Optical response in multi-quantum wells under Bragg conditions

phys. stat. sol. (c) 1, No. 6, 1410–1419 (2004) / DOI 10.1002/pssc.200304078

© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Optical response in multi-quantum wells under Bragg conditions

L. Pilozzi*, 1, A. D’Andrea1, and K. Cho2

1 Istituto di Metodologie Inorganiche e dei Plasmi, CNR, I-00016 Monterotondo Staz. (Roma), Italy 2 Graduate School of Engineering Science, Osaka University Toyonaka, Osaka 560-8531, Japan

Received 2 October 2003, accepted 15 October 2003 Published online 19 February 2004

PACS 71.36.+c, 78.67.De

A microscopic description of matter polarization and electromagnetic field is used to study the linear opti-cal response of N quantum wells under Bragg condition at the resonant wavelength. In this system the ra-diative interaction among induced polarizations plays an essential role and the inclusion of this effect leads to a correct description of the size and internal structure dependence of the optical response. The spectral evolution of reflectivity and absorbance with increasing N from the super-radiance regime to the 1D photonic crystal limit (N → ∞) is shown. Polaritonic dispersion curves are calculated for the infinite system and compared with the photonic dis-persion curves due to the background dielectric function modulation. A decomposition of the polaritonic matrix of interaction is proposed in order to separate different contri-butions in the optical response and to achieve a comparison with the response of a cluster of non-interacting units.


1 Introduction

In the study of the optical properties of mesoscopic systems, matter coherence requires the use of a mi-croscopic self-consistent theory [1]. At variance of the macroscopic case, it allows to include the micro-scopic spatial structure of the electromagnetic field and of the induced polarization, giving a non-local relation between them. In this work the microscopic nonlocal response theory and its capability to describe the effects of the sample size is shown by studying resonant systems where the self-consistency between matter polariza-tion and electromagnetic field leads to light mediated interactions among the resonant units. This interac-tion induces a correction to the polaritonic self-energy and leads to a strong modification of the optical response of the system where super-radiant (SR) decay phenomena and 1D resonant photonic crystal behaviour can be observed by changing the number N of resonant units. The theory of this structure for the case of finite N value was developed in terms of SR mode in a number of papers [2, 3]. In these works the main result is that there exist just one super-radiant mode which has a lifetime N times that of a single quantum well Γo. The reflection coefficient for such a struc-ture has been given in the form:

( )

( ) ( )

2

2 2

o

o o

NR

N

Γ

ω ω γ Γ=

− + +

, (1)

where γ is the nonradiative decay costant.

* Corresponding author: e-mail: [email protected], Phone: +39 06 90672 223, Fax: +39 06 90672 238

phys. stat. sol. (c) 1, No. 6 (2004) / www.pss-c.com 1411


Theoretical [4] and experimental [5, 6] studies have confirmed the enhanced decay rate of superradi-ance in time and frequency domain. Moreover, in the N → ∞ limit, where equation (1) is no more cor-rect, the dispersion equation, obtained by using the transfer matrix approach, is given in the following form [7, 8]:

sin ( )

cos ( ) cos ( ) zz o

o

q dKd q d

iΓ

ω ω γ= −

− −

. (2)

