Decay rate and renormalized frequency shift of superradiant exciton in a cylindrical quantum wire

Decay rate and renormalized frequency shift of superradiant

exciton in a cylindrical quantum wire

Yueh-Nan Chen*, Der-San Chuu

Department of Electrophysics, National Chiao-Tung University, Hsinchu 30050, Taiwan Received 12 November 2002; received in revised form 13 December 2002

Abstract

The decay rate and renormalized frequency shift of superradiant exciton in a cylindrical quantum wire are studied. The transition behavior from 1D wire to 2D film is examined through the property of the radiative decay. Similar to the case in a quantum well, the decay rate of the higher mode exciton is larger than that of the lower mode one. Moreover, it is also found the decay rate and frequency shift do not show oscillatory dependence on wire radius because of the conservation of angular momentum.

r2003 Elsevier Science B.V. All rights reserved. PACS: 42.50.Fx; 32.70.Jz; 71.35.-y; 71.45.-d

Keywords: Quantum wires; Superradiance; Excitons; Frequency shift

Since Dicke [1] pointed out the concept of

superradiance, the coherent effect for spontaneous radiation of various systems has attracted exten-sive interest both theoretically and experimentally

[2–5]. In bulk crystal, the excitons will couple with

photons to form polaritons [6]-mixed modes in

which energy oscillates back and forth between the exciton and the radiation field. What makes the excitons trapped in the bulk crystal is the conservation of crystal momentum. If one

considers a thin film[7], the excitons can undergo

radiative decay as a result of the broken crystal symmetry along the normal direction of the film

plane. The decay rate of excitons in a thin film is

enhanced by a factor of ðl=dÞ2compared to a lone

exciton in an empty lattice, where l is the wave length of emitted photon and d is the lattice constant of the film.

Lots of investigations on the radiative linewidth of excitons in quantum wells have been performed. An abnormal increase of excitonic radiative life-time with the decrease of well width below 5 nm

for InxGa1xAs=InP quantum well was observed

by Cebulla et al.[8]. Brandt et al.[9]measured the

radiative lifetime of excitons in InAs quantum sheets and observed the increasing of radiative lifetime with the decreasing of well thickness.

Hanamura [10] investigated theoretically the

ra-diative decay rate of quantum dot and quantum well. The obtained results are in agreement with

that of Lee and Liu’s[7]prediction for thin films.

*Corresponding author. Tel.: +8863571212156177; fax: +88635725230.

E-mail addresses:[email protected] (Y.-N. Chen), [email protected] (D.-S. Chuu).

0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(03)00043-7

Knoester [11] studied the radiative dynamics crossover from the small thickness, superradiant exciton regime to bulk crystal, polariton regime. The oscillating dependence of the radiative width of the exciton-like polaritons with the lowest energy on the crystal thickness was found.

Recently, Bj.ork et al. [12] examined the

relation-ship between atomic and excitonic superradiance in thin and thick slab geometries. They demon-strated that superradiance can be treated by a unified formalism for atoms, Frenkel excitons, and

Wannier excitons. In Agranovich et al.’s work[13],

a detailed microscopic study of Frenkel exciton-polariton in crystal slabs of arbitrary thickness was performed.

For lower dimensional systems, the decay rate of the exciton is enhanced by a factor of l=d in a

linear chain[14]. First observation of superradiant

short lifetimes of excitons was performed by Ya.

Aaviksoo et al. [15] on surface states of the

anthracene crystal. Ivanov and Haug [16]

pre-dicted the existence of exciton crystal, which favors coherent emission in the form of

super-radiance, in quantum wires. Manabe et al. [17]

considered the superradiance of interacting Fren-kel excitons in a linear chain. Recently, we have also shown the superradiant decay of the quantum wire exciton is greatly enhanced in a planar

microcavity [18]. In this paper, the exciton is

assumed to be confined in a hollow cylindrical quantum wire. The crossover of the superradiant decay rate of the exciton from a wire with small radius to the 2D limit is explicitly obtained. Moreover, the crossover of the coherent frequency shift from narrow wire to thin film is also investigated by using the method of renormaliza-tion[19,20].

