The Composition Effect on the Dynamics of Electrons in Sb-Based QD-SOAs

In 1961, it is suggested that stimulated emission can occur in semiconductors by recombination of carriers that are injected across a p-n junction. This is the backbone in the concept of semiconductor lasers and amplifiers [1]. This concept is connected with the “one dimensional electron in a box”, a problem discussed by quantum mechanics text books, to generate a new field of confined semiconductor structures known thereafter as quantumwell (QW), quantum-wire (QWi) or quantum dot (QD) structures where the carriers are confined in semiconductor in one, two, or three directions, respectively [2]. Semiconductor optical amplifiers (SOAs) and lasers performance may be substantially improved by using the QD-SOAs characterized by a low threshold current density, high saturation power, broad gain bandwidth and week temperature dependence as compared to bulk and multiquantum well devices [3]. QD active region results from reducing the size of conventional (bulk) crystal to a nanometer size scale in all the crystal directions. This results in a complete quantization of energy states. The density of states becomes comparable to a delta-function. In order to cover losses of the waveguide of the SOA, QDs are then grown by a large number of dots grown on a WL which is already in the form of QW layer, see Fig. 1. In this chapter, we examine the effect of changing composition of the layers constructing the Antimony (Sb)based quantum dot semiconductor optical amplifiers (QD-SOAs). We startoff in sections 2-3 a general description of QD-SOA and the importance of SOAs, especially Sb-based SOAs, in solving problems. Manufacture imperfection in the shape of QDs are taken into account in the gain calculations in section 4, where they are represented by an inhomogeneous function. Along with the gain description in this section, a global Fermifunction is used where the states in the barrier layer, wetting layer (WL) and ground and excited states of the dots are included in the calculations of Fermi-energy. This mathematical formulation coincides with our subject of study since we would like to see the effect of all the QD-SOA layers. QD-SOA is modeled in section 5 using rate equations (REs) model for the barrier layer which is assumed to be in the form of separate confinement heterostructure layer (SCH), wetting layer (WL), ground state (GS) and excited state (ES) in the QD region of Sb-based structures and then solved numerically. A 3 REs model is discussed first where the


Introduction
In 1961, it is suggested that stimulated emission can occur in semiconductors by recombination of carriers that are injected across a p-n junction. This is the backbone in the concept of semiconductor lasers and amplifiers [1]. This concept is connected with the "one dimensional electron in a box", a problem discussed by quantum mechanics text books, to generate a new field of confined semiconductor structures known thereafter as quantumwell (QW), quantum-wire (QWi) or quantum dot (QD) structures where the carriers are confined in semiconductor in one, two, or three directions, respectively [2]. Semiconductor optical amplifiers (SOAs) and lasers performance may be substantially improved by using the QD-SOAs characterized by a low threshold current density, high saturation power, broad gain bandwidth and week temperature dependence as compared to bulk and multiquantum well devices [3]. QD active region results from reducing the size of conventional (bulk) crystal to a nanometer size scale in all the crystal directions. This results in a complete quantization of energy states. The density of states becomes comparable to a delta-function. In order to cover losses of the waveguide of the SOA, QDs are then grown by a large number of dots grown on a WL which is already in the form of QW layer, see Fig. 1. In this chapter, we examine the effect of changing composition of the layers constructing the Antimony (Sb)-based quantum dot semiconductor optical amplifiers (QD-SOAs). We startoff in sections 2-3 a general description of QD-SOA and the importance of SOAs, especially Sb-based SOAs, in solving problems. Manufacture imperfection in the shape of QDs are taken into account in the gain calculations in section 4, where they are represented by an inhomogeneous function. Along with the gain description in this section, a global Fermifunction is used where the states in the barrier layer, wetting layer (WL) and ground and excited states of the dots are included in the calculations of Fermi-energy. This mathematical formulation coincides with our subject of study since we would like to see the effect of all the QD-SOA layers. QD-SOA is modeled in section 5 using rate equations (REs) model for the barrier layer which is assumed to be in the form of separate confinement heterostructure layer (SCH), wetting layer (WL), ground state (GS) and excited state (ES) in the QD region of Sb-based structures and then solved numerically. A 3 REs model is discussed first where the

Semiconductor optical amplifier
Transmitters, receivers, electronic switches and routers limits the capacity of optical communication systems that can exceed 10 Tb/s by the speed of their electronic components [3]. Optical fibers suffers from attenuation (which limits transmission distance) and dispersion (leading to an increase in the system bit error rate, BER). 3R method (reshapingretiming-retransmission) is then used to regenerate the optical signal in optical fibers. It has a number of disadvantages. This involves low optical and electrical transparencies in addition to network unreliability. These limitations may be overcome by using SOAs. The

