Garnett W. Bryant, N. Malkova, and J. Sim

Atomic Physics Division and JQI, NIST, Gaithersburg, 

M. Zielinski and W. Jaskolski

NRC, Ottawa, Canada and UMK, Torun, Poland

J. Aizpurua

DIPC, San Sebastian, Spain

Reengineering the Optics of Quantum Do

Mechanical Strain
Self-assembled quantum dots

... molecular beam epitaxial growth... self-organizing: aInAs = 6.04 Å, aGaAs = 5.65 Å

... growth driven by interfacial strain ... integrability

GaAs

InAsGaAs

Strained quantum dots
Externally imposed strain to modulate dots and co

Internal Strain Internal Strain•Interfaces in core/shell nanocrystals and self-assembled QDsExternal strain External strain•Stressors above self-assembled QDs•Vibrations in nanomechanical cantilevers or bridges with embedded self-assembled dots•Surface acoustic waves interacting with QDs

(Physics Tod

(Zoller gro

s

How much control over SADs or couple QDs

R.L. WillGrowth 2

B. Grandidier, et al., Phys.Rev. Lett. 85, 1068 (2000)
Entanglement

Key resource of quantum science Key resource of quantum science

•Anisotropic splitting•Exchange (shape anisotropy)•State control

–Annealing–Electric fields–Magnetic fields

•Alloy effects as big as shape effects

[Mlinar and Zunger, PRB 79, 115416 (2009)]

Goal: control states with external strain...what is possible
Uniaxial stress on excitons in quantum 

[Seidl, et al., APL 88, 203113 (2006)]

Control splitting with external strain
Modulation of single dot levels with surface acous

[Gell, et al., APL 93, 081115 (2008)]

Control energy levels with SAWs: single photon sour

Modulation

Scheme

Antibunching
1D coupled dots

... vertically stacked self-assembled dots... laterally coupled dot

(by accident or design)

R.L. Williams, et al. Growth 223, 321 (20

B. Grandidier, eRev. Lett. 85, 10

Coupled quantum dots
Novel optical cavities: Rolled up struct

[Kipp et al. PRL 96, 77403 (2006)]

Modulating QD prop

with rolled up waveg

radius of curvature 

S = a: radius of curvatu

Seidl et al.: 1 GPa uniaxial strain at breaking...~10 meV shifts f
... more uses for nanomechanical/QD 

hybrids

Local strain gauge for

... nanomechanical energy harvesting... mass sensing... motion sensing

Mechanical computing

... nanomechanical memory... coupling to nanoelectronics and nanooptics... coupling to solid state QI systems

Optomechanics [Painter group, Caltech: Rosenberg et al., arXiv:0
Outline

• Modeling nanomechanical/QD hybrids 

– Atomistic tight-binding models for QDs– Valence force field– Model for external strain

• Self-assembled dots under external strain

– Electron and hole levels, band gaps– Strain relaxation– Charge redistribution

• Vertical and lateral double dots

– Electron and hole levels– Strain-induced tunneling

• Excitons

– Fine structure – Strain-induced mixing– Polarization rotation

B. GranRev. Le

R.L. WilliamGrowth 223
Atomic geometry

... precise definition of size, shape, surfacesModel: sp3s* or sp3s*d5 atomic orbitals, nearest-neighbor coupling, sStrain: empirical valence force fieldEmpirical tight-binding parameters... sp3s*: Vogl, Hjalmarson and Dow, J. Phys. Chem 44, 365 (1983) 

... sp3s*: CdS, ZnS: Lippens and Lannoo, PRB 39, 10935 (1989)

... sp3s*d5: Jancu et al., PRB 57, 6493 (1998)

... sp3s*d5 : Sapra, Shanti, and Sarma, PRB 66, 205202 (2002)

Optical response: dipole matrix elements from Slater orbitals, on-si

Empirical Tight-Binding Calculations: Atomistic App
Self-assembled quantum dots in nanomechanica

Results for small bridge with pyramidal Q

s

- atomistic tight-binding theory: spin-orbit interaction, no p- relaxation: valence force field - external strain: fixed, bent bridge 

- internal lattice relaxation due to InAs/GaAs lattice misma

and applied strain

- excitons: configuration interaction approach

L=81 nm

25 nm

81 nm height 3 n

base 7 n

Symmetric bend (biaxial deformation) Antisymmetric bend 

x

z

y
Caveats

One QD size (small ~1/2 size)Symmetric QDStrain applied only along x (100)

