Effects of Quantum well Size Alteration on Excitonic Population Oscillation Slow Light Devices Properties

doi:10.4236/opj.2013.32B070

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Optics and Photonics Journal, 2013, 3, 298-304

doi:10.4236/opj.2013.32B070 Published Online June 2013 (http://www.scirp.org/journal/opj)

Effects of Quantum well Size Alteration on Excitonic

Population Oscillation Slow Light Devices Properties

Hassan Kaatuzian, Hossein Shokri Kojori, Ashkan Zandi, Reza Kohandani

Photonics Research Laboratory (PRL), Department of Electrical Engineering, Amirkabir University of Technology

(Tehran Polytechnic), Tehran, Iran

Email: [email protected]

Received 2013

ABSTRACT

This paper investigates the effects of quantum well size changes on center frequency and slow down factor of an slow

light device. In this way, we consider the quantum well size alteration effects on oscillator strength and binding energy

of exciton. First, we investigate the variations in oscillator strength of exciton due to different quantum well size. Sec-

ond, exciton binding energy level shift due to size of quantum well is investigated. According to this analysis, we have

developed a new method for tuning slow light device bandwidth center frequency and slow down factor. Analysis and

simulation of a basic GaAs/AlGaAs quantum wells optical slow light device based on excitonic population oscillation

shows that size of quantum wells could tune both of the frequency properties and slow down factor of an optical slow

light device. In our simulation with 34 quantum wells each with the width of 60Å, we have received the slow down

factor of more than 60,000. These achievements are useful in optical nonlinearity enhancements, all-optical signal

processing applications and optical communications.

Keywords: Slow Light; Coherent Population Oscillation; Quantum Well; Exciton Oscillator Strength; Exciton Binding

Energy

1. Introduction

During past years several experiments have shown the

possibility of controlling the speed of light [1]. Optical

buffers have lots of potential scientific and engineering

applications [2].

The applications of such an optical buffers (OB) could

be 1) bit level synchronization; 2) data packet synchro-

nization; 3) Optical gates and 4) boosting nonlinear in-

teraction [3-4-5]. There are several methods to slow

down the speed of light. Each of these methods are dif-

ferent in their mediums and principles. Slow light could

be realized by methods such as electromagnetically in-

duced transparency (EIT) in an ultra-cold atomic cloud,

waveguide in photonic crystals, coupled resonator optical

waveguide [6-7]. By considering the principle of the

slow light, these techniques are categorized in WG dis-

persion and material dispersion [3]. In this paper, our

focus will be on material dispersion.

For achieving the slow light in semiconductors, large

material dispersion should be obtained. For this goal,

coherent population oscillation (CPO), stimulated Bril-

louin and Raman scattering, four wave mixing (FWM)

and EIT can be used [1-5]. There are many research re-

ports about semiconductor optical buffer because of their

advantages in compactness, integration with other optical

devices, ability of tuning the buffer parameters and wide

range of working temperature. Recently, there are huge

uses of CPO and FWM in quantum well (QW) and

quantum dot (QD) semiconductor to change in absorp-

tion or gain spectra [2-8].

The excitonic population oscillation has been success-

fully presented by Ku et al [1]. The experimental setup

consists of a linearly polarized pump tuned to the reso-

nance energy of 1s heavy-hole (HH) exciton in quantum

wells which is incident in the QW growth direction, as

shown in Figure 1. A weak signal with parallel or or-

thogonal polarization to the pump propagates along QW.

Pump causes variation of absorbance and phase delay [9].

Dephasing mechanisms exhibit observable features in

absorption spectra. However the effect of excitation in-

duced dephasing and spin flip should be considered in

formulation for the dephasing of population in semicon-

ductor quantum structure under intense pump. According

to these considerations, slow light base on excitonic

population has shown in [1].

