analysis and design of lead salt pbse/pbsrse single quantum well in the infrared region

IJASCSE, VOL 1, ISSUE 4, 2012

www.ijascse.in 1

Dec. 31

Analysis and Design of Lead Salt PbSe/PbSrSe Single Quantum Well In the Infrared Region

Majed F. Khodr Electronics and Communication Engineering

American University of Ras Al Khaimah

Ras Al Khaimah, UAE

Abstract— There is a considerable interest in studying the

energy spectrum changes due to the non parabolic energy

band structure in nano structures and nano material

semiconductors. Most material systems have parabolic

band structures at the band edge, however away from the

band edge the bands are strongly non parabolic. Other

material systems are strongly parabolic at the band edge

such as IV-VI lead salt semiconductors. A theoretical

model was developed to conduct this study on PbSe/Pb 0.934

Sr0.066 Se nanostructure system in the infrared region. Moreover,

we studied the effects of four temperatures on the analysis and

design of this system. It will be shown that the total losses for the

system are higher than the modal gain values for lasing to occur

and multiple quantum well structures are a better design choice.

Index Terms—Semiconductor device modeling,

Nanotechnology, Modeling, Semiconductor lasers, Semiconductor

material

I. INTRODUCTION

Recently, IV-VI lead salts quantum well lasers which

exhibit strong quantum optical effects, have been used to

fabricate infrared (IR) diode lasers with wide single-mode

tunability, low waste heat generation, and large spectral

coverage up to about 10 µm. In this region, these IV-VI lasers

may play a key role in IR spectroscopy applications such as

breath analysis instruments, air pollution monitoring and IR

integrated optics and IR telecommunication devices.

In this work we focus on breath analysis as a promising

application and diagnostic tool that should perform well in

clinical settings where real time breath analysis can be

performed to assess patient health [1]. Based on literature

reports, health conditions such as Breast cancer and Lung

Cancer have biomarker molecules in exhaled breath at

wavelengths in the infra-red (IR) region. A new technique that

may play a key role in detecting these biomarkers is Tunable

Laser Spectroscopy (TLS) [1]. PbSe/Pb 0.934 Sr0.066 Se quantum

well laser structures, as part of TLS system, can be used to

generate these critical wavelengths that can be absorbed by the

various biomarkers molecules and hence detecting their

presence in parts per million (ppm). Laser emission at these

critical wavelengths is related to several system parameters

[1,2].

In this work analysis and design are done on PbSe/Pb 0.934

Sr0.066 Se single quantum well (SQW) laser structure. The

developed model is being used to perform energy level

calculations, modal gain-current density relation, and threshold

current–cavity length relation to determine the critical

parameters of interest to the desired design structure. The

effects of band structure this material system and temperature

are included in this model and studied extensively.

II. ENERGY LEVEL CALCULATIONS

It is very well known that the energy levels in the bands can

be calculated in the approximation of the envelope wave

function which can be determined to a good approximation by

the Schrodinger-like equation [3,4]. By solving this equation

for the finite well case, one can exactly determine the

quantized energy levels and their corresponding wave

functions for electrons in the conduction band and holes in the

valence band. Because of the inversion symmetry around the

center of the well, the solution wave functions can only be

even or odd.

For a well material with parabolic bands in the growth

direction (z-direction), the effective masses in the

Schrodinger-like equation are at the extreme of the bands and

are independent of the energy. For a well material with non-

parabolic bands in the z-direction, two methods can be used to

solve for the energy levels [4,5]. The first method uses the

"effective mass" equation, also known as the Luttinger-Kohn

(LK) equation and the second method is the "energy-

dependent effective mass" (EDEM) method. The energy level

shifts due to non-parabolicity effects differ depending on the

method and system parameters used. Throughout this work,

the effective mass of the barrier material is considered constant

and independent of energy.

The lead salts, such as PbSrSe, are direct energy gap

semiconductors with band extreme at the four equivalent L


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points of the Brillouin zone. Because the conduction and

valence bands at the L points are near mirror images of each

other, the electron and hole effective masses are nearly equal.

