Far East Journal of Electronics and Communications 17 Pushpa Publishing House, Allahabad, India http://www.pphmj.com http://dx.doi.org/1.17654/ec1761319 Volume 17, Number 6, 17, Pages 1319-136 ISSN: 973-76 ODAL GAIN AND CURRENT DENSITY RELATIONSHIP FOR PbSe/PbSrSe QUANTU WELL NORAL AND OBLIQUE DEGENERATE VALLEYS Electrical, Electronics and Communication Engineering Department American University of Ras Al Khaimah P. O. Box 11, Ras Al Khaimah United Arab Emirates Abstract PbSe/PbSrSe is a promising lead salts material system that can be utilized in several near infrared applications. This material system has normal and degenerate oblique valleys with different effective masses and emission wavelengths. In this work, we investigate the confinement factor, maximum gain, modal gain and threshold current density for both the valleys at K using modified multiple quantum well structure. It was found that both the valleys have indistinguishable gain and threshold current density values, and the only effect of degeneracy is the emission wavelength. Also, it was concluded that the current density values are high and this could be due to thermionic leakage current across the barrier. 1. Introduction Direct band-gap IV-VI semiconductor layers epitaxially grown on silicon are promising materials for several near infrared applications. These materials are arguably one of the most compatible direct band gap materials Received: June 3, 17; Accepted: September 1, 17 Keywords and phrases: confinement factor, modal gain, threshold current, degeneracy, lead salts, mid-infrared.
13 that can be epitaxially grown on silicon [1]. The crystalline quality of the IV-VI semiconductor materials has not exhibited degradation when subjected to repetitive thermal cycling []. Detailed procedures for BE growth and characterization of PbSrSe/PbSe QWs on (111)-oriented BaF and silicon have been described elsewhere [3-8]. Important results from this work show the removal of L-valley degeneracy [4, 8]. The lowest energy intersubband transition is between heavier effective mass electron and hole states in L- valleys that are normal to the (111) plane in reciprocal space, while the next lowest transition is between lighter effective mass electron and hole states in the three oblique valleys that lie along the equivalent 111 directions. Figure 1. Three-dimensional schematic of the vertical optical cavity structure epitaxially grown by BE [1]. In a more recent publication, the author calculated the effective masses, first energy levels, and the emitted wavelength for the normal valley and oblique valleys at K [9]. In this work, we investigate the confinement factor, maximum gain, modal gain, current density, and threshold current for PbSe Pb.934 Sr. 66Se modified multiple quantum well (QW) structure with PbSe as the well material, and Pb.934 Sr. 66Se as the barrier material (Figure 1). The structure is designed to have a well width of 3nm, barrier width 5nm, number of wells 8, and number of barriers 9. At these well and
odal Gain and Current Density Relationship for PbSe/PbSrSe 131 barrier widths and based on previous publication [1], the effect of band nonparabolicity is negligible as will be confirmed in the current investigation.. ethod Used The reader is referred to reference [9] for the calculated transition energy levels between the conduction and valence bands for each valley. These values have been used to calculate firstly the maximum gain from the analytical gain γ ( ω ) as a function of current density, secondly, the optical confinement factor Γ for both the valleys for the TEo mode, and finally the modal gain is obtained by multiplying the maximum gain values by the confinement factor. The calculations of the confinement factor for QW well structure Γ can be found considering the structure as a three region waveguide with identical cladding layers (same n r, c ), plus a center layer of average thickness t and average index of refraction n r. These average quantities for the QW structures are given by the following equations: and t = N w N B (1) w + B n r Nwwnr, w + NBBnr, B =, () t where N w and N B are the number of wells and barriers, respectively, whereas w and B are their thicknesses. The confinement factor then is given by: where D N w Γ = w, (3) D + t t D = π ( n r n r, c ). (4) λ