Now two considerations must be made: i) the interpretation in terms of the SR mode becomes invalid even before the N → ∞ limit; ii) equation (2) has generally three solutions except for the exact Bragg condition /o c dω π= and for vanishing nonradiative decay constant γ. This result seems to be not physi-cally correct because the nonradiative damping can affect the form but not the number of the dipersion curves. Therefore it is necessary to establish a consistent picture connecting the two regimes for small and large N and to derive the dispersion equation for the N → ∞ limit. Moreover our method, based on the orthonormality property of the exciton envelope function in the “site picture”, is well suited also for 2D and 3D cases where the transfer matrix method is no more avail-able and it is completely equivalent to more rigorous approaches [9], based on the Bloch waves expan-sion. We consider a 1D array of N quantum wells with inter-well distance satisfying the Bragg condition / 2d λ= . This kind of resonant Bragg reflector, represents (for N → ∞) a particular example of photonic crystal [10] in which the frequency of the exciting light lies near a resonance of the system itself. Altought the loss of the gap scalability in these systems seems a disadvantage, respect to the dielectric photonic crystals, the resonance condition allows to achieve: a) a strongly enhancement of the dielectric contrast value b) an increase in the radiation-matter coupling for selected periodicities of the sample. Moreover, the correct description of the dielectric function allows to obtain the right dispersion curves also for photon energies far from the resonant energy of the system. In a recent paper of one of the author [11] the dependence of the optical response on sample size has been investigated. The analysis of the transition from SR to photonic crystal regime in a resonant Bragg reflector as a function of quantum well number N is performed. In the present formulation the driving field of the exciton polarization is the total field of Maxwell equation, while in the framework used in Ref. [11] only the transverse electromagnetic field is consid-ered. We will point out that the two different procedures gives the same physics due to the selfconsis-tency of the method. In the present work a general procedure for computing the optical response in a cluster of size N of a periodic non-local medium is applied in the framework of the self-consistent ABC free theory. The ad-sorbance of the resonant Bragg system in two different regimes, namely: i) super-radiance regime (for small N values), and ii) resonant Bragg reflector limit is computed and discussed. The optical response of the system, respect to the case of N non-interacting quantum wells is shown by decomposing the po-laritonic matrix of interaction. The paper is organised as follows. In Section 2 the semiclassical microscopic non-local self-consistent theory is adopted to obtain the optical response of a general multi-quantum well cluster of size N. Sec-tion 3 is devoted to the N → ∞ limit, while Section 4 contains the discussion of the results and conclud-ing remarks.

2 Optical response of a multi-quantum well cluster

In nanoscale systems, the microscopic character of the polarization reflects in the induced EM field, making necessary the self-consistent determination of the two quantities and leading to a non-local treatment of the problem.

1412 L. Pilozzi et al.: Optical response in multi-quantum wells under Bragg conditions


For the system we are going to describe, a 1D array of N quantum wells centred in the z d=� posi-

tions of the sample, extending in the z > 0 half-space, we consider a cubic dielectric tensor of the form:

( , '; ) ( ') 4 ( , '; )wz z z z z zα αε ω ε δ πχ ω= − + (3)

and solve the integro-differential Maxwell equations along the lines of Ref. [12] In eq. (3) wε is the bulk dielectric constant of the wells that we choose equal to that in the barriers bε ,

, ,x y zα = marks the three different polarizations T, L and Z respectively, and in RWA

( ) ( ')

( , '; ) ( ) ( ) ( ) ( ')o

Z ZZ Z S Z Z

iα α α

ϕ ϕχ ω ω χ ω ϕ ϕ

ω ω Γ

∗

∗

= ≡

− −

∑ ∑� �

� � �

� ��

(4)

Notice that in equation (4) only one resonant level for each well has been considered and ( )Zϕ�

and ω

�� are the eigenfunction and the eigenvalue of the � -th site ( � =1,2,...N). Imposing the Maxwell's boundary conditions to the electromagnetic field at the surface Z=0, we obtain the reflectivity of the system in the form:

( )

( )1 ( )

o

o

rr

r

α

α

α α

α

∆ ωω

∆ ω

−

=

−

(5)

where the term orα

is the contribution due to the ,o bε ε dielectric function discontinuity, and the quantity:

( )'

2( ) 1

'2 ''

( )( ) ( ) ( ) ( )

2ZiK d d o o

Z ZZ

e K D KiKc

α

α

α

ω χ ω∆ ω ϕ ω ϕ

+ −

= ∑ � �

�

�

��

��

� (6)

is the reflectivity due to the , ( , ')b z zε ε discontinuity. Moreover it is easy to obtain the transmittivity:

( )'

2( ) 1

'2 ''

( )( ) 1 ( ) ( ) ( )

