Effect of Mechanical Strain on the Optical Properties of Quantum Dots:

Controlling Exciton Shape, Orientation, and Phase with a Mechanical Strain

Garnett W. Bryant,1,* M. Zielin ́ski,2 Natalia Malkova,1 James Sims,3 W. Jasko ́lski,2 and Javier Aizpurua4

1

Atomic Physics Division and Joint Quantum Institute, National Institute of Standards and Technology,

100 Bureau Drive, Gaithersburg, Maryland 20899-8423, USA

2

Instytut Fizyki UMK, Grudzia ̧dzka 5, 87-100 Torun ́, Poland 3

Information Technology Laboratory, National Institute of Standards and Technology,

100 Bureau Drive, Gaithersburg, Maryland 20899-8911, USA 4

Centro Mixto de Fı ́sica de Materiales CSIC-UPV/EHU and Donostia International Physics Center,

Paseo Manuel Lardizabal 4, Donostia-San Sebastia ́n 20018, Spain

(Received 22 March 2010; revised manuscript received 21 May 2010; published 5 August 2010)

We show how a nanomechanical strain can be used to dynamically reengineer the optics of quantum

dots, giving a tool to manipulate mechanoexciton shape, orientation, fine structure splitting, and optical

transitions, transfer carriers between dots, and interact qubits for quantum processing. Most importantly, a

nanomechanical strain reengineers both the magnitude and phase of the exciton exchange coupling to tune

exchange splittings, change the phase of spin mixing, and rotate the polarization of mechanoexcitons,

providing phase and energy control of excitons.

DOI: 10.1103/PhysRevLett.105.067404 PACS numbers: 78.67.Hc, 73.21.La, 85.35.Be

Quantum dots (QD) have attracted great interest for

applications in photonics, sensors, quantum information,

and precision measurement. Passive control of QDs is

obtained by tailoring dot size, shape, and composition via

growth. Dynamical control of exciton energies, polarization, and phase is highly desirable for QD nanophotonics.

For example, a two photon cascade from the QD biexciton

state can generate entangled photons in scalable devices

[1]. However, anisotropic exchange splitting (AES) of QD

excitons, induced by the asymmetry of typical as-grown

QDs, inhibits entanglement. Annealing [2], magnetic, and

electric fields [3–6], and dressing excitons with optical

fields [7] are being used to modify AES. An imposed

nanomechanical strain [8–11] provides a route to dynamically reengineer QD structural symmetry to control excitations, polarize transitions, tune exchange splitting,

induce entanglement, or modify coupling between QDs.

These are capabilities needed to use QDs in nanophotonics,

quantum information processing, and in optically active

devices, such as optomechanical cavities [12,13] and semiconductor nanotubes [14–17].

Nanomechanics is being studied for mass sensing

[18,19], mechanical computing [20,21], and energy harvesting [22]. Structures are being cooled to approach the

quantum limit for metrology and to provide coherent transducers that couple classical machines to quantum devices

[23–26]. Surface acoustic waves (SAW) are being used to

manipulate carriers in dynamically created QDs [27–32].

Local probes and control of nanomechanics and SAWs are

needed. Sideband cooling via optical absorption by QDs in

nanomechanical structures could drive structures to the

quantum limit [25]. An experiment shows that QD levels

are sensitive to a local strain produced by SAWs and

mechanical deformations [8–11]. QD response could be a

local strain gauge for nanomechanics.

To exploit hybrid nanomechanical-QD devices, a fundamental understanding is needed. This entails understanding

connections between the strain from lattice mismatch,

imposed nanomechanical strain, electron and hole states

of QDs in the nanomechanical device, and strained excitons (mechanoexcitons) in excited QDs. We study pyramidal InAs QDs in a GaAs nanomechanical bridge using

atomistic tight-binding theory. The bridge is bent to simulate an external strain applied to mechanoengineer QDs. A

bend in a nanomechanical structure is analogous to an

electric field, inducing Stark-like energy shifts. Electrons

and holes redistribute vertically along the QD growth axis,

or horizontally in the plane of the QD, depending on how

strain is applied. This behavior correlates with bendinduced changes in the local band profile. Strain-induced

charge redistribution in closely spaced QDs can induce

tunneling between dots. Most importantly, nanomechanical strain reengineers the magnitude and the phase of the

exciton exchange coupling via, primarily, strain-induced

hole redistribution. This leads to large changes in exciton

exchange splitting and polarization of bright excitons.

Exciton phase control is achieved because the exchange

reengineering changes the phase of spin mixing in the

mechanoexcitons.