We consider a free exciton in a hollow cylindrical quantum wire lying on the z-axis with

simple cubic structure (Fig. 1). For the physical

phenomenon we are interested in, we shall concentrate on the investigations of semiconductor quantum wires rather than existing carbon wires

[21,22]since the wavelength of the emitted photon is much larger than the diameter of the single-wall carbon tubes. The crossover behavior from 1D wire to 2D film may not be examined on carbon systems. However, we hope our model can serve as

a first step toward the understanding of the exciton decay in carbon nanotubes.

If the difference between the inner radius (r_o)

and outer radius ðr_>Þ is much smaller than the

effective Bohr radius of the exciton, i.e. DL5aex;

one may approximate that the exciton is confined

on the cylindrical surface with radius

rðEr_oEr_>Þ: This means the exciton is trapped

in an infinite deep and narrow quantum wire. The radius r is about Nd=2p; where N is the number of lattice points in the circumference direction and d is the lattice spacing. If one further assumes the radius of the wire is much larger than the effective Bohr radius of the exciton, variations of the wire radius only cause few changes on the Wannier exciton wavefunction. In this case, the main contributions to the superradiant decay rate and frequency shift still come from the bandgap energy and the number of the lattice points within a

wavelength of the emitted photon[20]. Therefore,

one can first consider the exciton as a particle with angular momentum n and longitudinal momentum

kz: After figuring out the decay rate and frequency

shift of a Frenkel exciton, the corresponding ones of a Wannier exciton can be obtained by replacing the single-atom dipole matrix element v with the

effective dipole matrix element [7]. Thus, the

Fig. 1. Schematic view of the quantum wire structure and its defining potential profile.

Hamiltonian for the exciton is Hex¼ X nkz Enkzc w nkzcnkz; ð1Þ

where cw_nkzand cnkzare the creation and destruction

operators of the exciton, respectively. The Hamil-tonian of free photon is

Hph¼ X q0_n0_k0 z _cðq02_{þ k}02 zÞ1=2b w q0_n0_k0 zbq 0_n0_k0 z; ð2Þ where bw_q0_n0_k0 z and bq 0_n0_k0

z are, respectively, the

creation and destruction operators of the photon.

The wave vector k0of the photon is separated into

two parts: k0

zis the parallel component of k

0_{on the}

z direction such that k02_{¼ q}02_{þ k}02

z:

The interaction between the exciton and the photon can be expressed as

H0¼X i X q0_n0_k0 z e mc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p_c ðq02_{þ k}02 zÞ 1=2_v s ðeq0_n0_k0 z  piÞ  ½bw_q0_n0_k0 zH ð1Þ n0 ðq0rÞ exp ðin0j_iþ ik_z0liÞ þ h:c:; ð3Þ

where m is the electron mass, (r_i; j_i; li) is the

position of the electron i in the cylindrical wire, p_i

is the corresponding momentum operator of the electron i; eq0_n0_k0

z is the polarization vector of the

photon, and H_nð1Þ0 ðq0rÞ is the Hankel function.

The essential quantity involved is the matrix

element of H0 between the ground state jGS and

the exciton state jn; kzS: We know that the

interaction matrix elements of H0can be written as

/n; kzjH0jGS ¼X l;j /c; ðl; jÞ; v; ðl; jÞjUn nkzðl; jÞH 0_{jGS; ð4Þ}

because the exciton state can be expressed as

jn; kzS ¼

X

l;j

Un

nkzðl; jÞjc; ðl; jÞ; v; ðl; jÞS; ð5Þ

in which the excited state jc; ðl; jÞ; v; ðl; jÞS is defined as

jc; ðl; jÞ; v; ðl; jÞS ¼ aw_c;ðl;jÞav;ðl;jÞjGS; ð6Þ

where aw_c;ðl;jÞða_v;ðl;jÞÞ is the creation (destruction)

operator of an electron in the conduction (valence) band at lattice site ðl; jÞ: The expansion coefficient

