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Light controlled optical Aharonov-Bohm oscillations in a single quantum ring Robert Anthony Taylor, Heedae Kim, Seongho Park, Rin Okuyama, Kwangseuk Kyhm, Mikio Eto, Gilles Nogues, Le Si Dang, Marek Potemski, Koo Chul Je, Jong Su Kim, Ji-Hoon Kyhm, and Jin Dong Song Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b02131 • Publication Date (Web): 18 Sep 2018 Downloaded from http://pubs.acs.org on September 18, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

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Light controlled optical Aharonov-Bohm oscillations in a single quantum ring a¶

†,‡

Heedae Kim,

Seongho Park,

∗,‡

Robert A. Taylor,

k

Gilles Nogues,

@

Jongsu Kim,

†School ‡Clarendon

Rin Okuyama,

§

Kwangseuk Kyhm,

k

Le Si Dang,

Jihoon Kyhm,

4

⊥

Marek Potemski,

and Jindong Song

∗,¶

§

Mikio Eto,

Koochul Je,

#

4

of Physics, Northeast Normal University, Changchun 130024, China

Laboratory, Department of Physics, University of Oxford, Oxford, OX1 3PU, U.K

¶Department

of Opto-mechatronics, Cogno-mechatronics, Physics Education, Pusan Nat'l University, Busan 609-735, South Korea

§Faculty

of Science and Technology, Keio University, Yokohama 223-8522, Japan

kDepartment

of NANOscience, Institut Néel, CNRS, rue des Martyrs 38054, Grenoble, France

⊥Laboratoire

National des Champs Magnetiques Intenses, CNRS-UJF-UPS-INSA, F-38042, Grenoble, France

#Institute

of Physics, Faculty of Natural Science and Mathematics, Sts Cyril and Methodius University, 1000 Skopje, Macedonia

@Department 4Center

of Physics, Yeungnam University, Gyeonsan 712-749, South Korea

for Opto-Electronic Convergence Systems, KIST, Seoul, 136-791, South Korea

E-mail: [email protected]; [email protected]

a equal

contribution with rst author

Abstract 1

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Page 2 of 19

We found that optical Aharonov-Bohm oscillations in a single GaAs/GaAlAs quantum ring can be controlled by excitation intensity. With a weak excitation intensity of 1.2 kWcm−2 , the optical Aharonov-Bohm oscillation period of biexcitons was observed

to be half that of excitons in accordance with the period expected for a two-exciton Wigner molecule. When the excitation intensity is increased by an order of magnitude (12 kWcm−2 ), a gradual deviation of the Wigner molecule condition occurs with decreased oscillation periods and diamagnetic coecients for both excitons and biexcitons along with a spectral shift. These results suggest that the eective orbit radii and rim widths of electrons and holes in a single quantum ring can be modied by light intensity via photo-excited carriers, which are possibly trapped at interface defects resulting in a local electric eld. keywords: quantum rings, excitons, Aharonov-Bohm eect, photoluminescence, light excitation The Aharonov-Bohm (AB) eect has been investigated mostly by electrical experiments in mesoscopic ring structures, where oscillations of the conductance and persistent current with external magnetic eld (B ) were observed at extremely low temperature (< due to the fragile quantum coherence in sub-micron sized systems.

13

100 mK)

Recently, the AB eect

became accessible to optical experiments by using type-II quantum dots (QDs) and quantum rings (QRs), where quantum coherence on a nanometer scale (∼ temperatures up to three orders of magnitude higher (<

50 nm)

is observed at

100 K). 412

While either an individual electron or hole rotates in the shell of a type-II QD, both electrons and holes orbit around the circumference of a QR. When a strong Coulomb interaction is present between the electron and the hole in a QR, the exciton (X) can be approximated by a single particle. However, charge neutral Xs require dierent orbital radii for the electron and the hole (Re

6= Rh )

in order to give rise to the AB eect.

between the magnetic uxes threading their orbits (Φ of a radially polarized X (φ optically as

B

= eΦ/¯ h).