Using an atomistic model is critical for an accurate

description of nanomechanical-QD hybrids with atomic

scale variations in composition and shape, and significant

local and imposed strain [33]. We use tight-binding theory

[33–35] for electron and hole states with an sp3s orbital

model, nearest-neighbor coupling, spin-orbit effects, strain

from lattice mismatch, and imposed mechanical strain.

PRL 105, 067404 (2010) PHYSICAL REVIEW LETTERS week ending

6 AUGUST 2010

0031-9007=10=105(6)=067404(4) 067404-1  2010 The American Physical Society
Relaxation of local and imposed strain is included via

atomistic valence force field theory. Exciton states are

determined with a configuration-interaction treatment

[36–38]. An atomistic model is essential for describing

AES [39–41]. While AES arises if the QD has geometrical

asymmetry [42], AES can arise even if the QD has in-plane

geometrical symmetry, because the atomic lattice can

break symmetry [39,40].

We consider a QD in a nanomechanical bridge, as shown

in Fig. 1. The bridge is clamped at each end and bent along

[100] by shifting the bridge vertically along [001]. After

the structure is bent, surface atoms are held fixed while

atoms inside the bridge relax to minimize imposed strain

and lattice mismatch. The maximum bend amplitude S we

discuss produces a beam elongation of 0.25% and lattice

distortions a tenth of the shifts due to lattice mismatch. For

the simulations, the bridge is 80 nm wide in the lateral

directions and 25 nm thick, with 10 million atoms. A small,

square pyramidal QD (height 3 nm, base 7 nm) and the

wetting layer are located near the middle of the bridge. The

QD is symmetric, so AES is induced by atomic symmetry

breaking. We consider two bends: a bend symmetric about

the QD that distorts biaxial deformation from lattice mismatch, and a bend antisymmetric about the QD that gives

vertical shear, as shown in Fig. 1. Our results are sensitive

to bend geometry and dot location. Calculations for different bends, boundary conditions, and dot positions must be

done to build a complete picture of nanomechanical strain

effects [8,11]. We chose the bend along [100] to focus on

effects of mechanical deformation rather than piezoelectric

effects that can arise for other bend directions. For a small

QD, piezoelectric effects are small [34]. We ignore these

piezoelectric effects here.

QD electron energy shifts in a bent bridge are shown in

Fig. 2. A symmetric bend raises (lowers) the electron

energy when the bridge is bent up (down). Few meV shifts

are possible for the S shown. The 1S-1P splitting is nearly

constant, suggesting that the main effect of the bend is a

conduction-band profile shift. This bend has similar effects

on hole levels. For a shearing bend, the lowest confined

electron energies increase until they cross wetting layer

states for S 2a (a is the GaAs lattice constant), indicating that confined levels are shifted laterally out of the QD

by shear. In contrast, a shearing bend reduces hole energies. As illustrated in Fig. 2, the band gap Eg can increase

or decrease substantially for symmetric bends, but Eg

decreases only slightly for shearing bends (because hole

energies change more than electron energies). Figure 2 also

shows how states redistribute (e;v) inside the dot when

the bridge is bent. A symmetric bend shifts both electrons

and holes vertically. A shearing bend pushes electrons and

holes horizontally in opposite directions. This spatial control is important for coupled dots. Calculations for a pair of

QDs in a bridge show that a symmetric bend can transfer

charge between vertically stacked QDs, while a shear bend

can transfer charge between laterally coupled QDs.

Effects of mechanical strain can be explained by straininduced changes to band profiles [34,43]. For electrons, the

change is primarily via the hydrostatic deformation potential. For holes, there is also a shear contribution. Here we

discuss the simpler case for electrons. Figure 3 sketches the

biaxial deformation of the QD due to lattice mismatch.

When the bridge is bent symmetrically, internal reaction

tries to undo the bend, providing vertical expansion (compression) near the wetting layer and compression (expansion) near the QD apex for S < 0 (S > 0). As shown in

Fig. 3(d), this relaxation lowers (raises) the conduction

band near the wetting layer and raises (lowers) the band

near the QD apex for S < 0 (S > 0). The response is not a

uniform band shift. Local variation of the QD band profile

shifts the energies and redistributes electron density. The

effect of a shearing bend on electrons arises from internal

relaxation which pushes up one side of the dot and pushes

down the other side in reaction to the bend, providing a

local profile which is lower on one side of the dot and

higher on the other.

Models for QDs coupled to mechanical strain which

mechanically deform the QD without the internal reaction

will be suspect. Calculations done by taking a flat beam

with relaxation due to lattice mismatch, bending it, but not

allowing additional reaction, predict energy shifts opposite

to those in Fig. 2. The local, nonlinear lattice reaction must

be included. In contrast, SAW structures with weaker strain

FIG. 1 (color online). Schematic of a bend applied to a QD in a

nanobridge: (left) symmetric (biaxial deformation) bend with a

QD at an antinode, and (right) vertical shearing bend with a QD

at a node. S is the bend amplitude for clamped ends.

