Superradiant and Aharonov-Bohm effect for the
quantum ring exciton
Y.N. Chen*, D.S. Chuu
Department of Electrophysics, National, Chiao-Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan Received 26 December 2003; accepted 18 February 2004 by K.-A. Chao
Abstract
The Aharonov-Bohm and superradiant effect on the radiative decay rate of an exciton in a quantum ring is studied. With the increasing of ring radius, the exciton decay rate is enhanced by superradiance, while the amplitude of AB oscillation is decreased. The competition between these two effects is shown explicitly and may be observable in time-resolved experiments.
q2004 Elsevier Ltd. All rights reserved.
PACS: 42.50.Fx; 32.70.Jz; 71.35. 2 y; 71.45. 2 d Keywords: A. Quantum ring; D. Exciton; D. Superradiance
With the advances of modern fabrication technologies, it has become possible to fabricate the ring-shaped dots of
InAs in GaAs[1]. In the circumstances of Aharonov-Bohm
(AB) effect, one of the important features is the periodic
dependence of interference patterns on magnetic flux F[2].
Most of the measurements, however, are available only from the transport experiments on metallic rings in the
meso-scopic regime[3]. Very recently, optical detection of the AB
effect on an exciton in a single quantum ring has become
possible [4]. This makes it more interesting to study the
optical properties of the quantum ring exciton.
On the other hand, the electron-hole pair is naturally a candidate for examining the spontaneous emission. How-ever, as it was well known, the excitons in a three-dimensional system will couple with photons to form polaritons—the eigenstate of the combined system consist-ing of the crystal and the radiation field which does not
decay radiatively[5]. Thus, in a bulk crystal, the exciton can
only decay via impurity, phonon scatterings, or boundary effects. The exciton can render radiative decay in lower dimensional systems such as quantum wells, quantum wires,
or quantum dots as a result of broken symmetry. The decay rate of the exciton is superradiant enhanced by a factor of
l=d in a one-dimensional (1D) system[6]and ðl=dÞ2for 2D
exciton-polariton[7], where l is the wave length of emitted
photon and d is the lattice constant of the 1D system or the thin film. In the past decades, the superradiance of excitons in these quantum structures have been investigated
inten-sively[8].
Although many investigations have been focused on superradiance of the quantum confined excitons, the coherent radiation together with the AB effect for an exciton in the ring geometry has received little attention so far. In this paper, we investigate the decay properties of a neutral exciton in the 1D quantum ring. It is found that there is a competition between the superradiant and AB effect for the exciton decay rate.
Consider first an exciton in a quantum ring with radius
r , Nd=2p; where d is the lattice spacing and N is the
number of the lattice points. In our model, the circular ring is joined by the N lattice points, and we also assume the effective mass approximation is valid in the circumference direction. The validity of these assumptions will be discussed later. Therefore, the state of the exciton can be
specified as ln; n; ml, where n is the exciton wave number. n
and m are quantum numbers for internal structure of the
0038-1098/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2004.02.051
Solid State Communications 130 (2004) 491–494
www.elsevier.com/locate/ssc
* Corresponding author. Tel.: þ 886-357-121-2156177; fax: þ 886-357-252-30.
exciton, and will be specified later. Here, n takes the value of an integer. The matter Hamiltonian can be written as
Hex¼ X nnm Ennmc † nnmcnnm; ð1Þ
where c†nnm and cnnm are the creation and destruction
operators of the exciton, respectively. The Hamiltonian of free photon is Hph¼ X q0k0 zl "cðq02 þ k0z2Þ1=2b†q0k0 zlbq 0k0 zl; ð2Þ where b†q0k0 zl and bq 0k0
zl are the creation and destruction
operators of the photon, respectively. The wave vector k0of
the photon were separated into two parts: k0z is the
perpendicular component of k0on the ring plane such that
k02 ¼ q02 þ k0z2:
The interaction between the exciton and the photon can be expressed as H0¼X i X q0k0 zl e mc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p"c ðq02 þ k0 z2Þ1=2v s ½b†q0k0 zlH ð1Þ n ðq0rÞexp ðin0wiÞ þ hcðeq0k0 zl·piÞ; ð3Þ
where Hð1Þn is the Hankel function, ðr; wiÞ is the position of
the electron i in the ring, piis the corresponding momentum
of the electron i operator, and eq0k0
zlis the polarization vector
of the photon. The using of Hankel function in Eq. (3) means the wave which generated by the recombination of the
exciton moves outward to infinity[9]. For large radius, the
Hankel function behaves like eiq0r:
The exciton state in a quantum ring can be expressed as
ln; n; ml ¼ X
we;wh
Un;n;mp ðwe; whÞlc; weþ wh; v; whl; ð4Þ
and the interaction matrix elements can be written as
kn; n; mlH0l Gl ¼ X we;wh kc; we þ wh; v; whlU p n;n;mðwe; whÞH0lGl; ð5Þ
in which the excited state lc; weþ wh; v; whl is defined as
lc; weþ wh; v; whl ¼ a†c;w
eþwhav;whlGl; ð6Þ
where a†c;w
eþwh (av;wh) is the creation (destruction) operator
for an electron (hole) in the conduction (valence) band at
site weþ whðwhÞ: The expansion coefficient U
p
n;n;mðwe; whÞ is
the exciton wave function in the quantum ring:
Upn;n;mðwe; whÞ ¼ p1NffiffiexpðinrcÞFnmðweÞ; ð7Þ
where the coefficient 1=pffiffiffiN is for the normalization of the
state ln; n; ml; and rc¼ mpeðweþ whÞ þ m p hwh mp eþ m p h
is the center of mass of the exciton. Here, mpe and m
p hare,
respectively, the effective masses of the electron and the
hole. FnmðweÞ is the hydrogenic wavefunction in the ring and
will be calculated later.