6

In this case, a dierence

= Bπ|Re2 − Rh2 |) determines the phase

Therefore, oscillations in the X energy can be observed

increases. When a one-dimensional (1D) model is assumed for a QR structure,

2

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the X AB oscillation period On the other hand, when

∆BX = (h/e)/(π|Re2 − Rh2 |) N

can be estimated.

electrons are strongly correlated in a ring structure, they can

behave as a composite rigid body, called a Wigner molecule (WM). oscillation period of an

∆Be = (h/e)/(πRe2 ), 1720

a two-electron WM has been claimed in a large quantum dot

13

In this case, the AB

N -electron WM with applied B becomes fractional (∆BN e = ∆Be /N )

when compared to that of a single electron

otube

1316

1416

and the presence of

and a clean carbon nan-

by electrical measurements. In an optical analogy, biexcitons (XXs) in a QR can be

a two-X WM.

9

If a pair of Xs located at opposite positions in a QR are strongly correlated

like a rigid dumbbell, the AB oscillation period of the XX WM (∆BXX ) is expected to be half that of the X (∆BX ), i.e.

∆BX /∆BXX = 2.

Although the orbits of electrons and holes in a QR are often simplied into 1 dimensional loops, the rim width is considerable in comparison to the QR radius.

In this case,

the wavefunctions and the Coulomb interaction of electrons and holes in a nite rim width need to be considered to predict optical AB oscillations of X and XX, and the two-X WM criterion is also known to be vulnerable to Coulomb interactions. Nevertheless, this subtle range has barely been investigated experimentally so far. In addition, the presence of a local electric eld (ELoc ) is quite likely at the interface between a QR and the capping barrier material.

Suppose that carriers excited in a capping barrier aect

ELoc

at the interface,

the wavefunction modication of a QR might be susceptible to optical excitation intensity changes. In this work, we have found that optical AB oscillations of both Xs and XXs in a single GaAs/AlGaAs QR, can be controlled by the optical excitation intensity, whereby the two-X WM criterion is scrutinised in terms of

∆BX /∆BXX .

For the growth of a GaAs QR on AlGaAs substrates, gallium molecular beams are supplied initially without arsenic ambient, leading to the formation of droplet-like clusters of gallium atoms on the substrate.

21,22

As an arsenide ux is supplied in the chamber, the

gallium droplet becomes crystallised into GaAs ring structures, where the morphology is governed by the interplay between the migration of surface adatoms and their crystallisation. In

3

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Figure 1:

Page 4 of 19

(a)

(b)

(c)

(d)

AFM images of a single GaAs QR in a top (a) and side (b) view. (c) With an 12 kWcm−2 (10P ), the micro-PL spectra of a single GaAs/AlGaAs QR

excitation intensity of

were observed as a function of

B , where the Zeeman doublet spectra of X and XX were tted

by Gaussian functions (d).

4

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Fig. 1(a) and (b), the rim height of an uncapped GaAs QR grown by droplet epitaxy

21,22

anisotropic, where a dierence between the maximum and minimum rim height along and [110] is

∼ 2 nm,

and the lateral size is also elongated along

[1¯10].

is

[1¯10]

In this volcano-like

structure, the wavefunction is likely to be localised in a crescent-like shape unless sucient

B

magnetic eld

is applied,

2326

and the two separate localised states become merged be-

yond a characteristic magnetic eld (Bc ). As a result, AB oscillations of a QR begin when

B ≥ Bc . 5 As the excitation intensity (Iex ) increases, an additional photoluminescence (PL) peak becomes signicant at by the quadratic decay time (τXX

Iex

∼ 1 meV below the X PL spectrum,

which has been attributed to XX

dependence of the integrated PL intensity (∼

∼ 370 ps)

are rarely seen at either

compared to that of X (τX

∼ 4 meV

below and

∼ 2 meV

∼ 820 ps).