13 The three region waveguide with n r, c, nr and t is an accurate model for an QW laser when the waveguide supports only the fundamental mode, and the model decreases in accuracy as the device parameters are modified such that higher order modes propagate [1]. The derivation for the analytical gain expression is given by the following expression [1]: πe ρ γ( ω ) = red QW, n avg [ f ( ω ) ( ω )] ( ω ω ) ε ω c fv H n n cm w r, w n= 1 (5) and the radiative component of the carrier recombination is found from the spontaneous emission rate: R sp( ω ) = e nr, w ωρred 3 πm ε c w conv avg c v n= 1 f ( ω ) [ 1 f ( ω )] H ( ω ω ). n (6) From this, the radiative current density is calculated by the following equation: J = ew ( ω ) Δ ω, (7) R sp where e is the charge of the electron, m is the electron free mass, c is the speed of light, w is the well width, n, is the index of refraction at the lasing frequency ω, ε is the permittivity of free space, r w QW, n avg is the transmission matrix element, ρ red is the reduced density of states, f c, v( ω ) are the Fermi-Dirac distribution functions, H ( x) is the Heaviside function that is equal to unity when x > and is zero when x <, and ωn is the energy difference between the bottom of the n-subband in the conduction band and the n-subband in the valence band. In order for the laser oscillation to occur, the modal gain at the lasing
odal Gain and Current Density Relationship for PbSe/PbSrSe 133 photon energy ωl must equal the total losses α total. The laser oscillation condition is given as: gmod ( ω ) = Γ γ ( ω ) = α. (8) l max The threshold current needed to compensate for the total losses is calculated by the usual formula: I l total = J Area = J L width. (9) th th th The threshold current density J th that corresponds to the modal gain value that satisfies the oscillation condition can be obtained from the modal gaincurrent density plot. 3. Results and Discussion Plotting the confinement factor as a function of well width in Figure for the PbSe PbSe.934Te. 66 QW structure with 8 wells (at K) shows that Γ increases in value. Its value at N w = 8 equals.35, i.e., 35% of the optical power can be confined to the active region. Figure. The confinement factor values as the function of number of wells. Plotting the maximum gain values obtained from equation (5) as a function of the corresponding current density values from equation (7) in Figure 3, one can notice the nonlinear relationship between these two variables. As the current density increases, the maximum gain value
134 increases up to the saturation level where all the states below the Fermi level are filled. ultiplying the maximum gain values in Figure 3 with the confinement factor value of.35, the modal gain values can be obtained. These values as the function of current density are shown in Figure 4. Figure 3. The maximum gain values as the function of current density. Figure 4. The modal gain values obtained by multiplying the maximum gain values with the confinement factor value at N w. Finally, for a cavity width of μm, length of 6μm, and mirror reflectivities fixed at R 1 =. 4 and R =.4, the estimate total loss for the system under investigation was found to be approximately 1 (1/cm). This total loss has to be equal to a modal gain value from equation (8), and from
odal Gain and Current Density Relationship for PbSe/PbSrSe 135 Figure 4 one can obtain the corresponding threshold current density which is around 1A/cm. Applying this value to equation (9) with cavity length of 6μm, and cavity width of μm, the threshold current value would be 1A. This high operating value could be due to the leakage current for the QW and further investigation is required. The effect of non-parabolicity is included for comparison purposes, and since the well width value if high, the effects of non-parabolicity can be ignored. From [9], the first energy levels for both normal and oblique valleys are.44ev and.48ev, respectively. Hence, the confinement factor value, maximum gain values, modal gain values, and current density values for both the valleys were found almost indistinguishable to show as separate graphs. Hence we show here the normal valley value and not the oblique valley ones. 4. Conclusion For the number of wells used, the confinement factor reduced the maximum gain values and has no effect on the threshold current density. The estimated total loss for the system was within the modal gain values, however, the high threshold current calculations obtained indicate that there can be a thermionic leakage carrier that needs to be addressed. The confinement factor values, maximum gain values, modal gain values, and threshold current density values for the normal and oblique valleys are indistinguishable. This is due to the small difference between both the first energy state levels. References [1] L. A. Elizondo, Y. Li, A. Sow, R. Kamana, H. Z. Wu, S. ukherjee, F. Zhao, Z. Shi and P. J. ccann, Optically pumped mid-infrared light emitter on silicon, J. Appl. Phys. 11 (7), 1454. [] H. Zogg, S. Blunier, A. Fach, C. aissen, P. üller, S. Teodoropol, V. eyer, G. Kostorz, A. Dommann and T. Richmond, Thermal-mismatch-strain relaxation in epitaxial CaF, BaF /CaF, and PbSe/BaF /CaF layers on Si(111) after many temperature cycles, Phys. Rev. B 5 (1994), 181-181.
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