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

IJASCSE, VOL 1, ISSUE 4, 2012
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 presence in parts per million (ppm). Laser emission at these
energy spectrum changes due to the non parabolic energy critical wavelengths is related to several system parameters
band structure in nano structures and nano material [1,2].
semiconductors. Most material systems have parabolic
In this work analysis and design are done on PbSe/Pb 0.934
band structures at the band edge, however away from the
Sr0.066 Se single quantum well (SQW) laser structure. The
band edge the bands are strongly non parabolic. Other
developed model is being used to perform energy level
material systems are strongly parabolic at the band edge
calculations, modal gain-current density relation, and threshold
such as IV-VI lead salt semiconductors. A theoretical
current–cavity length relation to determine the critical
model was developed to conduct this study on PbSe/Pb 0.934
Sr0.066 Se nanostructure system in the infrared region. Moreover,
parameters of interest to the desired design structure. The
we studied the effects of four temperatures on the analysis and effects of band structure this material system and temperature
design of this system. It will be shown that the total losses for the are included in this model and studied extensively.
system are higher than the modal gain values for lasing to occur
and multiple quantum well structures are a better design choice.
II. ENERGY LEVEL CALCULATIONS
Index Terms—Semiconductor device modeling, It is very well known that the energy levels in the bands can
Nanotechnology, Modeling, Semiconductor lasers, Semiconductor be calculated in the approximation of the envelope wave
material function which can be determined to a good approximation by
the Schrodinger-like equation [3,4]. By solving this equation
I. INTRODUCTION for the finite well case, one can exactly determine the
Recently, IV-VI lead salts quantum well lasers which quantized energy levels and their corresponding wave
exhibit strong quantum optical effects, have been used to functions for electrons in the conduction band and holes in the
fabricate infrared (IR) diode lasers with wide single-mode valence band. Because of the inversion symmetry around the
tunability, low waste heat generation, and large spectral center of the well, the solution wave functions can only be
coverage up to about 10 µm. In this region, these IV-VI lasers even or odd.
may play a key role in IR spectroscopy applications such as For a well material with parabolic bands in the growth
breath analysis instruments, air pollution monitoring and IR direction (z-direction), the effective masses in the
integrated optics and IR telecommunication devices. Schrodinger-like equation are at the extreme of the bands and
are independent of the energy. For a well material with non-
In this work we focus on breath analysis as a promising parabolic bands in the z-direction, two methods can be used to
application and diagnostic tool that should perform well in solve for the energy levels [4,5]. The first method uses the
clinical settings where real time breath analysis can be "effective mass" equation, also known as the Luttinger-Kohn
performed to assess patient health [1]. Based on literature (LK) equation and the second method is the "energy-
reports, health conditions such as Breast cancer and Lung dependent effective mass" (EDEM) method. The energy level
Cancer have biomarker molecules in exhaled breath at shifts due to non-parabolicity effects differ depending on the
wavelengths in the infra-red (IR) region. A new technique that method and system parameters used. Throughout this work,
may play a key role in detecting these biomarkers is Tunable the effective mass of the barrier material is considered constant
Laser Spectroscopy (TLS) [1]. PbSe/Pb 0.934 Sr0.066 Se quantum and independent of energy.
well laser structures, as part of TLS system, can be used to The lead salts, such as PbSrSe, are direct energy gap
generate these critical wavelengths that can be absorbed by the semiconductors with band extreme at the four equivalent L
various biomarkers molecules and hence detecting their

www.ijascse.in Page 11

Dec. 31
width is increased. Moreover, as this effect is higher for higher
quantized energy levels. As for the fourth energy level the