2 ) Iex

and the short PL

Although charged excitons

above the X PL spectrum, a charged

exciton-free QR was carefully selected. As shown in Fig. 1(c), the micro-PL spectrum of a single QR was measured at

4 K for increasing B

by using a resistive DC magnet with

bore diameter, where excitation by a continuous-wave Ar

+

52 mm

laser (488 nm) was introduced to

the position-controlled sample through a multi-mode ber, and the PL was collected through a single-mode ber. The wiggling of the energy with increasing

B

in Fig. 1(c) is claried

by plotting the PL peak emission energies of the Zeeman doublet (EPL (σ which were obtained by Gaussian tting with an accuracy of the Zeeman doublets of X and XX become resolvable for

+

∼ 2.5 µeV

B > 2T

)

and

(Fig.

EPL (σ − )), 1(d)).

As

(Fig.1 (d)), a ring shape

wavefunction is possibly formed from separate crescent-like ones, near

Bc ∼ 2 T.

Recently, we have studied a strongly localised exciton state in a GaAs QR,

26

and the

cross-sectional wavefunction shape perpendicular to the growth direction was shown to be crescent-like.

For a micro-PL spectrum of the strongly localised state, near

found both the excitonic

g -factor (∼ −0.2)

and diamagnetic coecient (γX

1.812 eV,

we

∼ 1.3 µeVT−2 )

were very small. Therefore, the Zeeman splitting was not signicant unless a strong magnetic eld (B

> 4 T)

is applied. On the other hand, when a relatively low energy micro-PL spec-

5

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Page 6 of 19

(a)

0T 0.6 T 1.0 T 1.6 T

X

XX

P=1.2 kWcm-2

1.721

1.722

1.723

1.724

1.725

PL intensity (arb. units)

Energy (eV)

(b)

0T 0.6 T 1.0 T 1.6 T

XX X

10P

1.721

1.722

1.723

1.724

1.725

Energy (eV)

PL intensity (arb. units)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

PL intensity (arb. units)

Nano Letters

(c)

X

B=0 T

1.721

Figure 2:

XX

P 10P

1.722

1.723

Energy (eV)

1.724

PL spectra of X and XX are measured under weak (P

strong excitation (10P ) (b) for increasing

B

inseparable. (c) When excitation is increased and XX show a blueshift in PL spectrum.

6

= 1.2 kWcm−2 )

1.6 T, where the Zeeman from P to 10P in the absence

up to

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1.725

(a) and

doublets look of

B,

both X

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1.732 eV) was investigated in a GaAs QR, 9

trum (∼

X and XX become separable near coecient (γX

∼ 1.3 µeVT−2 )

2 T,

we found the Zeeman doublet spectra of

and the excitonic

g -factor (∼ −1.3)

and diamagnetic

are enhanced by an order of magnitude compared to those for

the strongly localised state. Therefore, one may suggest that

Bc

seems to depend on the de-

gree of localisation and the QR anisotropy. While strongly localised states in a volcano-like QR require a large

Bc

to give rise to AB oscillations, a small

Bc

is expected for the weakly

localised states. As shown in Fig. 2(a) and (b), no measurable splitting can be seen in the Zeeman doublet photoluminescence spectra of X and XX until a magnetic eld of sucient strength is applied (B

> 2 T).

Therefore, these PL spectra can be attributed to localised

states in a QR. In Fig. 2(c), both X and XX show a blueshift for increasing excitation in the absence of

B.

As carriers are excited non-resonantly in the barrier, the blushift is possibly

associated with excitation-induced local eld The

B -dependence

ELoc .

of the average energy

[EPL (σ + ) + EPL (σ − )]/2

of X (Fig. 3(a),(b),

and (c)) and XX (Fig. 3(d), (e), and (f )) are plotted for three dierent excitations of

5P ,

10P ,

and

respectively.

