VIEWS: 24 PAGES: 6 CATEGORY: Research POSTED ON: 1/8/2013
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.
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 IJASCSE, VOL 1, ISSUE 4, 2012 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 www.ijascse.in Page 12 IJASCSE, VOL 1, ISSUE 4, 2012 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. www.ijascse.in Page 13 IJASCSE, VOL 1, ISSUE 4, 2012 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]. www.ijascse.in Page 14 IJASCSE, VOL 1, ISSUE 4, 2012 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 thL 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]. www.ijascse.in Page 15 IJASCSE, VOL 1, ISSUE 4, 2012 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. 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