2ZiK d d o o

Z ZZ

e K D KiKc

α

α

α

ω χ ωΤ ω ϕ ω ϕ

− −

= − −∑ � ��

� ��

��

� (7)

and the absorbance of the system 2 2

( ) 1 ( ) ( )A α α

αω ∆ ω Τ ω= − −

In order to separate in all these quantities the contribution due to the isolated wells from that due to the interactions among them, let us remind that in the ( )D

αω

�

matrix the well-well interaction is represented by the off diagonal terms while the diagonal elements refer to the exciton state in the single quantum well located in � -site of the sample. A useful description of the optical response of the system is then in terms of the diagonal and off diagonal elements of the ( )D

αω

�

matrix:

2

' , ' '2( ) ( ) ( ) ( )D M

cα α

α

ωω δ χ ω ω= +

��

� (8)

with *' '( ) ' ( ) ( , '; ) ( ')M dZdZ Z G Z Z Zα

ω ϕ ω ϕ= ∫�� and , *( ) ( ) ( ; )o oI dZ Z E Zα

αω ϕ ω= ∫� �

.

The diagonal elements of this matrix can be written in the form:

,

( )( ) poli

Di

αω ω Γ Σ ω

ωω ω Γ

− − +

=

− −

� �

� �

� �

� �

� �

(9)

where 2

,2( ) '( ) "( ) ( )pol oi S M

cα

ωΣ ω Σ ω Σ ω ω= + =

� � is the polariton self-energy for the single quantum

well. In particular, for a 2D exciton function of the form ( )2

( ) cos ( ) /Z N Z d Lϕ π = − � � �

we obtain



'

2

, ,

' '

1( ) Re ( ) ( )

2

1( ) ( ) ( )

2z

oz

z

iK d do oz z

z

M M KiK

M K K eiK

ω ω ϕ

ω ϕ ϕ−

≠

= +

=

� �

� � � � �

� � � �

(10)

3 Photonic crystal limit: N → ∞

The polariton dispersion curves are computed, for the infinite periodic system, imposing the conditions

Fig. 1 Polaritonic dispersion curve (solid line) for 12.6w bε ε= = . Dashed line represents the photonic dispersion curve in a homogeneus bulk. of continuity at the surfaces of one of the quantum well in the array and the periodicity condition of the electric field as for the photonic Kronig and Penney model. We observe that the electric field computed at the well surfaces in the site “ � ” and that in the site “ '� ” differ only for a phase translation due to the Fourier transform of the exciton envelope function. All the other quantities, as the self-energy and the z-

dependent function ( , ) ' ( ') ( , '; )G z dz z G z zω ϕ ω= ∫� � are site independent due to the |z-z’| difference



present in the “bulk” Green function '1( , '; )

2zik z z

z

G z z eik

ω−

= . In this scheme, the polariton dispersion

curves are given by the real part of the following complex equation:

( ) {

}

2

( ) '' 2 '' ''

( ) '' 2 '' ''

cos ( ) ( ) ( ) 2 ( )8

( ) ( ) 2 ( )

z w z w z w

z w z w z w

z z iq d L ik L ik Lo o

z z

iq d L ik L ik Lo o

k qKd e e i r i e i r

q k

e i e r e i i r

ω Σ ω ω Σ ω Σ ωε

ω Σ ω ω Σ ω Σ ω

− −

− − −

+ = − − + −

+ + − − +

� ��

� ��

(11)

where z zo

z z

k qr

k q

−

=

+

and ' ( )oω ω ω Σ ω= − +�� . If we neglect the exciton contribution to the dielec-

tric susceptibility, we recover the Kronig and Penney solution:

( )( )

( )2 2

cos ( ) cos( )cos sin ( ) sin2

z z

z w z w z w z wz z

k qKd k L q d L k L q d L

q k

+ = − − −

(12)

while, for the special case of equal dielectric constant for wells and barriers eq. (11) is strongly simpli-fied:

''cos ( ) cos ( ) ( )sin ( )z zKd q d q dω ω Σ ω= +� �� (13)

This condition gives the polariton dispersion curves shown in Fig. 1 and computed for the parameter values given in Tab. 1.