P,

All show clear AB oscillations along with a diamagnetic en-

ergy increase, where the diamagnetic coecients (γ ) were obtained by a quadratic tting

∼ γB 2 .

In Fig. 3(g),

tting function of XX (gXX by the

[EPL (σ + ) − EPL (σ − )]/2

gµB B

enables us to estimate the

∼ −1.31 ± 0.08).

B -dependence

for X and XX are also plotted, and a linear

g -factors

of X (gX

∼ −1.61 ± 0.05)

and

A subtle deviation from a linear dependence may be explained

of the

g -factor,

and this is possibly associated with a wavefunction

modication in a QR structure. On the other hand, the diamagnetic coecients of both X (γX ) and XX (γXX ) decrease by

1.2 kWcm−2 (P )

to

20 %

12 kWcm−2 (10P ).

(Fig. 3(h)) as the excitation intensity increases from Because a diamagnetic coecient is proportional to

its cross-sectional area in the plane perpendicular to

B,

the decrease of

γX

and

γXX

implies

a shrinkage of the wavefunction rim width with excitation. Additionally, both X and XX show a blueshift of intensity from

P

to

∼ 0.2 meV 10P .

at

B = 2T

for an order of magnitude increase of excitation

Therefore, excited carriers seem to change the potential energy,

7

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(a)

(d)

(b)

(e)

(c)

(f)

2

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Page 8 of 19

(g)

(h)

[EPL (σ + ) + EPL (σ − )]/2 of X (a,b,c) and XX (d,e,f ) with B are plotted to obtain the diamagnetic coecient γ for P , 5P , and 10P excitation, respectively. (g) [EPL (σ + ) − EPL (σ − )] of X (lled circles) and XX (open circles) are assumed to be gµB B in order to obtain the g -factor for P , 5P , and 10P excitation, respectively. (h) The diamagnetic Figure 3:

(a)

coecients of X and XX decreased with increasing excitation power.

8

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possibly via

ELoc ,

resulting in a modication of the ring orbit wavefunctions.

As a prerequisite condition for an optical AB eect, a built-in electric eld in a QR is known to play a crucial role in the charge separation (Re

6= Rh ). 6 Provided that the tunneling

towards the direction away from the ring centre is dominated by light electrons, the heavy hole resides near the ring centre. In this case,

Re > Rh

(Fig. 4(a)) can be assumed. As strain

due to indium is known to induce an internal electric eld in InAs and InGaAs QRs,

27,28

it

is advantageous for the charge separation. Although GaAs QRs are strain-free during the gallium droplet formation and crystallisation, a surface surfactant and low substrate temperature are used in order to suppress the migration length of gallium atoms.

Therefore,

Ga-rich growth conditions and the lack of As-interdiusion may give rise to defects at the interface between the GaAs QR and AlGaAs barrier such as Ga-antisites and As-vacancies. Consequently, trapped carriers at the defects result in a localised electric eld optically excited carriers aect

ELoc .

Suppose

ELoc , a potential energy change can be induced in a QR. This

process may give rise to an excitation intensity dependence of the PL energy, diamagnetic coecient, and AB oscillation period. When an electron-hole pair in a QR is considered (Fig. mation), the total orbital angular momentum (`X

= `e + `h )

4(a) and Supporting Inforis given by the sum of the

individual orbital angular momentum of an electron and a hole, and the X orbit radius

1/2

RX = [(me Re2 + mh Rh2 )/(me + mh )]

is dened in terms of the orbit radii (Re,h ) and the ef-

fective masses (me,h ) for an electron and a hole. According to the simplied one-dimensional (1D) model, the eigenstates can be given by

Ψ`XX (θX , φ) = ei`X θX ψ `X (φ), where two azimuthal

angles are dened in the centre-of-mass (CM) coordinate (θX tive coordinate (φ

= φe − φh )

=

2φ me Re2 φe +mh Rh h ) and the rela2 me Re2 +mh Rh

with the individual azimuthal angles of an electron and a hole

(φe,h ) respectively. The Coulomb interaction in the relative coordinate is given by

φe − φh e2 =− (Re − Rh )2 + 4Re Rh sin2 ( ) 4π 2 "

Veh

9

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#−1/2

.