points of the Brillouin zone. Because the conduction and
valence bands at the L points are near mirror images of each model calculated the energy level including the effects of non
other, the electron and hole effective masses are nearly equal. parabolicity and it seems that this level does not exist
Furthermore, the bands are strongly non parabolic [7]. Due to assuming parabolic bands. Therefore it is important to include
limitation in using the Lutting-Kohn equation [3], the energy- the effects of non paraboliciyt to be able to calculate all the
dependent effective mass method was adopted in this work for energy levels for the system. Similar results can be obtained
all calculations and analysis. for the valence band.
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)
Fig. 1. The effects of non parabolicity on the conduction band energy levels
where the barrier is Pb 0.934 Sr0.066 Se with Eg=0.46 eV and at 300K.
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 The emitted wavelength values at 300K for the system are
material. The difference in the energy gaps between the well show in Fig. 2 where the effects of band non parabolicty are
material and the barrier material is assumed to be equally included and compared to those excluding the effects of band
divided between the conduction and valence bands. The offset non parabolicity. One notice that the emitted wavelength
energy or the barrier potential for this system is 0.09 eV. This values are higher including non-parabolicity and this
assumption is made because measurements on the offset difference is higher for smaller well widths and decreases as
energy for this system have not been made. the well width increases. For applications that require critical
wavelength calculation such as Breath Analysis Technique
In addition, experimental data on similar IV-VI material [1,8-12], it is important to include the effects of non
QW structures showed that the conduction and valence band parabolicity to be able to obtain the desired accurate results for
offset energies are equal [7]. It was shown that, for a first detecting the existence of volatile compounds at their
approximation, the effective mass to be directly proportional corresponding wavelengths.
to the energy gap and the conduction and valence-band
mobility effective masses in the well are equal and the Therefore, in what follows, the effects of non parabolicty
calculated values are shown in terms of the free electron mass are included in all calculation of the system. However, we
[7]. In this study, the conduction and valence-band mobility included in our calculations the first energy levels transitions
effective masses in the well are assumed equal and the between the conduction and valence bands.
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


Dec. 31

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
Fig.2. The effects of non parabolicity on the emitted conventional in that one solves for the TE modes in a three
wavelengths at 300K. region dielectric optical waveguide [14]. A planar SQW
The emitted wavelengths as a function of five temperatures: structure is commonly represented as a three layer slab
77K, 200K, 150K, 250K, and 300K are shown in Fig. 3. For a dielectric waveguide where the guiding layer corresponds to
fixed well width, the emitted wavelengths decreases with the active layer and the cladding layers correspond to the
increasing temperature and increases with increasing well passive layers [14]. If the structure is symmetrical (i.e., the
width at the same temperature. cladding layers have the same index of refraction), then the
waveguide will always support at least one propagation mode
This graph is important for investigators who are using this [14]. The index of refraction for the well material PbSe is
material system in tunable diode laser absorption spectroscopy 4.865 and the index of refraction for the barrier material
to measure certain markers in exhaled breath which are Pb 0.934 Sr0.066 Se is 4.38 and they are considered in this work
correlated with certain diseases [8]. Examples include the independent of wavelength and temperature [2].
measurement of exhaled nitric oxide for Asthma at 5.2 m
[9,10], Acetone for Diabetes at 3.4 m [11], Acetaldehyde for The radiation confinement factor is one crucial parameter in
Lung Cancer at 5.7 m [12]. 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:

D2
o  2 (3)
D 2
where
w
D  2 ( ) (nr2,b  nr2, w ) , (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/ PbSe0.934Te0.066 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
Fig.3. The effects of temperature on the emitted wavelengths .
index of refraction of the well material is fixed at nr , w =4.865
The calculated values include the effects of non-parabolicity.


Dec. 31
IV. MODAL GAIN AND CURRENT DENSITY CALCULATIONS

and that of the cladding layer at nr ,b =4.38 [2]. The effect of Within the framework of Fermi's Golden Rule, the two
major components of gain calculations are the electron and
non-parabolicity on the confinement factor and thus on modal hole density of states, and the transition matrix element
gain is noticeably very small and therefore it can be neglected describing the interaction between the conduction and valence
for all well widths as it is shown in figure 4. band states. The derivation for the analytical gain expression
This is expected because including the non-parabolicity is given by the following expression [4,17]:
effects for this system shifts the first energy levels toward the
2
band extreme and thus, slightly increases the emitted photon
wavelength  which decreases o as seen from Eq.(3). The
e 2  red M QW ,n
 (o )  avg