There could be several choices to control an optical

buffer. These control procedures can be divided in two

sections. The first one is before the fabrication and the

second is after the fabrication. Before fabrication, the

techniques such as changing the QW geometry, the dop-

H. KAATUZIAN ET AL. 299

ing, the alloy percentage and the materials of QW could

be employed to design a practical optical buffer [7]. Af-

ter fabrication, current injection and pump intensity are

the choices for tuning of OB. Pump intensity change the

Rabi frequency of QW material and the population of

carriers; so, these parameters affect the profile of absorp-

tion spectra which could change the steep of refractive

index in resonance frequency [5].

In this paper, the effects of changing quantum well

size on tuning slow device properties will present. Con-

sidering the effect of change in quantum well width and

dielectric constant mismatch on binding energy and os-

cillator strength, we developed a new technique for con-

trolling the slow dawn factor (SDF =/

cv) and the

bandwidth center frequency. According to simulations,

for the constant effective length of QWs, decreasing in

QW width and as a result increasing in the number of

QWs will cause two effects. First one is slow down fac-

tor enhancement. So, increasing in oscillator strength of

exciton will improve the slow down factor. Second effect

is enhancement in binding energy and downshift of cen-

ter frequency. So, increasing in binding energy of exciton

will shift frequency response of a quantum well slow

light device in frequency domain.

This paper is organized as follows. In section 2 theory

of slow light via excitonic population oscillation, the

effect of quantum well width on oscillator strength and

binding energy of excitons are presented in three subsec-

tions. Section 3, shows the modified model for descrip-

tion of a slow light device to investigate oscillator

strength effects. The results and discussion is presented

in section 4, also in this section we present our new tun-

ing technique. Section 5, concludes the paper.

2. Theory

Figure 1 shows typical schematic of a multiple quantum

wells (MQW) structure which is contained pump and

probe signal for slow light purpose.

Figure 1. MQW in presence of pump and signal for slow

light purpose.

For investigation the effect of exciton oscillator

strength (EOS) and binding energy on slow light device,

three experiments in GaAs/AlGaAs multiple quantum

wells were considered. The first experiment is about slow

light using excitonic population [1]. The second experi-

ment introduces the dependence of EOS on quantum well

width [10-11]. The last one, considers the effects of QW

size on binding energy of excitons [12]. It is worth men-

tioning that this topology was selected as a test bench

and the results of analyses can easily be used in other

implementations of slow light devices as well.

2.1. Dependence of EOS on Quantum Well

Width

Absorption of excitons can be described in terms of ex



(dielectric constant) as follows [9]:

exexr exi

EEiE







  (1)

where exr



and exi



are respectively the real and

imaginary parts of ex



, ex is the transition energy of

excitons,



and E are linewidth (FWHM) and the pho-

ton energy, respectively. Parameter

is related to os-

cillator strength per unit area. Absorption coefficient



is related to oscillator strength per unit area f by [13]:

4π(

ef EE

nm cL





) (2)

where

()

2()(/2

EE EE





 

)

(3)

where 0 is the free electron mass and is the well

width. is the light velocity in vacuum and is re-

fractive index of the material. The oscillator strength per

unit area can be obtained as follow by comparing equa-

tions (1) and (2):

8π

mAL

 (4)

According to [11], for narrow wells, the factor A is

directly proportional to oscillator strength, f, and related

to well width as 2

L. As a result, the EOS is re-

lated to width of quantum well inversely.

Figure 2 shows the behavior of EOS for ground state

HH1-CB1 according to quantum well width. The pa-

rameters which are used to simulate are the same with

[10-11].

2.2. Dependence of Binding Energy on Quantum

Well Width

The exciton bound-state energies and wave functions as a

function of spatial dimension calculated according to He

H. KAATUZIAN ET AL.

300

Figure 2. Well width dependence of the excitonic oscillator

strength in GaAs/AlxGa(1-x)As with different alloy percent-

age x.

model [14]. The discrete bound-state energies and orbital

radii are given by [14]:

()



 



(4)

(







)

(5)

where is quantum number, and n0



are effec-

tive Rydberg constant, 0

(/((/ )))



)( /



))

, and effec-

tive Bohr radius, 00

(( /



 

, respectively.

and



are the Rydberg and Bohr radius constant,

is the free electron mass, μ is exciton reduced mass, 0



is dielectric constant.

