InP (311)B quantum dot lasers with different QD size

Accepted Manuscript Title: Investigation of Dynamical Characteristics and Modulation Response Function of InAs/InP (311)B Quantum Dot Lasers with Different QD Size Authors: Kaveh Kayhani, Esfandiar Rajaei PII: DOI: Reference:

S1569-4410(17)30094-9 http://dx.doi.org/doi:10.1016/j.photonics.2017.03.010 PNFA 585

To appear in:

Photonics and Nanostructures – Fundamentals and Applications

Received date: Revised date: Accepted date:

9-5-2016 13-3-2017 28-3-2017

Please cite this article as: Kaveh Kayhani, Esfandiar Rajaei, Investigation of Dynamical Characteristics and Modulation Response Function of InAs/InP (311)B Quantum Dot Lasers with Different QD Size, Photonics and Nanostructures - Fundamentals and Applicationshttp://dx.doi.org/10.1016/j.photonics.2017.03.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Investigation of Dynamical Characteristics and Modulation Response Function of InAs/InP (311)B Quantum Dot Lasers with Different QD Size Kaveh Kayhani and Esfandiar Rajaei

Department of Physics, University of Guilan, Rasht, Iran Highlights     

Increase in transition energy with reduction of size and the effect of diameter on separation of levels. The preferred size for 1.55 μm emission from GS level and ES level. For smaller diameter threshold gain is at lower current and saturation gain is higher. Resonance frequency and modulation bandwidth of GS level is larger for more height QDs but for ES level is larger for less height QDs. Diameter decrement leads to larger modulation bandwidth while threshold injection current is smaller.

Abstract We have investigated the effect of size of InAS/InP (311)B quantum dot (QD), both height an diameter, on electronic levels and hence transition energies through k.p model, with focus on application in QD lasers. Therefore the results have been included in dynamic analysis based on coupled differential rate equations, to obtain variation of laser properties such as gain, photon number and modulation bandwidth in accordance to different experimental outcomes. This demonstrates that larger modulation bandwidth of ground state is given by QDs with smaller diameter but larger height however for excited state larger modulation bandwidth is given by smaller diameter and smaller height.

Keywords:

Quantum Dot; Quantum Dot lasers; k.p; Band Structure; Rate Equations; Modulation Bandwidth.

1. Introduction The growth of Quantum Dot (QD) semiconductors with Stranski-Krastanow method attracts many attentions due to producing spatial confinements which results in separation of electronic levels of electrons and holes [1]. These artificial atoms have many applications in semiconductor based devices like QD lasers with special characteristic such as lower threshold current [2], less sensitive to temperature [3] and zero-chirped modulation bandwidth [4, 5] than Quantum Wells, in view of quantum confinement. Accordingly, QD laser is a good choice for high-speed light

source in optical communications, however two main issues have to be considered, the material and the size. As regards 1.55 μm wavelength is a requisite in long-haul optic fiber, reaching this wavelength is difficult with InAs/GaAs structure because of large band gap. Small lattice mismatch in InAs/InP QDs leads to small strain and thus small band gap, made them good candidate for 1.55 μm emission [6]. Nevertheless, small lattice mismatch construct large QDs with small surface density which brings about insufficient gain. High density array of QDs with limited size dispersion has been developed on high index substrate (311)B InP, reducing threshold current either [79]. The other issue is the roll of QD size that has been investigated in this paper. Literally, height and width of QD, are tuning parameters of electronic levels determining emission wavelength [10]. On the other hand, modulation band width, limits to 10-12 GHz [11], is determined by resonance frequency and damping factor, both depending on gain compression and the capture dynamics of carriers [11-13]. Indeed, gain compression depends on electronic levels and the capture time, throughout carriers relax from wetting layer (WL) into the dot levels and the relaxation time, carriers relax from excited state (ES) to ground state (GS) in the dot, depend on ratio of WL surface to surface covered by QDs, in another word QD diameter [13, 14]. According to experiment InAs QDs on high index (311)B InP substrate typically have 8 nm height and 35 nm diameter and surface density of QDs 5.5 × 1010 𝑐𝑚−2 [6]. The height of dot can be controlled by double cap (DC) growth, so 1.55 μm emission can be obtained [6, 15, 16]. Simultaneously the height dispersion has been reduced that narrows the line width broadening of optic emission [6, 11, 15]. In this method the height is approximately equal to first cap layer. However, DC method only varies the height of QDs, Caroff et al [17] represented that with carful control of As flux the growth of QDs with smaller size, typically 4.8 nm height and 24 nm diameter, is achievable. Arrays constructed with this method have higher density of QDs 16 × 1010 𝑐𝑚−2 with smaller dispersion of size that results in more narrow line width, 50meV compared to 60meV [17] in typical method. Indeed, in experimental investigations the crystallite size can be calculated with Scherrer equation From X-ray diffraction method either [18].