non-parabolicity effects are expected to be more obvious for  o nr , wcm w2
o o (5)
higher quantized energy levels. 
[ f c (o )  f v (o )] H (o  n )
n 1
and the radiative component of the carrier recombination is
found from the spontaneous emission rate[3]:

e 2 nr , w  o  red
Rsp ( o ) 
2
M conv avg
m  o  c w
2
o
2 3
(6)

 f c ( o )  [1  f v ( o )]   H ( o   n )
n 1
From this, the radiative current density is calculated by the
following equation [3]:

J  ew Rsp (o )o , (7)

Fig.4. The effects of non-parabolicity on the confinement where e is the charge of the electron, mo is the electron free
factor calculations at 300K. mass, c is the speed of light, w is the well width, nr , w is the
The effects of temperature on the confinement factor are
index of refraction at the lasing frequency  o ,  o is the
shown in Fig 5. The confinement factor increases with
2
temperature at a fixed well width and this is due to the effects permittivity of free space, M QW ,n is the transmission
of temperature on the emitted wavelength as seen from Fig. 3 avg

and Eq 3. matrix element ,  red is th reduced density of states,
f c ,v (o ) 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
Fig 5. The effects of temperature on the confinement factor as for the non- parabolic bands is similar to that of the parabolic
a function of well width. The effects of non-parabolcity are case except in the reduced density of states and the quasi
included in the calculations. 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.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:
g mod (l )  o max (l )   total , (8)

The threshold current needed to compensate for the total loss
is calculated by the usual formula [19]:

I th  J th  Area  J thL  width (9)

The threshold current density J th that corresponds to the
Fig 6 Modal gain as a function of current density at 300K. the
modal gain value that satisfies the oscillation condition can be
inset showes the maximum gain as a function of current
obtained from the modal gain-current density plots. The
density.
threshold current calculations are performed assuming the
The behavior of the modal gain vs. current density values at width has a constant value of 20  m, the cavity length L as an
five different temperatures: 77K, 150K, 200K, 250K, and 300 independent variable L and the mirror reflectivities fixed at
K and including the effects of non-parabolicity are shown in R1=0.4 and R2=0.4 . The estimate total loss for the system
Fig. 7. From this figure one notice that the transparency under investigation at cavity length of 600 m was found to be
current J0 (intercept at gain =0) increases with increasing approximately 46 (1/cm), which is higher than the modal gain
temperature. Moreover, the slope of the gain versus current values shown in Fig. 7. Therefore, a modification to the design
density plot decreases with increasing temperature. These two of the system is needed were multiple quantum well structures
quantities are important in calculating the characteristic are required.
temperature T0 for the system.
The modal gain-current density relation can be deduced from
The threshold current values and characteristic temperature that of a single quantum well by multiplying the modal gain
calculation are left for future publication. 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
[16] D. Botez, "Near and far-field analytical approximations for the
fundamental mode in symmetrical waveguide DH lasers," RCA Rev.,
39, 577 (1978).
V. SUMMARY AND CONCLUSION [17] R. H. Yan, S. W. Corzine, L. A. Coldren and I. Suemune, "Corrections
to the expression for gain in GaAs," IEEE J. Quantum Electron., 26, 213
In this work we analyzed PbSe/Pb 0.934 Sr0.066 SQW structure (1990).
by calculating the quantized energy levels, confinement factor, [18] M. F. Khodr, B. A. Mason, P. J. McCann, "Optimizing and Engineering
EuSe/PbSe0.78Te0.22/EuSe Multiple Quantum Well Laser Structures”
maximum gain and modal gain current density relationships. IEEE Journal of Quantum Electronics, 34, 1604 (1998)
The effects of band non parabolicty was studied and it was [19] M. F. Khodr "Effects of non-parabolic bands on nanostructure laser
shown that non parabolicity will have small effect on devices”” Proceedings of SPIE, 8102, (2011)
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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