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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
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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.
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 (<
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
= eΦ/¯ h).
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,
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
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
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.
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
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.
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
[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:
coecients of X and XX decreased with increasing excitation power.
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,
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.
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 , 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
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
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 ),
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
states, and the X orbital angular momentum changes from
`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
`XX EXX (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
This can be explained by
EX`X =−1 (B)
`XX EXX (B).
EX`X =0 (B)
`XX =−1 EXX (B)
EX`X =−1 (B),
abrupt decrease of XX PL energy corresponds to the energy dierence between and
due to the long AB oscillation
compared to that of the minimum
selection rule enforces the XX emission transition from
For example, consider-
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
Ve,h (r) ∼
h2 ¯ 4 (r 2me,h We,h
− Re,h )2 ,
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).
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, · · ·)
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)),
(Fig. 3(b,e)), and
(Fig. 3(c,f )), respectively. The excitation dependence of
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),
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◦
b As shown in Fig. 4(i), we found that the XX binding energy (EXX ) in the absence of
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
to 10P ,
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
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
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
emerges for strong excitation.
In summary, we have found that both X and XX in a single GaAs/AlGaAs QR show
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
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
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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