(1)

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(a)

Page 10 of 19

(b)

(c)

(f)

(d)

(g)

(e)

(h)

(i)

Figure 4:

(j)

Schematic diagram of X (a) and XX (b) in a QR. When the X in a QR is

considered as a single particle, the X orbit radius for the electron and the hole (Re

6= Rh ),

RX

is determined by dierent orbit radii

and a pair of Xs in a QR are localised at opposite

positions as a consequence of the dipole-dipole interaction. The size (shaded area) of the X and XX can be estimated in terms of the X localisation angle (ξX ) and the quantum uctuation angle of a two-X Wigner molecule (ξXX ), respectively. Optical AB oscillations of X (c,d,e) and XX (f,g,h) were obtained after removing the tted quadratic functions for −2 an excitation of P = 1.2 kWcm , 5P , and 10P . Theoretical AB oscillations (solid line) were also compared. With increasing excitation intensity, both the XX binding energy in the absence of

B

(i) and

∆BX /∆BXX

(j) change.

10

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mX = me + mh ,

Provided that the e-h pair is tightly bound with the total mass

ψ `X (φ) ∝ exp{−φ2 /(2ξX2 )}

motion can be described by a Gaussian function, second order approximation for

φ.

the relative

through the

If the X localisation angle is small enough (ξX

the X energy in the relative coordinate

(EXrel )

becomes independent of

`X .

< 2π ),

Therefore, the

electron-hole pair in a QR can be considered as a single particle, and the energy of a rotating X in a QR under an external

B

becomes dominated by the energy in the CM coordinate as

h ¯2 π(Re2 − Rh2 )B = ` + X 2 2mX RX h/e "

EX`X (B)

where

Erel

contributes as a

B -independent

#2

+ EXrel ,

(2)

constant.

Likewise, a XX in a QR can be considered with a pair of localised Xs (Fig.

4(b) and

Supporting Information), where the total XX orbital angular momentum (`XX

= `X1 + `X2 )

is given by the sum of the two individual X orbital angular momenta (`X1 and

`X2 ).

In this

case, the dipole-dipole interaction between a pair of X dipoles can be described as

2 θ e2 (Re − Rh )2 1 + sin 2 Vdd (θ) = 3 4πRX 8 sin3 2θ

which depends on the azimuthal angle dierence (θ

ξXX ≈ [RX /(Re − Rh )]5/4 ξX uctuation size is small (ξXX considered

`XX -independent.

θ=π

as

θ = π.

Thus, the XX wavefunction

2 ψ(θ) ∝ exp[−(|θ| − π)2 /(2ξXX )],

< π/2),

and

rel the XX energy of the relative motion (EXX ) can be

Consequently, the total XX energy is also dominated by the

rel EXX

becomes a

B -independent

h ¯2 π(Re2 − Rh2 )B = ` + 2 XX 2 2(2mX )RX h/e "

By plotting Eq.

(3)

can be used to characterise the two-X WM. If the quantum

energy of two Xs in the CM coordinate, and

`XX EXX (B)

0

= θ1 − θ2 ) of the two Xs. Vdd (θ) becomes

minimised when the two Xs are maximally separated, i.e. of the relative motion becomes localised near

(2) as a function of

B,

constant as

#2 rel + EXX .

(4)

a series of parabolas overlap for various

`X

11

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Page 12 of 19

states, and the X orbital angular momentum changes from

`X

to

`X − 1

on the cusps of X

AB oscillations to minimise the energy. Therefore, the X AB oscillation period is given by

∆BX = (h/e)/(π|Re2 − Rh2 |).