Table 1 Parameters used in the calculation

dielectric function well width resonance energy

εb=12.6 Lw= 8 nm 1.5152 eVoω =�

Throughout all the Brillouin zone, Fig. 1a, the polaritonic dispersion curve (represented with a solid line) is similar to the one in a homogeneous bulk (reported in the picture as a dotted line). Differences there appear at the boundary of the Brillouin zone, Fig. 1b, c, where gaps open in the polaritonic curve due to periodicity of the dielectric function induced by the exciton contribution. The photon bulk disper-sion curves cross the boundary Brillouin zones at on ω� due to the imposed periodicity. In the polaritonic dispersion curves appears a third curve very flat in energy at the exciton resonance

Fig. 2 Polaritonic (solid line) and photonic dispersion curves for 12.6 , 3w bε ε= = .and periodicity ( , ) / 2.o bd λ ω ε=



renormalized by the real part of the radiative contribution ' ( )Σ ω that goes to zero for /K dπ→ (Fig. 1d). In the present case, where the background dielectric constants are the same for wells and barriers, the dispersion curves behaviour is due to the interplay between the background dielectric constant and the dispersive part of the non-local exciton dielectric function. This dispersive component is large at the

Fig. 3 Polaritonic dispersion curves for 12.6 , 3w bε ε= = .and periodicity ( , ) / 2.o wd λ ω ε=

resonance energy, but modifies the curves in the whole energy range. This fact underlines the importance of the use of a well behaved dielectric function also for photon energy far from the resonance. For the general case of different background dielectric constants in wells ( wε =12.6) and in barriers ( bε =3.0) two different situations are considered here: i) the lattice periodicity is ( , ) / 2o bd λ ω ε= and ii) it is ( , ) / 2o wd λ ω ε= . For the i) situation the system is in the Bragg condition for the resonance energy; its photonic (dashed line) and polaritonic (solid line) bands are shown in Fig. 2. The photonic curve shows the gaps due to the background dielectric function modulation. The polaritonic curve presents the same characteristics of the previous case except for the gaps width that is enhanced by the bakground contribution. When the periodicity of the system does not coincide with the Bragg condition for the reso-nant wavelength the exciton-photon interaction gives rise to a polariton splitting near the exciton reso-nance since the first gap is moved in the higer energy range (Fig. 3a, b). An interesting situation to study could be that of a second exciton state dropping in the photonic gap.

4 Results and discussion

As stated in the previous sections, in MQW systems, the self-consistency between matter polarization and radiation field gives rise to light mediated interactions among different wells that lead to a radiative

Fig. 4 Reflectivity spectral evolution for 1 < N < 400. Fig. 5 HWHM of the reflectivity for 1 < N < 500.

-0.01 0.00 0.01

0.0

0.2

0.4

0.6

0.8

1.0

N = 200

N = 300

N = 400

reflectivity

ω−ω0 (eV)

N = 1

N = 2

N = 10

N = 20

N = 40

N = 50

N = 80

N = 100



correction (shifts and widths) to their eigen-energies. This correction, the exciton-polariton self-energy, strongly modifies the optical response of the system and depends on the sample geometric structure. To study this correction we have considered a system composed of N identical quantum wells with inter-well distance satisfying the Bragg condition and physical parameter values given in Tab. 1. The reflectivity spectrum and its half-width at half-maximum (HWHM) as a function of quantum well number (N=1 ÷ 500) are given in Fig. 4 and Fig. 5 respectively. The HWHM behaviour can be explained dividing the full range in three different zones. In the first one (1 < N < 80) the linear increase is a consequence of the existence of a SR mode that monopolizes the response of the system. In the saturation zone, where the reflectivity curve is no more Lorentzian (Fig. 4), the constant HWHM value represents the width of the high reflectivity band of the MQW system, that in this regime behaves like a Bragg reflector. In the transition zone, that connects the SR and the Bragg reflector regime, there is a redistribution of the oscillator strength of the system between the modes that for smaller N values were dark.