-6 -4 -2 0 2 4 6

Bend amplitude (a)

1.16

1.2

1.24

1.28

1.32

E (eV)

1S

1P

(a)

0 0.5 1 1.5 2 2.5 3

Bend amplitude (a)

1.1

1.15

1.2

1.25

1.3

1.35

E (eV)

1S

1P

(b)

FIG. 2 (color online). Electron 1S and 1P energies for (a) a

biaxial deformation and (b) a shearing bend. Blue or red band

gap shift Eg and redistribution e;v of the lowest conduction

(e) and highest valence state (v) for different bends.

PRL 105, 067404 (2010) PHYSICAL REVIEW LETTERS week ending

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do couple linearly to QDs and can be accounted for via

band shifts (results not shown).

For the QDs considered, the lowest excitons are made

from the lowest-energy electron-hole pair states with little

mixing of higher pair states. The lowest electron and hole

states are spin-degenerate, so the lowest pair state is fourfold degenerate. Coulomb and exchange interaction split

the pair ground state into two lower-energy dark excitons

(DE) and two bright excitons (BE). Figure 4 shows the

energy splitting of these four mechanoexcitons from the

lowest pair state. The two DEs are nearly degenerate and

split from the pair ground state by the binding energy. The

binding energy increases (decreases) for a biaxial deformation with S > 0 (S < 0) because the electron and hole

are pushed more into (out of ) the QD [see Fig. 2]. The

binding increases for a shearing bend, even though the

electron and hole are separated laterally, because the vertical electron-hole separation is reduced by the shearing

bend. The BEs are split from the DEs by exchange interaction. The BEs are split by the AES. Binding energy and

DE-BE exchange splitting vary slowly as mechanical strain

is applied. However, mechanical strain significantly alters

AES, with reduced AES and an apparent anticrossing for a

biaxial deformation with S > 0 and increased AES for a

shearing bend.

When mechanical strain is applied, DEs remain dark and

BEs remain bright, because DE transitions, with change in

total z spin Jz  1, are not strongly mixed with BE

transitions with Jz 1⁄4 1. The two BEs remain orthogonally polarized after bending. However, BE polarization is

rotated significantly by bending. In a flat bridge, BEs are

polarized along QD diagonals, as in Fig. 4(d). For a biaxially deformed bridge, the lowest BE becomes polarized

along the [100] bend for S > 0 and approaches this polarization for S < 0. For a shearing bend, the polarization of

the lowest BE rotates instead toward [010], becoming

perpendicular to the bend for S ’ 2a. Strain-induced distortion of the single-particle dipole moments cannot explain the rotation of exciton polarization. The polarizations

of single particle transitions change only by about 10% for

these strains (not shown here). As we will discuss, the

strain reengineers the exciton exchange coupling to modify

not only the exchange splitting but also the phase of the

spin mixing that determines the polarization.

The BEs are mixtures of the two degenerate, Jz 1⁄4 1

pair states that are excited by circularly polarized light

(Jz 1⁄4 1). The (real) Coulomb matrix element VCoul

determines the binding energy, but does not strongly mix

Jz. DE-BE exchange splitting is determined by a (real)

exchange matrix element which also conserves spin,

Vexch;sc. AES between BEs is determined by the magnitude

of the (complex) off-diagonal exchange interaction

Vexch;mix which does mix Jz. VCoul and Vexch;sc are much

larger than jVexch;mixj, but strain-induced changes in VCoul

and Vexch;sc are comparable to jVexch;mixj, as shown in

Fig. 5. VCoul, Vexch;sc, and jVexch;mixj follow, respectively, the change in binding energy, the DE-BE exchange

splitting, and the AES shown in Fig. 4(c). VCoul and Vexch;sc

vary smoothly under strain while jVexch;mixj shows the

apparent anticrossing. This is a matrix element effect rather

than a true anticrossing, because the exciton character does

not switch near the anticrossing.