On the other hand, a saw-like abrupt decrease of the XX energy

is seen at the rst transition magnetic eld where the selection rule for the transition between

`X

`XX EXX (B)

changes. and

EX`X (B).

ing the XX energy near the transition magnetic eld where minimise energy,

EX`X =0 (B)

period of the minimum

still remains less than

EX`X (B)

This can be explained by

EX`X =−1 (B)

`XX

changes from

`XX EXX (B).

EX`X =0 (B)

to

−1

to

`XX =−1 EXX (B)

to

Therefore, the

EX`X =−1 (B),

abrupt decrease of XX PL energy corresponds to the energy dierence between and

0

due to the long AB oscillation

compared to that of the minimum

selection rule enforces the XX emission transition from

For example, consider-

and the

EX`X =−1 (B)

at the transition magnetic eld (Supporting Information).

Although the 1D model provides an intuitive picture for the X in a QR, it needs to be rened with a quasi 1D QR model, i.e.

2Wh < Re,h ,

where the nite rim widths of electrons

(2We ) and holes (2Wh ) are considered. Provided that the radial connement potential energy for electrons and holes can be approximated

29,30

as

Ve,h (r) ∼

h2 ¯ 4 (r 2me,h We,h

− Re,h )2 ,

the

n,m n,m eigenenergy (Ee,h ) and eigenfunction (Ψe,h ) of the single-particle states can be obtained analytically for electrons and holes, which are represented by a radial quantum number (n) and an angular momentum (m).

30

In order to consider the Coulomb interaction, these states

were used to diagonalise the total Hamiltonian numerically. By using the exact diagonal-

`XX `X isation method, the eigenenergies of X (EX (B)) and XX (EXX (B)) can be obtained for various ring orbital angular momentum states (`X,XX

= 0, ±1, ±2, ±3, · · ·)

(See Supporting

Information). Experimental AB oscillations of X (Fig. 4(c,d,e)) and XX (Fig. 4 (f,g,h)) were obtained after removing the tted quadratic functions for an excitation of 3(a,d)),

∆BX

5P

and

(Fig. 3(b,e)), and

∆BXX

10P

(Fig.

(Fig. 3(c,f )), respectively. The excitation dependence of

is seen clearly. For three excitations

tion periods of X (∆BX

P = 1.2 kWcm−2

= 1.97 T, 1.85 T,

and

1.77 T)

12

P , 5P ,

and

10P ,

and XX (∆BXX

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both the AB oscilla-

= 0.92 T, 0.68 T,

and

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0.48 T)

were observed to decrease. As the AB oscillations of X and XX are sensitive to the

wavefunction change,

∆BX

to excitation intensity.

and

∆BXX

can be a measure of the wavefunction susceptibility

ELoc

Although the detailed wave modication mechanism by

not been claried, we suggest that the increased excitation possibly enhances

ELoc

has

via the

trapped carriers. Suppose the trapped charge distribution is random and non-uniform, the excitation dependence can be considered in terms of Because the observed

∆BX

plausible to assume large the decrease of

γX

and

and

Re,h

γXX

∆BXX

Re,h

and

We,h

as eective parameters.

in a QR decrease with increasing excitation, it is

regarding

∆BX = (h/e)/(π|Re2 − Rh2 |).

On the other hand,

with excitation (Fig. 3(h)) implies a decrease of

We,h .

In order

to reproduce the experimental AB oscillation periods and diamagnetic coecients of X and XX for three excitations

P , 5P , and 10P , we found the optimum parameters of Re (32.0 nm,

38.4 nm,

and

48.0,nm), Rh (20.0 nm, 24.0 nm,

3.6 nm),

respectively. Note that our model does not explain how dierent excitations give

and

30.0,nm)

and

We,h (4.0 nm, 3.8 nm,

and

rise to a modication of the wavefunction in a QR. Nevertheless, we used our calculations as a reference when experiments were compared. Given the calculation results for

EX`X (B)

and

`XX EXX (B)

with the optimum parameters

(Supporting Information), the AB oscillations (solid lines in Fig. obtained after removing the quadratic

∆BXX

B -dependence

terms.