Furthermore, the bands are strongly non parabolic [7]. Due to

limitation in using the Lutting-Kohn equation [3], the energy-

dependent effective mass method was adopted in this work for

all calculations and analysis.

In order to solve for the energy levels, it is necessary to

specify the potential barrier, the effective masses for the

carriers in the well, and in the barrier for the particular single

quantum well structure of interest. The system of interest in

this work is PbSe/Pb 0.934 Sr0.066 Se. The energy gap and

effective masses of Pb 1-x Sr x Se system dependence on

temperature according to these relations [2 ]:

(1) and the empirical equation for the longitudinal mass:

(2) where the barrier is Pb 0.934 Sr0.066 Se with Eg=0.46 eV and

effective mass=0.142 m0, and the well is PbSe with its

Eg=0.28 eV and effective mass=0.08 m0 at 300K. In this

study we ignored the non-parabolicity effects of the barrier

material. The difference in the energy gaps between the well

material and the barrier material is assumed to be equally

divided between the conduction and valence bands. The offset

energy or the barrier potential for this system is 0.09 eV. This

assumption is made because measurements on the offset

energy for this system have not been made.

In addition, experimental data on similar IV-VI material

QW structures showed that the conduction and valence band

offset energies are equal [7]. It was shown that, for a first

approximation, the effective mass to be directly proportional

to the energy gap and the conduction and valence-band

mobility effective masses in the well are equal and the

calculated values are shown in terms of the free electron mass

[7]. In this study, the conduction and valence-band mobility

effective masses in the well are assumed equal and the

effective masses of the carriers outside the well are assumed

constant.

The energy level calculations for the system were calculated

using the EDEM method. The conduction band energy levels

calculation assuming parabolic and non-parabolic bands are

shown in Fig. 1. As shown in the figure, the energy levels

including the effects of non parabolicity are lower than those

excluding the effects of non-parabolicity and this difference is

higher for small well width values and decreases as the well

width is increased. Moreover, as this effect is higher for higher

quantized energy levels. As for the fourth energy level the

model calculated the energy level including the effects of non

parabolicity and it seems that this level does not exist

assuming parabolic bands. Therefore it is important to include

the effects of non paraboliciyt to be able to calculate all the

energy levels for the system. Similar results can be obtained

for the valence band.

Fig. 1. The effects of non parabolicity on the conduction band energy levels

at 300K.

The emitted wavelength values at 300K for the system are

show in Fig. 2 where the effects of band non parabolicty are

included and compared to those excluding the effects of band

non parabolicity. One notice that the emitted wavelength

values are higher including non-parabolicity and this

difference is higher for smaller well widths and decreases as

the well width increases. For applications that require critical

wavelength calculation such as Breath Analysis Technique

[1,8-12], it is important to include the effects of non

parabolicity to be able to obtain the desired accurate results for

detecting the existence of volatile compounds at their

corresponding wavelengths.

Therefore, in what follows, the effects of non parabolicty

are included in all calculation of the system. However, we

included in our calculations the first energy levels transitions

between the conduction and valence bands.



Dec. 31

Fig.2. The effects of non parabolicity on the emitted

wavelengths at 300K.

The emitted wavelengths as a function of five temperatures:

77K, 200K, 150K, 250K, and 300K are shown in Fig. 3. For a

fixed well width, the emitted wavelengths decreases with

increasing temperature and increases with increasing well

width at the same temperature.

This graph is important for investigators who are using this

material system in tunable diode laser absorption spectroscopy

to measure certain markers in exhaled breath which are

correlated with certain diseases [8]. Examples include the

measurement of exhaled nitric oxide for Asthma at 5.2 m

[9,10], Acetone for Diabetes at 3.4 m [11], Acetaldehyde for

Lung Cancer at 5.7 m [12].

Fig.3. The effects of temperature on the emitted wavelengths .

The calculated values include the effects of non-parabolicity.

III. CONFINEMENT FACTOR CALCULATION

A principal feature of the QW laser is the extremely high

optical gain that can be obtained for very low current densities.