The validity of this description is confirmed by the evolution of the absorbance of the system. Increas-ing the quantum wells number till a limit Nt we observe the presence of only one absorbance peak coin-

cident with the only bright mode of the system. The SR character of this mode is confirmed by its radia-tive damping AΓ that increases linearly with the well number as shown in Fig. 6.

Above the threshold value Nt, other modes became bright and in coincidence with their eigenfrequen-

Fig. 6 HWHM of the absorbance in the super-radiant regime

Fig. 7 Reflectivity (solid line), trasmittivity (dashed line) and absorb-ance (filled curve) for the N=200 system.

0 10 20 30 40 50 60 70 80

0

1

2

3

4

5

N

ΓA (m

eV)

1.343575 1.343600 1.419036 1.4190481.06515 1.06530

0.0

0.2

0.4

0.6

0.8

1.0

1.448100 1.448105 1.462070 1.462075 1.469802 1.469805

intensity

ω−ωo (µeV)

N=200



cies the absorbance spectrum shows additional peaks. They modify the spectrum for energy values close to oω� . In Fig. 7 some of these peaks are shown for selected energy values, for the case with N=200, together with the reflectivity and trasmittivity curves. In order to have a better description of the reflectivity lineshape, let us consider the simple case of two equal QWs and separate the reflectivity amplitude as a sum of three different contributions:

( ) ( ) ( ) ( )S D ODα α α α

∆ ω ∆ ω ∆ ω ∆ ω= + + (14)

In this decomposition, the quantity:

( ) ( )

22

1 2,2

1,2 1,2

( )( ) ( ) ( ) ( )

2z

oz i K d

Sz

KD e

iKcα α α α

ϕω∆ ω χ ω ω ∆ ω

−

= =

= ≡∑ ∑��

� � � �

� �

(15)

is the sum of polaritonic contributions of the l-th well in interaction with the homogeneus electromag-netic field; it gives the same contribution to the radiative correction of the oω� eigenstate as that of the

single well. In fact, neclecting every term of interaction between the wells, we obtain the reflectivity of

the system, normalized with the well number 2

/ 2Sα

∆ , equal, in position and amplitude, to that of the

single well. Obviously this property is still valid for N > 2. The terms of interaction ( )D

α

∆ ω , ( )ODα

∆ ω introduce a correction that change the shape of the reflectiv-

ity amplitude. In fact the diagonal and off diagonal terms have additional poles with respect to the single well case:

[ ][ ]

42 2

124

( ) ( )o

D So S OD o S OD

S Mc

i iα α

ω

∆ ω ∆ ωω ω Γ Σ Σ ω ω Γ Σ Σ

=

− − + + − − + −� � � �

(16)

[ ][ ]

1 2( )4 2 212

4

2( )

2 ( ) ( )

zik d do

ODz o S OD o S OD

S e M

ik i icα

ω ϕ∆ ω

ω ω Γ Σ Σ ω ω ω Γ Σ Σ ω

+

= −

− − + + − − + −� � � �

(17)

They are shifted in symmetric positions with respect to the single well pole of the quantity Re( )ODΣ

where 2

122( )OD oS M

c

ωΣ ω ≡ is the contribution of self-energy due to the well-well interaction. This term,

in this case, has the form [ ] 2 1

2( )

112( ) Im zik d d

OD oi S M ec

ωΣ ω

−

= − and depends from the inter-well distance.