According to Equation (5), the binding energy of the

1s exciton is given by:

()



ES (6)



changes between 2 and 3 in real quantum well struc-

ture. Reduction in well width causes compression in en-

velope function of electron and hole pairs.

Consequently the fractional dimension



decreases

from 3 towards 2 [12]. Although in a very narrow wells

because of spreading of envelope function in barrier level

the spatial extents begin to increase as the well width

decreases.

According to Figure 1 which describes the geometry

of a slow light device, the well width of QW is finite.

When the well width decreases below a specific value,

the envelope function spread into barriers. This phe-

nomenon restores the three dimensional character of ex-

citons. As a result, fractional dimension



should be

expressed as a function of spreading of envelop function.

As a result the fractional dimension



is described as

follows [12]:

2(2/ )/2

3bw





 (7)

In above equation and



are respectively, the

wave vector in barrier and the mean value of the three

dimensional Bohr-radius which is defined as follows:







 (8)

where



is a mean value of the three dimensional re-

duced mass of exciton. The binding energy of the com-

bined exciton in a finite quantum well is defined as fol-

lows [12]:

(2/)/2 2

(1 0.5)

bkL









(9)

where E*

is the mean value of the effective Rydberg

energy for the three dimensional exciton. The details of

solutions and the exact definition of parameters can be

found in [12].

Figure 3 shows the behavior of binding energy for

ground state HH1-CB1according to quantum well width.

The parameters which are used to simulate are the same

with [12-14]. It is important to note that Figure 3 does

not show the binding energy under 50 Å well width. Un-

der 50 Å, the behavior of binding energy will change and

because of spreading the envelop function in barrier level;

the amount of binding energy will decrease. However in

this paper, we don’t consider this phenomenon because

in quantum well slow light device the width of quantum

well is more than 50 Å.

2.3. Slow Light Using Excitonic Population

Oscillation

In this subsection, the primary attention is description of

general theory of slow light in semiconductors. “Equa-

tion (11) shows the group velocity of an optical pulse in a

medium with refractive index ()n



[4-5]”.

()

vdn n



 







(10)

Figure 3. Well width dependence of the binding energy in

GaAs/AlxGa(1-x)As with different alloy percentage x.

H. KAATUZIAN ET AL. 301

Slow light in semiconductor can be obtained by in-

creasing the refractive index. One possible way for re-

ducing group velocity is using materials in which

/dn d



is as large as possible [14]. The refractive index

of such sample is typically small [15-16]. But it varies

rapidly in the region of the resonance so /dn d



and

hence ng can be large. However, the absorption near

resonance is very high. So, all of light is absorbed in the

length of the device [16]. But, the effect of absorption is

canceled by using coherent optical effect. Laser beam

called pump laser, is tuned to one of the resonance fre-

quency of material. It causes a narrow dip in absorption

in profile [17].

In materials with saturable absorption, when a strong

pump and a signal beam of slightly different frequencies

interact, the population of ground state will oscillate in

time at the beat frequency of two input beam. This phe-

nomenon is called coherent population oscillation [16].

In the rest of this section, we consider a slow light

system with excitonic population oscillation which is

described in [1]. In a simple two level system, the co-

herent population beating induced by the frequency dif-

ference between the signal and the pump can produce a

dip in the absorption spectrum if dephasing time is much

shorter than the relaxation time of the population differ-

ence [18]. For investigating of dispersion in such system,

usually semiconductor Bloch equations is used. Moreover,

only the first C-like and HH-like quantized bands are

taken into consideration [15]. Table 1 shows the defini-

tions of parameters which are used in this subsection.