In this paper energy levels, dynamical characteristics and response function of modulation have been calculated and compared for two types of QDs with different growth procedure. Type A InAs dot growth on (311)B InP substrate with typical DC method [6, 15]. The diameter has been considered as 32 nm and the height has been determined by the height of first cap layer [6]. Type B is similar to type A with consideration the effect of controlled As flux during growth, thus the diameter is 24 nm [17]. For each type we considered samples with different first cap layer height, having samples with different QD heights, from 2 nm to 4 nm. The band gap energy in all samples is 0.47 eV. Theoretical methods for investigation of electronic levels of III-V systems include atomic models such as empirical pseudopotential method (EPM) [19, 20] and empirical tight-binding (ETB) [21, 22] and continuum models like k.p [6, 23]. Due to small lattice mismatch (about 3%) for this structure the k.p method has responsible accuracy and at the same time low computational cost and limited number of required known material parameters. Therefor eight-band k.p model with envelope function approximation has been applied and parameters were extracted from Refs [24-27]. Also the effect of strain distribution has been added to Hamiltonian according to Stier et al [25], however in experimental investigations strain could be calculated from XRD results using Williamson Hall equation [27]. Description of optical properties has been obtained by dynamic analysis which is based on numerical model consist of coupled differential rate equations model (REM) [11, 14]. A sinusoidal small-signal analysis allows us to extract modulation response function through these equations [11]. In this model the effect of gain compression and the ES lasing have been considered in the differential equations. In the rest of paper, in section 2 first we have introduced k.p model and the inclusion of envelope function approximation to make the model compatible for nanostructures. In the next subsection the results of k.p calculations have been represented for two types of QDs and the variation of transition energies has been discussed. In section 3 the first subsection represents REM and the included parameters then in second subsection first we have validated the model for steady dc current then the modulation response function has been obtained for a sinusoidal current. Discussions on variation of bandwidth and relaxation oscillation

frequency were represented in this subsection. Finally, results were summarized in the conclusion. 2. Electronic level with k.p model 2.1. k.p model for nanostructures As semiconductor nanostructures show significant quantum mechanical behavior the Schrödinger equation should be solved to obtain global electronic structure [28]. From mean field and adiabatic approximations (fixed atomic core) [29] singleparticle equation corresponding for all manybody interactions is written as 𝒑2 ℏ2 ( + 𝑉0 (𝒓) + 2 2 (𝛔 × 𝛁𝑉0(𝒓)). 𝒑) 𝝍𝑛𝒌 (𝒓) = 𝐸𝑛 (𝒌)𝝍𝑛𝒌 (𝒓) 2𝑚0 4𝑚0 𝑐

(1)

in which the relativistic correction term for spin-orbit coupling has been included. Considering Bloch eigenfunctions the single-particle Hamiltonian in Bloch coefficients bases is ′ (𝐻0 + 𝐻𝒌 + 𝐻𝒌.𝒑 + 𝐻𝑠.𝑜. + 𝐻𝑠.𝑜. )𝑢𝑛𝒌 (𝒓) = 𝐸𝑛 (𝒌)𝑢𝑛𝒌 (𝒓)

