
100
100
T=200K
T=4.2K
80
Photocurrent (pA)
Photocurrent (pA)
80
60
40
20
-8.0V
-6.0V
-4.0V
-1.5V
-2.0V
0
1.0
1.1
-1.0V
0V
1.2
Photon Energy (eV)
1.3
60
40 -7.0V
-6
20 -5
-4
-3
-2
-1
0
1.0
0V
1.1
1.2
Photon Energy (eV)
1.3
Carrier Escape Mechanisms
-1.5
Photocurrent (pA)
60
External Bias Voltage (V)
0.5
2.5
4.5
6.5
8.5
50
5K
80K
120K
160K
40
30
240K
200K
20
10
0
0
50
100
150
200
250
300
350
Electric Field (kV/cm)
• 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
Bias Voltage (Volts)
1000
Energy (meV)
230 kV/cm
600
-1
-2
-3
-4
1020
1280
1040
1260
1060
1240
1080
1100
1220
1120
1200
20 40 60 80 100120140160180200220
Electric Field (KV/cm)
400
Bias Voltage (Volts)
E
0
-1
-2
-3
-4
1.0
200
0.8
0
1200
1240
1280
Energy (meV)
15 kV/cm
1320
e-h Overlap
Photocurrent (pA)
800
0
0.6
0.4
0.2
0.0
40
80
120
160
200
Electric Field (KV/cm)
Wavelength (nm)
1300
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 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
Exciton energy (eV)
2nm
1.32
1.30
1.28
1.26
1.24
1.22
1.20
-60
-40
-20
0
20
Applied Field (kV/cm)
40
60