Un

nkzðl; jÞ is the exciton wave function in the

cylindrical tubule: Un nkzðl; jÞ ¼ 1 ffiffiffiffiffi N p 1ffiffiffiffiffiffi N0 p einjþikzl_{; ð7Þ}

where the coefficient 1=pffiffiffiffiffiffiN0 _{is for the}

normal-ization of the state jn; kzS:

After summing over l and j in Eq. (4), we have

/n; kzjH0jGS ¼X q0_gnn e mc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p_c ðq02_{þ ðk} zþ gÞ2Þ1=2v s  ½bq0_;mþn n;kzþgðeq0;nþnn;kzþg Anþnn;kzþgÞ  H_nþnð1Þ_nðq0rÞ þ h:c:; ð8Þ where Anþnn;kzþg ¼pffiffiffiffiffiffiffiffiffiNN0 Z

d2s exp ðiðkzþ gÞtzþ iðn þ nnÞtjÞ

 wcðsz; tjÞði_=Þwvðsz; tjÞ; ð9Þ

wcðsz; tjÞ and wvðsz; tjÞ are, respectively, the

Wannier functions for the conduction band and

the valence band at site 0, and g ðnnÞ is the

reciprocal lattice in kz ðkjÞ direction. Hence the

interaction between the exciton and the photon (in the resonance approximation) can be written in the form H0¼X nng X q0_nk z Dq0_;nþn n;kzþgbq0;nþnn;kzþgc w nkzþ h:c:; ð10Þ where Dq0_;nþn n;kzþg ¼ H_nþnð1Þ_nðq0rÞeq0_;nþn n;kzþg Anþnn;kzþg  e mc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p_c ðq02_{þ ðk} zþ gÞ2Þ1=2v s : ð11Þ

Now, we assume that at time t ¼ 0 the exciton is

in the mode n; kz: For time t > 0; the state jcðtÞS

can be written as jcðtÞS ¼ f0ðtÞjn; kz; 0S þ X q0_n ng fG;q0_;nþn n;kzþgðtÞjG; q 0_{; n þ n} n; kzþ gS; ð12Þ

where jn; kz; 0S is the state with an exciton in the

mode n; kz in the cylindrical quantum wire,

jG; q0_{; n þ n}

n; kzþ gS represents the state in which

the electron-hole pair recombines and a photon in

the mode q0; n þ nn; kzþ g is created, and f0ðtÞ and

fG;q0_;nþn

n;kzþgðtÞ are, respectively, the probability

amplitudes of the state jn; kz; 0S and jG; q0; n þ

nn; kzþ gS:

In the resonance approximation, the probability

amplitude f0ðtÞ can be expressed as [7]

f0ðtÞ ¼ exp ðiOnkzt  1 2gnkztÞ; ð13Þ where g_nkz ¼ 2pX q0_n ng jDq0_;nþn n; kzþgj 2_dðo q0_;nþn n;kzþgÞ ð14Þ and Onkz ¼ P X q0_n ng jDq0_;nþn n;kzþgj 2 oq0_;nþn n;kzþg ð15Þ with oq0_;nþn n;kzþg ¼ Enþnn;kzþg=_  c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q02_{þ ðk} zþ gÞ2 q

: Here g_nkz and Onkz are,

respec-tively, the decay rate and frequency shift of the exciton. And P means the principal value of the integral.

If we neglect the Umklapp process, the exciton decay rate in the optical region can be calculated straightforwardly and is given by

gnkz ¼ 3 2p2g0 r dk10djH ð1Þ n ðr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 n k2z p Þj2_{; k} zokn; 0; otherwise; ( ð16Þ where kn¼ Ekzn=_ ¼ k0þ_ 2 n2=2mr2; g₀¼4e 2_{_k} 0 3m2_c2jvj 2_{; ð17Þ} and v ¼ Z d2swcðsz; tjÞði_=Þwvðsz; tjÞ: ð18Þ

Here, m is the effective mass of the exciton, vn

represents the single-atom dipole matrix element for an electron jumping from the excited state in the conduction band back to the hole state in the

valence band, and g₀ is the decay rate of an

isolated atom. We see from Eq. (16) that g_kzn is

proportional to 1=ðk0dÞ and r=d: This is just the

superradiance factor coming from the coherent

contributions of atoms. If kzis larger than kn; these

excitonic modes (trapped modes) are not capable of radiative decay. This is simply because the

energy _ckn of that exciton is not sufficient to

produce a photon.