After summing over wh; we have
kn; n; mlH0 lGl ¼ X q0k0 zl e mc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p"c ðq02 þ k0 z2Þ1=2v s ½bq0k0zlðeq0k0zl· AnnmÞHnð1Þþ hc; ð8Þ where Annm¼pffiffiffiNX we FnmðweÞ ð dwwcðw 2 weÞ exp in w 2 m p ewe mpeþ m p h !! 2i" › ›w wvðwÞ: ð9Þ
Here, wcðwÞ and wvðwÞ are, respectively, the Wannier
functions for the conduction band and the valence band. The essential quantity involved is the matrix element of
H0 between the ground state lGl and the exciton state
ln; n; ml: Hence the interaction between the exciton and the
photon (in the resonance approximation) can be written in the form H0¼X k0 znm X q0l Dq0k0 znnmbk0zq0lc † nnmþ hc; ð10Þ where Dq0k0 znnm¼ H ð1Þ n ðq0rÞ e mc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p"c ðq02 þ k0 z2Þ1=2v s eq0k0 zl·Annm: ð11Þ
By the method of Heitler and Ma in the resonance approximation, the decay rate of the exciton can be expressed as gnnm¼ 2p X q0k0 zl lDq0k0 znnml 2 dðvq0k0 znnmÞ; ð12Þ where vq0k0 znnm¼ Ennm=" 2 c ffiffiffiffiffiffiffiffiffiffiffiffi q02 þ k0z2 p :
The exciton decay rate in the optical region can be calculated straightforwardly and is given by
gnnm¼ e 2 " m2c r d ð lHnð1Þðq0rÞl2q0ð dðvq 0k0 znnmÞ ffiffiffiffiffiffiffiffiffiffiffiffi k0z2 þ q02 p leq0k0 zl· xnnml2dk0zdq0; ð13Þ where xnnm¼ X we Fnmp ðweÞ ð dwwpcðw 2 weÞ 2i" › ›w wvðwÞ: ð14Þ
Here, xpnnmrepresents the effective dipole matrix element for
an electron jumping from the excited Wannier state in the conduction band back to the hole state in the valence band.
As one can see from Eq. (13), the decay rate gnnm is
proportional to r=d: This is just the superradiant factor implying the coherent contributions. Furthermore, the
Y.N. Chen, D.S. Chuu / Solid State Communications 130 (2004) 491–494 492
asymptotic limit of gnnm(r ! 1) recovers the exciton decay
rate in 1D quantum wire: R1d¼
3p
2k0d
g0; where k0¼ 2p=l
and g0is the decay rate of an isolated two level atom.