Sb-based QD-SOA
Mid-to-far-infrared wavelength range (2000-30000 nm) has a variety of applications including remote sensing, medical diagnostic, free-space optical communications, atmospheric pollution monitoring, chemical sensing, thermal imaging, high power and midinfrared light sources [6]. Although HgCdTe material is already used to build devices working at these ranges, it suffers from slow response, high-power dissipation and undesired noise [7]. III-Sb based devices appear as a counterpart, since it is offering several advantages such as wide spectral range, low electron effective mass and high mobility at room temperature [8]. Although its research begins since the 70 s of the past century, Sbbased devices still need a lot of length of study especially when the active region is in the form of QD nanostructures. Using QD active region in the Sb-based devices adds the advantages resulting from QD nanostructures to that expected from the Sb-based infrared devices. QD-SOAs can be used in the building blocks of many photonic devices. Its response is controlled by their dynamic behavior which can be studied by the use of a three-level system of REs to describe QD-SOA states for ground state (GS), excited state (ES) and WL state, which is considered as a QW and then can be approximated by a single state. While quasi-Fermi levels in the conduction and valence subbands are typically calculated from the surface carrier density per QD layer, now in some recent researches, both WL and barrier layer are included [9]. The barrier layer considered here is assumed to be in the form of a separate confinement heterostructure layer (SCH). This layer used in semiconductor structures to assure carrier confinement, where the SCH layer must have a high bandgap compared to other layers constructing the semiconductor device. Accordingly, the barrier layer becomes included in the gain calculations. Due to this, we begin in our laboratory www.intechopen.com Selected Topics on Optical Amplifiers in Present Scenario 156 (NNRL) a series of studies for some of the characteristics of Sb-based QD devices [10]. Here, the carrier dynamics in III-Sb based QD-SOAs are considered using four-level REs system. QD, WL and SCH barrier compositions are examined to specify their characteristics. The results then, compared with those of three-level REs show the importance of including the barrier layer in the QD SOAs calculations. Because of the much larger effective mass of holes and lower quantization energies of the QD levels in the valence band, electrons behavior limits the carrier dynamics while holes in the valence band are assumed to be in quasithermal equilibrium at all times [11]. Thus, we determine carrier dynamics here by the relaxation of electrons in the conduction band only.   is the envelope function overlap between the QD electron and hole states.

Three rate equations model
QD-SOA characteristics can be studied using REs system constructed from three equations which describe the dynamics in WL, GS and ES of QD layer. The QD inhomogeneity due to www.intechopen.com Selected Topics on Optical Amplifiers in Present Scenario 158 fabrication imperfections is introduced through gain as discussed in section 3 above. Examples of three REs system can be found in [11,14]. In the three REs system, carriers are assumed to be injected with current density, J, in WL where the barrier layer effect is neglected. In WL, carriers are recombining at a rate ( (1 ) (1 ) 22 2 1 1 2 where q is the electron charge, L w is the effective thickness of the active layer, Г is the optical confinement factor and N Q is the surface density of QDs. The average signal photon density S av is given by [14] max   gPn si ng S av wLD L g c pw (9) P in is the input signal power to the SOA, n g is the group refractive index, ћ is the normalized Plank's constant, w p is the peak frequency, D is the strip width, L is the cavity length, c is the free space light speed, g max is the peak material gain taken at the peak frequency. The input signal wavelength is assumed to be injected to the QD SOA at a peak wavelength of GS for each structure. The single-pass gain of the structure is given by [11]   where α int is the loss coefficient. Although WL is assumed here to receive carriers by current injection or by escapes from QD ES, but the three RE models consider it as a common reservoir [10,15]. From the architecture of these three RE model, ES works as a common reservoir since it receives carriers from the states above and below it (WL and GS, respectively). So, ES is referred to as a reservoir for GS [8]. This is contradicted with the experimentally evidenced that carriers stay long in WL and short in ES [16]. Due to this and other reasons, discussed below, four REs system is more appropriate.