The coherent dynamic equations for Interband polari-

zation and carrier occupation numbers are as follows [1]:

,12

2, ,

,(0)

1,,, ,

()

[()] 2

()(

()

4Im[]

exex exex

exexsexex

iiNPi N

NNN NN







)













 



 











(11)

where (0)



is the respective population difference in

equilibrium. This equation, (12), is valid only in low ex-

citation regime. In this approximation the effect of elec-

tron hole plasma screening and phase space filling are

not include and the equation can be solved in the steady

state region. With relevant optical equation in semicon-

ductor, the linear relative permittivity tensor ()





experienced by the signal is obtained. Refractive index,

absorption and slow down factor are defined respectively

as follows:

() ()





 (12)

()2 Im[()]

ss ss



Re[ ()]

()Re[()] ss

ssss s











 (14)

Table 2 shows the experimental parameters which is

used in [1-15] for calculations. According to the absorp-

tion coefficient and real part of refractive index are

shown in [19], for resonance frequency the value of

/dn d



is high. So, the group index increases rapidly

and the effect of slow light can be seen in device.

3. Modified Fourier Transform Spectral

Interferometry Model for Slow Light

Device

For modeling a slow light device, the model which is

described in [20] is used. This model gives an analytical

expression for the full dielectric constant of Wannier

exciton of arbitrary dimensionality, d, [21]. The effective

dimensionality d, is a parameter that can vary between 2

and 3 in quantum well, depending on the ratio of the QW

thickness to the exciton Bohr radius. Moreover, d is also

related to the spatial extension of the exciton in the bar-

riers. The complex dielectric constant according to this

model is [20]:

Table 1. Definition of the symbols.

Table (2). Physical device parameter for GaAs/Al0.3Ga0.7As.



 (13)

H. KAATUZIAN ET AL.

302

0.5 1

() ([()][()

()

2[(0)])

EgEigE



 

















(15)

where

2( )

() 11

()(1 )

(cot[()]cot[(1)])

gdd





















(16)

where ()

 is Euler gamma function, and Eg are

the effective Rydberg and band gap energy,



linewidth, and is parameter which is related to exci-

ton oscillator strength. The function

(z)



is defined as,

0.5

() ))



( /(zRE

Considering the heavy and light hole, the total dielec-

tric constant for the QW is defined as follows [20]:

() 1()().

background heavylight

EEE





 (17)

It is worthwhile to note that in the case of excitonic

population oscillation for slow light purpose, only the

heavy hole contribution is considered.

Since we want to investigate the effect of oscillator

strength on slow light device, we changed the sign of

dielectric constant. Because of pump signal configuration

in quantum well and absorption cancellation, the steep

direction of refractive index near resonance frequency is

changed. So we have to change the sign of dielectric

constant to use this model for slow light devices. Base on

this modified model, the refractive index profile for QW

slow light device with different oscillator strength is

shown in Figure 4. EOS is change with aim of reducing

or increasing QW width. As it is clear from this figure,

increasing the oscillator strength will cause improvement

in steep of refractive index.

Figure 4. The refractive index profile for QW slow light

device with different oscillator strength. The solid line

shows experimental result of [1].

In Figure 4, the parameters values are , RH =

4.3 meV, Eg = 1.56 eV,

2.5d

0.5



 meV, according to

[20-22]. On the other hand according to “Equation (11)”,

slow down factor is directly proportional to derivative of

refractive index. As a result, increasing the EOS will

cause enhancement in slow down factor.

4. Results and Discussion

4.1. Binding Energy Effects

According to mentioned theory in section (2.2), the

binding energy will increase if the size of quantum well

decreases. “Equation (19) shows the dependence of ex-

citonic energy to binding energy”. This equation shows

that if binding energy is increased, the excitonic energy

will be decreased.

exbandgapbindingenergy

EE E





 (18)

Excitonic energy presents the center frequency of slow

light device base on excitonic population oscillation. As

a result, the size of quantum well in slow light device can

shift center frequency of device. This shift is upward

when the well width is increased and downward when

well width is decreased. With aim of this phenomenon,

we can tune the center frequency of our slow light de-

vice.