In particular at the resonant energy ODΣ is imaginary for the case of wells ( , ) / 2o bλ ω ε apart while it

is real for the ( , ) / 4o bλ ω ε case. This means that the correction to the polaritonic self-energy due to the

off diagonal elements has no effect on the energy position of the resonance but only on its radiative damping for the Bragg system while the contrary is true for the anti-Bragg case. For the Bragg case, where the condition 2 1( )zk d d π− = remains valid close to the resonance energy

oω� , the reflectivity assumes the form:

( )

[ ]

22

22

2 ''( )

( ) 2 ''

S

S

αΣ

∆ ω

ω ω Γ Σ

=

− + −��

(18)

that in the limit 0Γ+

→ is a Lorentzian curve centered in 'o Sω ω Σ= +�� and of HWHM = 2 ''SΣ .



In Fig. 8 the three reflectivity terms 2

( )Sα

∆ ω , 2

( )Dα

∆ ω , 2

( )ODα

∆ ω are shown togheter with the inter-

ference term 2 2 2 2

4 ( ) ( ) ( ) ( ) ( )S D ODα α α α α

∆ ω ∆ ω ∆ ω ∆ ω ∆ ω= − − −

Notice that for the parameter chosen the diagonal and out of diagonal terms assume the same value in all the energy range considered. In Fig. 9 the different terms contribute to produce as a result of their superposition the reflectivity peak of width 2 ''SΣ shown in Fig. 10, that is a fingerprint of superradiant condition.

-0,0004 -0,0002 0,0000 0,0002 0,0004

-3

-2

-1

0

1

2

3

4

∆2

S

∆2

D+∆

2

OD+∆

4

ω−ωo

(eV)

-0.0004 -0.0002 0.0000 0.0002 0.0004

0.0

0.2

0.4

0.6

0.8

1.0

∆2

ω−ωo (eV)

An analogue decomposition can be obtained for the general case of N QW and it is used for the discus-sion of the reflectivity spectral evolution in the transition from SR to photonic band regime [12].

References

[1] K. Cho, “Optical response of nanostructures” , Springer Series in Solid-State Science 139 (2003). [2] E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, Fiz. Tverd Tela 36, 2118 (1994) [Superlattices and Microstruc-

tures 16, 17 (1994)]. [3] M. P. Vladimirova, E. L. Ivchenko, and A. V. Kavokin, Fiz. Tekh. Poluprovodn. 32, 101 (1998) [Semiconduc-

tors 32, 90 (1998)]. [4] L. C. Andreani, G. Panzarini, A. V. Kavokin, and M. R. Vladimirova, Phys. Rev. B 57, 4670 (1998).

Fig. 8 Reflectivity terms 2

( )Sα

∆ ω , 2

( )Dα

∆ ω , 2

( )ODα

∆ ω and 4 .α

∆

Fig. 9 (left), Fig. 10 (right).

-0.0004 -0.0002 0.0000 0.0002 0.0004

0

1

2

3

4

∆2

S

ω−ωo (eV)

1.480 1.482 1.484 1.486 1.488 1.490 1.492 1.494

0.0

2.0x108

4.0x108

6.0x108

8.0x108

1.0x109

1.2x109

∆2

D=∆

2

OD

ω−ωo (µeV)

1.480 1.482 1.484 1.486 1.488 1.490 1.492 1.494

-2.5x109

-2.0x109

-1.5x109

-1.0x109

-5.0x108

0.0

∆4

ω−ωo (µeV)



[5] M. Hubner, J. Kuhl, T. Stroucken, A. Knorr, S. W. Koch, R. Hey, and K. Ploog, Solid State Commun. 105, 105 (1998).

[6] M. Hubner et al., Phys. Rev. Lett. 83, 2841 (1999). [7] E. L. Ivchenko, Sov. Phys. – Solid State 33, 1344 (1991). [8] L. I. Deych and A. A. Lisyansky, Phys. Rev. B 62, 4242 (2000). [9] S. Nojima Phys. Rev. B 59, 5662 (1999). [10] E. Yablonovich, Phys. Rev. Lett. 59, 2059 (1987). [11] T. Ikawa and K. Cho, Phys. Rev. B 66, 085338-1 (2002). [12] L. Pilozzi, A. D’Andrea, paper submitted to Phys. Rev. B.

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Optical response in multi-quantum wells under Bragg conditions