... minimal piezoelectric effect (ignore)... nanomechanical structures but not SAWs

Static distortion

... scattering of nanomechanical modes by QDs

(??)One type of boundary condition

... sensitivity to boundary conditions (?)
Internal biaxial strain Lattice relaxation [7% 

aGaAs = 5.65 Å; aInAs = 

X shift: z = 0

Compression in x,y (

ΔV > 0) 

Expansion in z (

ΔV < 0)

Bound states

12a 13a

CB

VB

~8%
Manipulating electron and hole energ

Electrons

Biaxial deformation•“Rigid” shift with fixed level ordering andstate symmetries•Shift: 1-10 meV•Electron and “hole” shift in same wayShear•Quadratic increase•Mixing, collapse to wetting layer•Electron and “hole” shift in opposite way

“Analog” to DC Stark effect: same or opposite c

1S

“1P”
Biaxial deformation

•“Rigid” shift with fixed level ordering andstate symmetries•Electron and “hole” shift in same wayShear•Quadratic decrease•Mixing

Holes

bend

bend down 

InAs QDs in GaAs bridges: coupling to bend
InAs QDs in GaAs bridges: coupling to bend

s

•Electron localized to QD•Hole more localized in WL under the dot

• E and h: down, out of QD, into WL• 4-10% charge shifts

• E QD• 4-shi

“Analog” to DC Stark effect

Hole ground 

state
InAs QDs in GaAs bridges: coupling to she

Electron and hole ground state

s

E: pushed left, then right and into WL H: pushed right and into 

50% polarization
Manipulating electron and hole stat

“Analog” to DC Stark effect: counter-rotating e and 

e

h

Electron H

Asymmetric stretching or squeezing
Importance of internal relaxation: symmetric bend

Electrons Holes

Biaxial deformation: internal relaxation is critical
Importance of relaxation

Z shift: x=y=0

Z shift: x=y

Conduction b

δZ sh

compression

Lattice deformation

Δ

Vdef > 0

Δ

Vdef < 0
Importance of internal relaxation: shear bend

Electrons Holes

Shear deformation: bending shear is dominant
Sensitivity to boundary conditions

Strain distribution (Tr

ε) for up and down bendings

bent down

bent upunbent

blue – comred - expa
Vertical double dot: electrons

bend

Biaxial deformation

bend down bend up

Shear

Transferring electrons and holes in coupl
Biaxial deformation: electrons

Down 6a Down 2a

Manipulating inter-dot transfer
Hole ground state: spin distribution in an un

y

y

x

x

x

y

Spin down

Spin up Pol

Po

x

z
Hole excited state: spin distribution in an un

y

y

x

x

x

y

Spin down

Spin up Pol

Po

x

z
Vertical double dot: holes

Spin-dependent coupling ??

s

Top dot Bottom d

•Initially, follows single dot•Strong transfer top to bott•Spin mixed from other dot

z
Excitons: biaxial 

deformation

Exciton follows pair ground stateBinding can increase or decrease by bending (charge shifting)Fine structureEx1 and Ex

2 are dark (“triplet”)

Ex

3 and Ex

4...exchange split 

bright states (“singlet”)Ex

3 and Ex

4...asymmetric 

exchange Bend-induce level crossing and polarization rotation

e 

h 
Excitons: 

biaxial deformation

Pair ground state weakly polarized, strain polarizes along yEx3(4) polarized along x-y (x+y) in 

unbent structureEx3(4) polarized along x (y) by bending
Excitons: shear

Exciton follows pair ground stateBinding increases by bending (hole squeezing)Fine structureEx1 and Ex

2 are dark (“triplet”)

Ex

3 and Ex

4...exchange split 

bright states (“singlet”)Ex

3 and Ex

4...asymmetric 

exchange and strong coupling Bend-induce level crossing and polarization rotation
Excitons: shear

Pair ground state weakly polarized, strain polarizes along xEx3(4) polarized along x-y (x+y) in 

unbent structureEx3(4) polarized along y (x) by bending (reverse of symmetric bend

)
Strained QDs

• DC Stark field (imperfect) analog

– Biaxial deformation:like Ez

• E and h shift the same way

– Shear bends: like Ex

• E and h shift the opposite way

– Internal relaxation and state distortion

• Excitons

– Dominated by e/h shift– Fine structure: mixing, level crossing and 

polarization rotation

• Coupled dots

– Strain-induced state crossing and transfer– Spin-dependent interdot coupling??

Final comments

e h