To verify the effect of quantum well size on center

frequency of device, we simulate the experimental device

which is presented in section (2.3) with different well

width sizes. The parameters for simulation are extracted

from [1-10-12].

Figure 5 shows the refractive index of a quantum well

slow light device with three different well width sizes.

As it is clear from this figure, the resonance frequency of

GaAs/Al0.3Ga0.7As QWs is shift with changing well size.

The refractive index profile broadening in Figure 5 is

Figure 5. The refractive index profile for QW slow light

device with different binding energy. The solid line shows

experimental result of [1].

H. KAATUZIAN ET AL. 303

very narrow because we want to show these three cases

in one coordinate. Beside, in Figure 5, the effect of QW

size on EOS is neglected. As far as we know, near reso-

nance frequency the derivative term in “Equation (11)”

increases rapidly and causes huge slow down factor.

Moreover, the transparency window is created which

causes cancellation of absorption effect. The size of well

width determines the location of transparency window in

frequency domain.

Table 3 shows the frequency shift and center of a slow

light with three different well widths. Experimental well

size of [1] is considered as origin. It is important to note

that there is obviously limitation to reducing QW size.

As mentioned in previous section, under specific well

width size which is about 50Å the behavior of binding

energy changes and other phenomenon happens.

Base on simulation results of this subsection, the cen-

ter frequency for QW slow light device will decrease if

the well width of quantum well decreases and the number

of QWs increases. The number of QWs must be in-

creased because the effective QWs width as mentioned in

Table 2 should be constant. For example if the well

width is divided by factor 2, then the number of QWs

must be increased by the same factor.

4.2. Exciton Oscillator Strength Effects

According to theory (2.1), we will use the effect of de-

creasing length of quantum well as parameter for in-

creasing the EOS.

Base on simulation results of section (3), the slow

down factor for QW slow light device will increase if the

well width of quantum well decreases and the number of

QWs increases. The number of QWs must be increased

based on the same reason which mentioned above. To show

the usefulness of the proposed method, the GaAs/Al0.3

Ga0.7As multiple quantum wells shown in Figure 1 was

simulated. But this time, we consider the effect of reduc-

ing QWs width.

Figure 6 shows the slow down factor for three differ-

ent QWs width. It can be seen that decreasing the well

width will increase the slow down factor.

5. Conclusions

We investigate the effects of excitonic oscillator strength

and binding energy of exciton on slow light devices.

Table 3. Frequency shift according to well width size.

Figure 6. The slow down factor for three different QWs

width. The solid line shows experimental result of [1].

According to described model in section (3), the slow

down factor can be increased to more than 60,000 in the

result of reducing well width. Moreover, the center fre-

quency of transparency window can be shift about

3.0476 × 1012 Hz with aim of binding energy. When QW

size is changed, both oscillator strength and binding en-

ergy of excitons are affected. Therefore, when the device

experiences downshift or upshift in its transparency

window, in the same time, the amount of slow down fac-

tor is increased or decreased respectively. Based on these

investigations, the new method for tuning slow light

properties is developed. This novel technique will help

designer to control properties of device. As far as we

know, semiconductor slow light devices suffer from nar-

row bandwidth. Moreover there are huge uses of mul-

tichannel system for slow light devices to overcome lim-

ited bandwidth problem. Multichannel systems need slow

light devices with different center frequencies. With the

aim of this technique, designers could fabricate devices

with different center frequencies and slow down factor.

Because of availability of center frequency shifting and

SDF controlling in this method, we could tune our slow

light device for optimal operational point. In addition,

this availability of frequency shifting in slow light device

could be interpreted as increasing of operational band-

width.

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