Now let us consider the AB effect for a superradiant exciton. For the 1D quantum ring, the exciton wavefunction
can be solved by Ro¨mer and Raikh’s approach[10]. The
wavefunction in the ground state ðn ¼ m ¼ 0Þ can be expressed as, F00ð0Þ ¼ V02 X1 N0¼1 1 ðEðeÞN0 þ E2NðhÞ02 D00Þ2 # " !21 ; ð15Þ where EðeÞN0 ¼ "2 2mer2 N02 F F0 and EðhÞ2N0¼ "2 2mer2 N0þ F F0
with the universal flux F0¼ hc=e: The constant V0, 0 is
defined as V0¼ 1 2p ð dwV½RðwÞ: ð16Þ
And the exciton energy D00takes the form
D0 0 10 !1=2 ¼ 2 pV0 10 sin 2p D0 010 1=2 cos 2p D 0 0 10 !1=2! 2 cos 2p F F0 ; ð17Þ where 10¼ "2 2r2 1 me þ 1 mh :
In the limit of large radius, the corresponding ground state energy is D0 0¼ 2 p2 V02 10 1 þ 4cos 2pF F0 exp 22p 2l V0l 10 ! " # : ð18Þ One should note V½RðwÞ is not specified since it describes the interaction between the electron and hole in
a realistic quantum ring. However, V0can still be extracted
from Eq. (18) in large radius limit, i.e. applying real
experimental data of a quantum wire exciton energy D0
0:
Besides, the exponential factor in Eq. (18) can be
represented by expð22pr=lÞ[9], where l is the decay length
of the wave function of the internal motion of electron and hole. Thus, the magnitude of the AB effect in the limit of large radius represents the amplitude for bound electron and hole to tunnel in the opposite directions and meet each other on the opposite side of the ring.
The dipole matrix element xnnmin Eq. (14) corresponds
to an average of dipole transitions between different sites,
weighted by the exciton wave function Fnmp ðweÞ: The sum in
Eq. (14) contains a we! 0 term in which the electron and
the hole are at the same site. If the corresponding integral does not vanish, this term dominates the sum. The effective dipole transition matrix element becomes
xnnm, Fpnmð0Þ
ð
dwwpcðwÞð2i"7ÞwvðwÞ ¼ F
p
nmð0Þxs; ð19Þ
where xsis essentially the dipole matrix element between
the atomic states at the same site. Combining Eqs. (13), (15), (17), and (19), one can obtain the AB effect for a
superradiant exciton. In Fig. 1 three curves of different
flux F are presented as a function of radius r: To plot the figure, we have assumed the wavelength of the emitted
photon l ¼ 8000 A˚ and lattice spacing d ¼ 5 A˚. The
dashed, solid, and dotted curves represent the cases of F ¼
0F0; 0:25F0; and 0:5F0; respectively. As can be seen, AB
effect becomes important in small radius limit. For F ¼
0:5F0; the decay rate decreases with the decreasing of ring
radius but reaches the minimum point as r is about 0:25a0
(where a0¼ 100 A˚ is the effective Bohr radius we assumed
in 1D limit). This is because the probability, for electron and hole to meet each other on the opposite side of the ring, increases with the decreasing of ring radius, while the coherent effect (superradiance) decreases with the decreas-ing of the radius. As a result, there is a competition between these two effects. One also notes the AB oscillation is not of
constant amplitude. In Fig. 2, relative decay rates
½gnnmðFÞ 2 gnnmðF ¼ 0Þ as a function of magnetic flux F
are plotted. The solid and dashed lines represent the cases of
r ¼ 1a0and r ¼ 0:5a0; respectively. The larger the radius,
the smaller the AB oscillation amplitude. As expected, the
superradiant decay rate is most enhanced for F ¼ 0:5F0;
and the oscillation period is equal to F0¼ hc=e:
Although present model considers the ideal 1D quantum ring, the physics discussed above can be applied to the
Fig. 1. Effect of Aharonov-Bohm on the radiative decay of a quantum ring exciton. The dashed ( – – ), solid, and dotted (X) curves correspond toF¼ 0F0; 0:25F0; and 0:5F0; respectively. In
small radius limit, Fp
nm depends strongly on radius r; and its
influence on the decay rate is evident. The vertical and horizontal units here are ð3p=2k0dÞg0 and ring radius (in units of a0),
respectively.
realistic quantum ring with finite width. The modified
quantity is the exciton wavefunction Fnm; which only
changes the amplitude of AB oscillation. In addition, the coherent radiation from the lattice points within a wavelength still holds as long as the angular momentum is preserved, i.e. not broken by impurities or phonons. This means a high quality quantum is required to observe the mentioned effects.
In summary, we have calculated the superradiant decay rate of an exciton in a quantum ring. Flux dependent oscillation of the superradiant exciton is shown explicitly. With the decreasing of ring radius, there is a competition between the superradiant and AB effects. The distinguishing features are pointed out and may be observed in a suitably designed experiment.
Acknowledgements
This work is supported partially by the National Science
Council, Taiwan under the grant number NSC 91-2120-M-009-002.
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Y.N. Chen, D.S. Chuu / Solid State Communications 130 (2004) 491–494 494