4(c,d,e,f,g,h)) were also

The experimental

∆BX

and

of both X and XX were successfully reproduced theoretically. However, a magnetic

eld dierence (δB ) is seen as a shift in the oscillation extremum. Because the wavefunction remains localised for

B < Bc

due to the anisotropic QR structure, the AB oscillations begin

at a nite magnetic eld. It is interesting that

δB

also changes with increasing excitation.

This result is also consistent with our conjecture, i.e. excitation dependent

ELoc

modies the

potential environment, whereby a full rotational motion around the rim becomes enabled at a dierent

Bc .

Our quasi-1 model is also useful to estimate the degree of localisation in wavefunctions of both electron and hole. Given the optimum parameters, we calculated the probability of

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Page 14 of 19

nding either an electron (ρee ) or a hole (ρhe ) at the position of a xed electron (See Supporting Information). For example, with the optimum condition (Re and

We,h = 4.0 nm),

ρhe

we found the

of X is comparable to

= 32.0 nm, Rh = 20.0 nm,

ξX ∼ 25◦

in Fig. 4(a). This

result is also in agreement with the 1D model, where an electron-hole pair in a QR can be considered as a single particle. On the other hand, in the case of XX, the four-body Coulomb interaction can be glimpsed through the distribution of the other electron (ρee ) and the two holes (ρhe ) with respect to a xed electron separately (Supporting Information), and this result is also in an agreement with our 1D model, i.e. a pair of Xs in our QR are localised at opposite positions as a consequence of the dipole-dipole interaction (Fig. 4(b)). Because the dipole-dipole interaction between the two Xs is relatively weak compared to the Coulomb between the electron and the hole,

ξXX ∼ 71◦

was obtained.

b As shown in Fig. 4(i), we found that the XX binding energy (EXX ) in the absence of

B

decreases with increasing excitation. This decrease is consistent with the enlarged orbits of XX in our model. The enlarged orbit radii result in a short AB oscillation period. As the excitation increases from

P

to 10P ,

∆BX

and

∆BXX

decrease by

10% and 40%, respectively.

According to the 1D model, the two-X WM criterion applies in the limited condition of a 1D structure, where the wavefunction localisation in a widthless rim is strong enough to assume a second order approximation for long as the pair of Xs fullls

φ

and

θ.

ξXX < π/2,

Specically, two Xs can be bound as a dimer as

whereby the total energy is dominated in the CM

coordinate and the decoupled motion in the relative coordinate becomes a constant. However, when nite rim widths are considered in the quasi-1D model, the two-X WM condition looks sensitive to the wavefunction shape. Although 1D model by using Eq. (2) and Eq. (4),

∆BX /∆BXX = 2 can be obtained in the

∆BX /∆BXX

increases up to 3.72 with increasing

excitation (Fig. 4 (j)). Therefore, the decreased XX binding energy seems to deteriorate the WM nature of the two-body rigidity, and this may explain why a subtle deviation from the XX WM condition (∆BX /∆BXX

> 2)

emerges for strong excitation.

In summary, we have found that both X and XX in a single GaAs/AlGaAs QR show

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an excitation dependence in the AB oscillation periods and diamagnetic coecients with a spectral blueshift. This can possibly be attributed to a modication of the wavefunctions by optically controllable

ELoc

at interface defects. While the XX binding energy decreases

for increased excitation, we also found that a gradual deviation from the XX-WM condition (∆BX /∆BXX

= 2)

occurs.

Acknowledgement This work was supported by Korean Grant (NRF-2017R1A2B4011594, Pioneer Research 2013M3C1A3065522), JSPS KAKENHI Grant (JP15H05870, JP26220711) and Fundamental Research Funds for Central University of China.

Supporting Information Available One-dimensional and quasi-one dimensional models of excitons and biexcitons in a QR are explained in detail, where the eigenenergies and the two-body probability density are calculated.

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