Equally important, however, in determining laser properties

are modal gain, determined by the optical confinement factor,

and the ability to collect injected carriers efficiently [13].

These latter factors prevent the improvement of laser

performance for arbitrarily thin QW dimensions unless

additional design features are added.

These design improvements include the use of multiple

QW's (MQW) and /or the separate confinement heterostructure

(SCH) scheme where optical confinement is provided by a set

of optical confinement layers, while carrier confinement

occurs in another embedded layer. In this work the focus will

be on SQW structure and the other design improvement are

kept for future publications.

The optical analysis of single quantum well lasers is

conventional in that one solves for the TE modes in a three

region dielectric optical waveguide [14]. A planar SQW

structure is commonly represented as a three layer slab

dielectric waveguide where the guiding layer corresponds to

the active layer and the cladding layers correspond to the

passive layers [14]. If the structure is symmetrical (i.e., the

cladding layers have the same index of refraction), then the

waveguide will always support at least one propagation mode

[14]. The index of refraction for the well material PbSe is

4.865 and the index of refraction for the barrier material

Pb 0.934 Sr0.066 Se is 4.38 and they are considered in this work

independent of wavelength and temperature [2].

The radiation confinement factor is one crucial parameter in

the laser design which can be calculated using the general

approximate solution that is valid for all well widths found by

Botez [15, 16]. The analytical approximation given by Botez

for calculating the optical confinement factor in a symmetrical

waveguide for the TEo mode is:

22

2

D

Do (3)

where

)()(2 2

,

2

, wrbr nnw

D

, (4)

and is the vacuum wavelength at the lasing photon energy

and D is the normalized thickness of the active region.

Plotting the confinement factor as a function of well width

in Fig. 4 for the PbSe/ 066.0934.0 TePbSe SQW structure (at

300K) shows that o decreases with decreasing well width w.

In this work, the variations of the index of refraction with

emitted photon wavelength are not considered. Therefore, the

index of refraction of the well material is fixed at wrn , =4.865



Dec. 31

and that of the cladding layer at brn , =4.38 [2]. The effect of

non-parabolicity on the confinement factor and thus on modal

gain is noticeably very small and therefore it can be neglected

for all well widths as it is shown in figure 4.

This is expected because including the non-parabolicity

effects for this system shifts the first energy levels toward the

band extreme and thus, slightly increases the emitted photon

wavelength which decreases o as seen from Eq.(3). The

non-parabolicity effects are expected to be more obvious for

higher quantized energy levels.

Fig.4. The effects of non-parabolicity on the confinement

factor calculations at 300K.

The effects of temperature on the confinement factor are

shown in Fig 5. The confinement factor increases with

temperature at a fixed well width and this is due to the effects

of temperature on the emitted wavelength as seen from Fig. 3

and Eq 3.

Fig 5. The effects of temperature on the confinement factor as

a function of well width. The effects of non-parabolcity are

included in the calculations.

IV. MODAL GAIN AND CURRENT DENSITY CALCULATIONS

Within the framework of Fermi's Golden Rule, the two

major components of gain calculations are the electron and

hole density of states, and the transition matrix element

describing the interaction between the conduction and valence

band states. The derivation for the analytical gain expression

is given by the following expression [4,17]:

1

2

,

2

,

2

)()]()([

)(

n

noovoc

o

avgnQW

owro

redo

Hff

M

wcmn

e

(5)

and the radiative component of the carrier recombination is

found from the spontaneous emission rate[3]:

)()](1[)(

)(

1

2

322

,

2

no

n

ovoc

avgconv

oo

redowr

osp

Hff

Mwcm

neR

(6)

From this, the radiative current density is calculated by the

following equation [3]:

oospRewJ )( , (7)

where e is the charge of the electron, om is the electron free

mass, c is the speed of light, w is the well width, wrn , is the

index of refraction at the lasing frequency o , o is the

permittivity of free space, 2

, avgnQWM is the transmission

matrix element , red is th reduced density of states,

)(, ovcf are the Fermi-Dirac distribution functions, H(x) is

the Heaviside function that is equal to unity when x> 0 and is

zero when x<0, and n is the energy difference between the

bottom of the n-subband in the conduction band and the n-

subband in the valence band.