Four rate equations model
Four REs system is used here to study Sb-based QD-SOAs and is built depending on the energy band diagram considered in Fig. 2. The advantages of using four rate equations system are twofold: 1) including of SCH barrier layer dynamics which is important in practice and 2) this RE model is more appropriate with the use of global Fermi-energy in the gain calculations which is described in section 2. In this four REs system, carriers are assumed to be injected with current density, J, in the SCH barrier layer then recombine at a rate (

Sb-based QD-SOA structures
Sb-based QD semiconductors suffer from the difficulty of QD growth methods, like the selfassembled growth method, due to kinetic effects. Sb-based QD crystals growth hinders due to large mismatch (>7 or 8%) with GaAs semiconductors. Now it is possible to grow GaSb QDs on GaAs substrate [18]. In addition, an interfacial misfit array method can be used now to grow Sb-based layers on a GaAs platform although the (8%) lattice mismatch between GaAs and GaSb [19]. Accordingly, we choose a large number of structures, it is classified here in a five types, as in Table 1. In these structures, we change the composition of QD ( structure Nos. 1, 4 and 5), WL (No. 2) or barrier layer (No. 3). This is to examine the effect of these layers on the QD-SOA and also to specify the possible spectral ranges in the Sb-based QD structures. In all of these structures there is a lattice mismatch between layers (dot and WL or WL and barrier) not exceeds (6%) and in some cases its value is very small. This is to intend these structures to the ease of QD growing. QD energy levels are calculated using parabolic band quantumdisc model [20] where the dots are assumed to be in the form of a disc with radius (14 ) nm and height (2) hn m  , unless states otherwise. Quantum well WL thickness is taken as (9 nm). The accuracy of the quantum disc model is checked by a comparison with the experimental data and numerical methods [20,21]. The parameters used in the calculations of Sb-based structures are stated in Tables 2 and 3. Thus, it is adequate to calculate energy levels without time consumption. An example of QD energy levels calculated using parabolic band model is shown in Fig. 3 for InAs 0.1 Sb 0.9 /GaAs .1 Sb .9 /Al .1 Ga .9 As QD-SOA. In the calculation of quasi-Fermi energy Eqs. (3) and (4) are used. Gain is then calculated, using Eq.1, and its peak value and peak wavelength for each structure is specified.

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The   Table 3. The relation used to calculate structure parameters (bandgap E g and effective mass m i ). Note that the subscript (i) with m i refers to conduction or valence band effective masses [22,23].