InFig. 2we plot the decay rate g_nkz as a function of wire radius. Our numerical results are obtained by employing the data of GaAs (band gap is 1:52 eV). For n ¼ 0 mode(solid line), the decay rate increases with the increasing of wire radius and approaches 2D limit as r is large comparing to the wavelength of the emitted photon in a 2D thin film. For higher modes (n ¼ 1; 2), the results

depend on the kinetic energy _2n2=2mr2: As can

be seen from the figure, the decay rate also approaches 2D limit for large radius. On the other hand, the decay rate increases as the wire radius is decreased until the maximum decay rate is attained. After that a further reducing of the radius leads to a sharp decrease of the decay rate. This is similar to the transition from 2D to 3D: the higher wave number modes have larger maximum

decay rate [12]. One also notes in small radius

regime, the decay rate of the higher mode is

smaller than that of the lower mode, i.e. g_n¼2;kz >

g_n¼1;kz > g_n¼0;kz: This phenomenon is also found in the quantum well system, and can be ascribed to

Fig. 2. Decay rate of the superradiant exciton as a function of the wire radius. The vertical and horizontal units are ps1_and

nm, respectively. And the solid, dashed, and dot-dashed lines represent the n ¼ 0; 1; and 2 modes, respectively. In this and following graph the parameters of a GaAs quantum well are chosen to obtain the numerical results.

the effects of interference[20]. Another interesting result is that the decay rate does not show any oscillatory dependence on radius while it shows oscillatory dependence on layer thickness in the

case of the semiconductor thin film [11,20]. The

reason is that the angular momentum of the exciton is conserved in a cylindrical system, but, in a semiconductor thin film, the momentum is not conserved in the direction of broken symmetry. If we sum over all the sites in Eq. (4), the phase will run through a circle and thus preserve the same value.

One might suspect the practical value of our numerical results because our model is based on the one-layer assumption and the idealized as-sumptions, e.g. perfect cylindrical symmetry and infinite confinement. However, as we mentioned above, for the superradiant decay rate and frequency shift one only needs to replace the dipole matrix element with an effective one. This is why our result agrees well with that of a realistic

GaAs quantum well in the large radius limit [10].

On the other hand, such an assumption may slightly deviate from the real case as the radius is

small (e.g. as the radius approaches aex). This is

because the wavefunction of the exciton becomes more compact in the quantum wire with small radius limit, and it causes the increasing of the dipole matrix element v in Eq. (17). Besides, the quality of the quantum wire also influences the observation of the crossover, i.e. the thickness fluctuations should be controlled well, otherwise it will destroy the coherence in the circular direction. Now let us turn to the results for the renorma-lized frequency shift. The frequency shift in Eq. (15) can be expressed as

Onkz ¼ pe2_ m2_c2_vP X q0_n ng jeq0_;nþn n;kzþg Anþnn;kzþgj 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q02_{þ ðk} zþ gÞ2 q  jH ð1Þ nþnnðq 0_rÞj2 Enþnn;kzþg=ð_cÞ  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q02_{þ ðk} zþ gÞ2 q : ð19Þ

As seen from above, the frequency shift suffers

from ultraviolet divergence when g and nn are

large, and has infrared divergence when the denominator approaches zero. Following the procedure as shown in the work of Lee, Chuu,

and Mei [19], the divergent problem is solved by

renormalization.