The excitation method that is of importance in this work is

injection of carriers into the active region by passing current

through the device. An increase in the pumping current leads

to an increase in the density of injected carriers in the active

region and with it, an increase in the quasi-Fermi levels [18,

19].

The gain, current density, and threshold current expressions

for the non- parabolic bands is similar to that of the parabolic

case except in the reduced density of states and the quasi

Fermi levels in the bands. More details about the model and

theoretical derivations can be found in reference [18].



Dec. 31

In laser oscillators, the concern is with the modal gain rather

with the maximum gain. The modal gain is obtained by

multiplying the maximum gain values given in Eq.(5) by the

confinement factor. The calculated maximum gain –current

density values are shown in the inset of Fig. 6 at 300K and

well width 7 nm.

The model gain values are small for this SQW system as can

be seen from Fig. 6.

Fig 6 Modal gain as a function of current density at 300K. the

inset showes the maximum gain as a function of current

density.

The behavior of the modal gain vs. current density values at

five different temperatures: 77K, 150K, 200K, 250K, and 300

K and including the effects of non-parabolicity are shown in

Fig. 7. From this figure one notice that the transparency

current J0 (intercept at gain =0) increases with increasing

temperature. Moreover, the slope of the gain versus current

density plot decreases with increasing temperature. These two

quantities are important in calculating the characteristic

temperature T0 for the system.

The threshold current values and characteristic temperature

calculation are left for future publication.

Fig.7: Modal gain calculations as a function of current density

at four different temperatures assuming non parabolic bands.

In order for laser oscillation to occur, the modal gain at the

lasing photon energy l must equal the total losses total .

The laser oscillation condition is given as:

totallolg )()( maxmod , (8)

The threshold current needed to compensate for the total loss

is calculated by the usual formula [19]:

widthLJAreaJI ththth (9)

The threshold current density thJ that corresponds to the

modal gain value that satisfies the oscillation condition can be

obtained from the modal gain-current density plots. The

threshold current calculations are performed assuming the

width has a constant value of 20 m, the cavity length L as an

independent variable L and the mirror reflectivities fixed at

R1=0.4 and R2=0.4 . The estimate total loss for the system

under investigation at cavity length of 600 m was found to be

approximately 46 (1/cm), which is higher than the modal gain

values shown in Fig. 7. Therefore, a modification to the design

of the system is needed were multiple quantum well structures

are required.

The modal gain-current density relation can be deduced from

that of a single quantum well by multiplying the modal gain

and the current density by the number of wells. Whether the

SQW or the MQW is the better structure depends on the loss

level. At low loss, the SQW laser is always better because of

its lower current density where only one QW has to be

inverted.

At high loss, the MQW is always better because the

phenomena of gain saturation can be avoided by increasing the

number of QW's although the injected current to achieve this

maximum gain also increases by the increase in the number of

wells. Owing to this gain saturation effect, there exists an

optimum number of QW's for minimizing the threshold current

for a given total loss [13].



Dec. 31

V. SUMMARY AND CONCLUSION

In this work we analyzed PbSe/Pb 0.934 Sr0.066 SQW structure

by calculating the quantized energy levels, confinement factor,

maximum gain and modal gain current density relationships.

The effects of band non parabolicty was studied and it was

shown that non parabolicity will have small effect on

quantized energy levels that are close to the band edge and it

will have a larger effect on those far above the band edge. The

confinement factor values for the first energy levels were very

small as expected for SQW structures with minimum or no

effects of non parabolicty. The effects of temperature on the

behavior of the system was analyzed and studied at four

different temperatures: 77K, 150K, 250K, and 300K. It was

concluded that for low loss values, SQW is a good choice to

be used.

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analysis and design of lead salt pbse/pbsrse single quantum well in the infrared region

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