QD-SOA gain
The REs system are solved numerically to see the gain change with input power P in , and to examine the dynamic effects. The parameters used in the numerical calculations are listed in Table 4. Fig. 4 shows input power-gain curves for Sb-based QD-SOAs studied. Fig. 4 (a) shows the effect of changing Sb mole fraction in the QD layer. The effect of WL, barrier and QD layers composition on gain is shown in Fig. 4 (b), (c), (d). Curve arrangement in these figures coincides with that in [12]. While both QD and WL composition are shown to give a respected changes in gain, the contribution due to changing Sb-composition in the barrier layer is minor as in Fig. 4 (c). The overall behavior of Fig. 4 is the same, gain saturates at low input power, then it declines at high input powers. This can be attributed to the carrier depletion in the QDs where the output power begins to increase. In Fig. 4 (d), InSb QD-SOA gives a higher gain than that obtained from GaSb QD-SOA. Comparing Fig. 4 (a) and (d) shows the effect of QD composition. To deal with composition effect, we must refer to the material confinement effect: the difference between bandgap in the dot and WL. The higher difference gives more carrier confinement in the QD layer. InSb is known as a lower band   Table 4. Parameters used in the calculations [11,17].
www.intechopen.com gap semiconductor. Thus, a higher gain is obtained from InSb than GaSb QD-SOAs due to the higher material confinement of the former (GaSb bandgap differ from InSb by ~0.56 eV). This reason is not enough to explain curves arrangement in Fig. 4 (a). Here gain curves are not arranged due to QD band gap where only very small differences result from varying Sb mole fraction in the QD region. Thus, in addition to material confinement, one must refer to the quantum confinement. Curves are arranged due to the gap (E c1 +E g +E v1 ) where E c1 , E v1 are the 1 st conduction and valence subbands and E g is the QD band gap. From Figs. 4 (a) and (d), the structure InSb/GaAs 0.7 Sb 0.3 /GaAs, is more appropriate for inline static amplification applications due to its maximum gain obtained and higher saturation power than InAs x Sb 1x /GaAs 0.1 Sb 0.9 /Al 0.1 Ga 0.9 As and GaSb QD-SOAs. The effect of QD size is shown in Fig. 4 (e) and (f). In Fig. 4 (e), it is shown that QDs with shorter height gives a higher gain. In Fig. 4 (f), QDs with longer radius gives higher gain.    Figure 5 (a) shows the carrier density dynamics in the barrier layer N s of InAs x Sb 1x /GaAs .1 Sb .9 /Al .1 Ga .9 As QD-SOAs at three x-mole fraction values (x=0.3, 0.7 and 0.9 Sbfraction). The energy subbands of the QD at each mole fraction are included through relaxation times (see Eqs. [15][16][17], thus a difference appears between curves. The carrier density in the barrier layer obtained here is in the range of (10 15 cm -3 ) which is near to the value of WL carrier density in [11] which assumes that carriers are injected directly to WL. N s curves are arranged according to the shallower Fermi energy level in the barrier layer of the QD SOA structure. According to this, the barrier layer in the structure with (0.9) Sb mole fraction filled earlier, since its quasi-Fermi energy is shallower, then the structure with (0.3) and finally, the structure with (0.7) Sb mole fraction. Fig. 5 (b) shows the carrier density in the WL (N W ). While N W obtained here is in the range of (10 12 cm -3 ), a (10 14 -10 15 cm -3 ) WL carrier density is obtained in [11] due to the neglect of barrier layer in that work. Here, N W curves are arranged according to the quasi-Fermi energy in their wetting layers. Fig. 5 (c) and (d) shows ES and GS occupation probabilities for the structures InAs x Sb 1x /GaAs .1 Sb .9 /Al .1 Ga .9 As at Sb mole fractions (0.3, 0.7 and 0.9) where the same probability is obtained in these structures for each ES and GS. This is due to very small differences between their relaxation times, i.e. the inclusion of QD subbands energies, not so much effects, see Eqs. (15)(16)(17). In Fig. 6 (a), the effect of changing mole-fraction in the WL of InAs .1 Sb .9 /GaAs x Sb 1-x /Al .1 Ga . 9 As QD structure is studied. WL is shown to be saturated at   higher carrier density for the structures with Sb mole fraction in WL (0.3 and 0.5). Here curves arrangement is appear according to shallower quasi-Fermi energy levels in the WL, where the effect of WL band gap energy is obvious in this arrangement. Fig. 6 (b) and (c) shows the dynamics of ES and GS occupation probabilities, respectively. Checking the parameters that arrange these curves shows that although the escape times to ES ( 12  and 2 w  ) are very short for the structure No. 2 with Sb mole fraction (0.7), it saturates after other mole fractions (0.3 and 0.5) in these structures. This can be explained if one follows the energy difference between QD ES and WL energy level where this difference is greater for the structure with (0.7) Sb mole fraction in the WL. Also for GS one must refer to the difference between QD GS subband and WL energy level. In Fig. 7, the WL dynamics are shown for In 0.1 AsSb 0.9 /GaAs 0.1. Sb 0.9 /Al x Ga 1-x As QD-SOAs. WL carrier density saturates at a higher value for the structure with (x=0.2) Al mole fraction in the barrier layer. Also the same reason for shallower quasi-Fermi energy in the WL can explain this arrangement. One can refer to effect of the main difference between structures here (energy gap of barrier layer E gb ) where a higher separation between N W curves is compared to the effect of WL energy gap (E gw ) as shown in Fig. 6 (a). Occupation probabilities in GS and ES coincides for both (x=0.2 and 0.4) structures and thus they are not drown. In Figs. (5)-(6), although the faster rate of transition between ES and GS, but they are not always the faster one reaching steady state. Both SCH barrier and WL get steady state faster although there is a longer rate of transition between the barrier and WL. This is because of the limited dynamics for these layers so they are in the steady state earlier. At all cases, the GS reaches steady state faster  www.intechopen.com than ES since the relaxation between ES to GS is very fast. Finally, a comparison is done between three and four REs system for QD-SOA as shown in Fig. 8 (a), (b) and (c) for WL, ES and GS respectively, where a three REs system is shown to be an overestimates the dynamics due to neglecting the effect of barrier layer. So, this layer must be included in the REs system used to study QD systems.  InAs 0.1 Sb 0.9 /GaAs 0.1 Sb 0.9 /Al 0.1 Ga 0.9 As Fig. 8. A comparison between three-and four-REs system for (a) WL carrier density, (b) ES and (c) GS occupation probabilities for InAs .1 Sb .9 /GaAs .1 Sb .9 /Al .1 Ga .9 As QD-SOA.

Conclusions
The progress towards SOAs, and then QD-SOAs, is reviewed. The importance of Sb-based structures is illuminated. To see the effect of QD-SOA layers, a REs for QD, WL and SCH layers are solved numerically. This make us specify gain-input power relation for Sb-based QD-SOAs and then examine layers dynamics. Both QD and SCH layers are shown to have an importance in the static characteristics due to their considerable effect in structure confinement. WL is shown to affect carrier dynamics especially in the ES and GS. Global quasi-Fermi energy (which included SCH, WL, QD ES and QD GS) is shown to be important in the explanation of results. At all cases, the inclusion of the SCH in the REs is shown to be essential.

Acknowledgment
We acknowledge our indebtedness to Prof. Mohammed Jasim Betti (Dept. of English, College of Education, Thi-Qar University) for proofreading the language of this work.