In the usual renormalization procedure used in quantum field theory, the infinite quantities in Green function is subtracted by some infinite quantities to make them finite. The subtraction procedure is possible by substituting some finite physical quantities, such as renormalized masses, charges, and wave functions. The finite physical quantities observed from bare infinite quantities can be visualized, in the case of bare charge for example, as the polarization of vacuum. This is equivalent to say that there is virtual photon and opposite charged electron cloud surrounding the bare charge making the charge measured from outside become finite—a phenomena similar to that of shielding in dielectric material. But the renormalization procedure in quantum field theory is for free electrons. In the case of the condensed matter, the electrons are confined by periodic potential and complex interactions. Borrowing from the concepts of renormalization used in quantum field theory, we have

Oren_nkz ¼ Onkz lim

k0-0;d-N

Onkz; ð20Þ

where the two limiting processes k0-0 and d-N

reduce the exciton to a free electron. In the limiting

process d-N; the ordinary exciton becomes a

lone exciton standing alone in an empty lattice with no interaction with other atoms. And the

process k0-0 means that there is no energy

difference between electron and hole.

We will now show that the ultraviolet diver-gence comes from the inclusion of Umklapp process. Define Onkzðl; jÞ ¼ X k2nj Jnkzðk2; njÞ exp ðik2l þ injjÞ ð21Þ with Jnkzðk2; njÞ ¼ pe2_ m2_c2_vP X q0 jeq0_n jk2 vj 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q02_{þ k}2 2 q  jH ð1Þ njðq 0_rÞj2 Enjk2=ð_cÞ  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q02_{þ k}2 2 q : ð22Þ

From the above equations, we have X

l;j

Onkzðl; jÞ exp ðikzl  injÞ

¼X k2nj Jnkzðk2; njÞ X lj eiðk2kzÞlþiðnjnÞj ¼ N0_N X gnn Jnkzðkzþ g; n þ nnÞ ¼ Onkz: ð23Þ

And from Eq. (21) we have lim_d-0Onkz ¼

Onkzðl ¼ 0; j ¼ 0Þ; so we can rewrite the

renorma-lized frequency shift Oren_nkz as

Oren_nkz ¼ Oren nkzð0; 0Þ þ O coh nkz ð24Þ with Oren_nkzð0; 0Þ ¼ Onkzð0; 0Þ  lim k0-0 Onkzð0; 0Þ ð25Þ and Ocoh_nkz ¼X la0 ja0 eiðkzlþnjÞ_O nkzðl; jÞ: ð26Þ

As can be seen from Eq. (25), the

renormaliza-tion affects only the part Onkzð0; 0Þ—the frequency

shift of the lone exciton in an empty lattice—and thus nothing to do with the correlation within the

crystals. The coherent part Ocoh_nkz in Eqs. (24) and

(26) is not touched by renormalization procedure.

The separation of Onkz into two parts as shown in

Eq. (24) is conceptually equivalent to singling out of the source term of quantum electrodynamical divergence in a correlated system.

Now we will investigate the origin of ultraviolet divergence. From Eq. (19) with the substitution P q0 -R q0_dq0_=ð2p=R2_{Þ and} P g -R dg=ð2p=LzÞ; we have Onkz C e2_N0 4pm2_c2P X nn Z q0dq0 Z dgjeq0;nþnn;kzþg vj 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q02_{þ ðk} zþ gÞ2 q  jH ð1Þ nþnnðq 0_rÞj2 Enþnn;kzþg=ð_cÞ  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q02_{þ ðk} zþ gÞ2 q : ð27Þ

It is noted that the result in Eq. (27) is the same as

Onkzð0; 0Þ from Eqs. (21) and (23). Hence we

conclude that the ultraviolet divergence of

Onkzð0; 0Þ is really the same as that from Umklapp

process of large g and nnthat contribute to the full

Onkz: Once Onkzð0; 0Þ of the lone exciton is rendered

finite by renormalization via Eq. (25), the Umk-lapp processes of large g and n that arise from the unrestricted sum over l and j will also be rendered

finite simultaneously. Accordingly, Oren_nkz can be

written as Oren_nkz ¼pe 2_{_N}0 m2_c2_vP X q0 jeq0_nk z vj 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q02_{þ k}2 z p  jH ð1Þ n ðq0rÞj 2 Enkz=ð_cÞ  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q02_{þ k}2 z p : ð28Þ

and thus the ultraviolet divergence problem is solved.