Because Jz 1⁄4 1 pair states are mixed to make BEs, the

phase of mixing is determined by the phase of Vexch;mix in

Fig. 5. BE polarization is determined by the phase of the

mixing. When phases for these pair states are chosen so

that their optical dipole moments are proportional to x^ 

iy^, then Vexch;mix is imaginary for S 1⁄4 0, ensuring that BEs

-6 -4 -2 0 2 4 6

S (a)

-70

-65

-60

-55

-50

Eex - Epair (meV)

BE2

BE1

DE2

DE1

(a)

-6 -4 -2 0 2 4 6

S (a)

0

2

4

6

8

Energy splitting (meV)

(c)

Asymmetric exchange 

 splitting

Exchange splitting

Binding energy/10

-6 -4 -2 0 2 4 6

S (a)

0

10

20

Absorption (arb. units)

BE1

x

y

x + y

x - y

(d)

0 0.5 1 1.5

S (a)

-70

-65

-60

-55

-50

Eex - Epair (meV)

BE2

BE1

DE2

DE1

(b)

FIG. 4 (color online). Exciton fine structure for (a) biaxial

deformation and (b) shearing bend. (c) Energy splittings for

biaxial deformation. (d) Absorption strength of the lowest BE

for biaxial deformation and polarization along QD diagonals

(x þ y, x  y) and QD sides parallel (x) and perpendicular (y) to

the bend.

-5 0 5 10 Z (a)

1.2

1.3

1.4

1.5

1.6

1.7

Conduction band edge (eV)

Flat

S = 5a

S = -5a

(d)

-5 0 5 10 Z (a)

-0.01

-0.005

0

0.005

0.01

∆zn+1 - ∆zn (a)

S = 5a

S = -5a

z shift from a relaxed, flat bridge

compression

expansion

(c)

FIG. 3 (color online). Schematics for (a) relaxation due to

lattice mismatch in a flat bridge and (b) additional relaxation

[grey (red) arrow] in a bent bridge. (c) Lattice shift along the

vertical axis through the QD center for biaxial deformation, zn

is the extra relaxation of atom n from its position in a flat bridge

for atoms ordered along the axis, and (d) conduction band profile

along the vertical axis for a flat bridge and biaxially deformed

bridges. The dotted curve is the unrelaxed band profile.

PRL 105, 067404 (2010) PHYSICAL REVIEW LETTERS week ending

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are polarized along QD diagonals in a flat bridge. When

strain is applied, the phase of Vexch;mix and the polarization

rotate. The sign of Vexch;mix determines whether polarization rotates toward x or y.

A picture of why strain controls the phase of Vexch;mix

emerges by looking at Coulomb and exchange matrix

elements. VCoul couples electron density at r [eðrÞ 1⁄4

jeðrÞj2, for electron state eðrÞ] with hole density at r0

,

[hðr0

Þ1⁄4jhðr0Þj2]. VCoul is real and spin independent.

Vexch;sc couples polarization density PðrÞ at r, with PðrÞ 1⁄4

eðrÞhðrÞ, and the conjugate polarization density at r0

,

Pðr0

Þ. Thus, Vexch;sc is real, depends mostly on the interaction weighted average of jPðrÞjjPðr0

Þj and weakly on the

phase difference between PðrÞ and Pðr0

Þ. In contrast, for

Vexch;mix, the Kramers degeneracy of pairs with opposite

spin ensures that the coupling is between PðrÞ and Pðr0

Þ.

Vexch;mix and Vexch;sc depend similarly on jPðrÞj, but

Vexch;mix depends on the sum of the phases of PðrÞ and

Pðr0

Þ. Mechanical strain manipulates exciton binding energy and DE-BE exchange splitting by reshaping the electron and hole densities. The smooth variation of binding

energy and exchange splitting reflect limited straininduced reshaping. The exciton polarization and phase of

spin mixing are manipulated by changing the relative phase

and orientation of the electron and hole pair. BE polarization varies for alloy QDs with the same composition but

different atom distributions [41]. This also points to the

connection between polarization, phase of spin mixing,

and relative orientation of the electron and hole.

In conclusion, nanomechanical strain can be used to

reengineer quantum dots. Electron and hole energies and

distributions shift together or in opposite directions depending on how strain is applied, giving control to tune

optical response. Strain can be used to transfer carriers

between dots, giving a tool to interact qubits for quantum

information processing. Changes in band gap, finestructure splitting, charge shifts, and polarization correlate

to applied strain, giving signatures to gauge local strain.

Internal strain from lattice mismatch, the nanomechanical

strain, and internal readjustment to undo the applied strain

must all be included to model strain effects. Most importantly, the applied strain can be used to manipulate

mechanoexciton phase and fine structure, and rotate polarization, providing both energy and phase control to modify

the inner workings of excitons.

*garnett.bryant@nist.gov

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-6 -4 -2 0 2 4 6

S (a)

-0.5

0

0.5

1

1.5

2

Coulomb, exchange coupling (meV)

∆VCoul

∆Vexch, sc

|Vexch, mix|

(a)

-6 -4 -2 0 2 4 6

S (a)

0

45

90

135

180

Phase of Vexch,mix (deg)

Biaxial deformation

Shear deformation

(b)

FIG. 5 (color online). (a) Change of VCoul, Vexch;sc, and

jVexch;mixj for a biaxial deformation. (b) Phase of Vexch;mix.

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