In the kzB0 and n ¼ 0 mode, the renormalized

result can be reduced as Oren_nkz ¼pe 2_{_N}0 m2_c2_vP X q0 jeq0_nk z vj 2 q0 jH₀ð1Þðq0_rÞj2 k0 q0 : ð29Þ

As seen from above, the frequency shift suffers

from infrared divergence when q0_{B0 or q}0_Bk

0:

This can be overcome by substituting i_r by

imcq0_{t [19] in Eq. (18) when q}0 _{is small. It is}

equivalent to the dipole interaction form, H0_{Br }

E: With this treatment, we have Oren_nkzB2pr d N 0 PX q0 Bq0_nk z jH₀ð1Þðq0rÞj2 k0 q0 ð30Þ with Bq0_nk z ¼ pe2_{_} m2_c2_vjeq0_nk z  vj 2_{; when q}0 _{is large;} pe2_q02 v jeq0nkz dj 2 ; when q0 is small; ( ð31Þ where d ¼ Z d2swcðsz; tjÞtwvðsz; tjÞ: ð32Þ

Eq. (30) can not be evaluated analytically. But for large radius, the asymptotic form of Hankel function is: H₀ð1Þðq0_rÞBpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_2=ðpq0_rÞeiq0_r

: And the renormalized frequency shift reduces to 2D limit

[19]: O2D¼ gsingle 1 k0d 2 ; ð33Þ

where g_single¼2e 2 _c En;kzB0 _ jk0dj2 ð34Þ

is the radiative decay rate of a single isolated exciton. Analogous to the decay rate, the renor-malized frequency shift is explicitly seen to be

coherently enhanced by the same factor ð1=k0dÞ2

as a result of the interaction of the phase-matched photon amplitude with the delocalized excitonic

amplitude in the plane. In Fig. 3, we numerically

calculated the frequency shift as a function of wire radius. One can see from the figure the renorma-lized frequency shift does not show oscillatory dependence on radius, either. For n ¼ 0 mode, unlike the behavior of decay rate, the magnitude of the frequency shift first increases with the decreas-ing of radius. After reachdecreas-ing a minimum point, the frequency shift approaches to zero rapidly.

For usual semiconductors, the superradiant

enhanced factor is about 106 _{for excitons in the}

optical range. However, due to the extreme

smallness of gsingle itself, observation of On;kz is

not expected to be easy. Its dependence on wire radius may be a useful feature to observe this

quantity. Recently, R.omer and Raikh studied

theoretically the exciton absorption shredded by

a magnetic flux F in a quantum ring [23]. They

found the oscillator strength of the exciton is most enhanced when F is equal to half of the universal

flux quantum F0¼ hc=e: And the oscillation

period is equal to F0: Owing to the similarity in

cylindrical geometry, the decay rate and the

frequency shift may be observed experimentally if one varies the magnetic flux or the wire radius.

In summary, we have calculated the decay rate of the superradiant exciton in a hollow cylindrical quantum wire. Similar to the case in a quantum well, the higher wave number modes are shown to have larger maximum decay rate. It is also found the decay rate does not show any oscillatory dependence on wire radius because of the con-servation of the angular momentum in the cylindrical system. On the other hand, the frequency shift of the exciton is properly renorma-lized by removing the ultraviolet and infrared divergence. It is found the shift does not shown oscillatory dependence on the radius, either. Some distinguishing features are pointed out and may be observed in a suitably designed experiment.

Acknowledgements

We would like to thank Professor M. F. Lin of National Cheng Kung University for a helpful discussion. This work is supported partially by the National Science Council, Taiwan under the grant number NSC 91-2112-M-009-012.

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