(2)

that 𝝍𝑛𝒌 (𝒓) = 𝑒 𝑖𝒌.𝒓𝑢𝑛𝒌 (𝒓) in which 𝑢𝑛𝒌 (𝒓) is a periodic Bloch coefficient 𝒑2 𝐻0 = + 𝑉0 (𝒓) 2𝑚0

(3)

𝐻𝒌 =

(4)

ℏ2 𝒌2 2𝑚0 ℏ 𝐻𝒌.𝒑 = 𝒌. 𝒑 𝑚0 ℏ2 (𝛔 × 𝛁𝑉0 (𝒓)). 𝒑 𝐻𝑠.𝑜. = 4𝑚02 𝑐 2

(5) (6)

In k.p approach, the Bloch coefficient usually expands in terms of complete set of Bloch coefficients around special k0 point in which the coefficients and eigenvalues have been considered to be known 𝑢𝑛,𝒌 (𝒓) = ∑ 𝑐𝑛,𝜇 (𝒌)𝑢𝜇,𝒌𝟎 (𝒓) 𝜇

(7)

that the eigenvalue equation for the expansion coefficients is ∑ Ĥ𝜈𝜇 (𝒌)𝑐𝑛,𝜈 (𝒌) = 𝐸𝑛 (𝒌)𝑐𝑛,𝜈 (𝒌) 𝜇

(8)

With Lowdin perturbation theory only limited number of bands and wavefunctions near this extrema point take part in the equation and the other bands were concluded as perturbation [30] Ĥ′𝜈𝜇 (𝒌) = Ĥ𝜈𝜇 (𝒌) + ∑ 𝛽

Ĥ𝜈𝛽 (𝒌)Ĥ𝛽𝜇 (𝒌) 𝐸𝜈 (𝒌0 ) − 𝐸𝛽 (𝒌0 )

(9)

where υ,μ run over bands lie close to Fermi energy. The finite dimension of Ĥ′𝜈𝜇 (𝒌), determines the type of multi-band k.p model. Matrix elements have been specified as empirical parameters that are fitted to match experimental properties of the bulk band structure. Different set of parameters were introduced by DresselhausKip-Kittel known as DKK parameters [31], Luttinger [32], Foreman [33] and others, also the relations between different set of parameters were well summarized in Ref [34]. Since, interruption of lattice periodicity by mesoscopic potential V(r) in nanostructures, the Bloch theory dose not valid anymore and the wavevector (k) is not a good quantum number anymore [28]. In envelope function approximation (EFA) wavefunctions are superposition of Bloch functions at extrema point [28, 35] 𝜓(𝒓) = ∑ 𝐹 (𝑛)(𝒓)𝜓𝑛,𝒌0 (𝒓)

(10)

𝑛

that expansion coefficients 𝐹 (𝑛) (𝒓) are called envelope functions. Through this approximation the k.p model for bulks become compatible for nanostructurs (𝜇) (𝒓) = 𝐸𝐹 (𝜈) (𝒓) ∑ Ĥ𝐸𝐹𝐴 𝜈𝜇 (𝒓) 𝐹 𝜇

(11)

Hamiltonian Ĥ𝐸𝐹𝐴 𝜈𝜇 (𝒓) corresponds to the bulk k.p Hamiltonian (8), Fourier transformed into real space (𝑘 → −𝑖∇) plus the external potential 𝑘.𝑝

Ĥ𝐸𝐹𝐴 𝜈𝜇 (𝒓) = Ĥ𝜈𝜇 (−𝑖𝜵) + 𝛿𝜈𝜇 𝑉(𝒓)

(12)

Furthermore, strain distribution effect in hetrostructures, developed by semiconductor materials with different lattice constant [36], has been included into the Hamiltonian according to continuum model improved by Stier [25]. The strain tensor components 𝜀𝑖𝑗 defined as 𝜀𝑖𝑗 =

1 (𝑒 + 𝑒𝑗𝑖 ) 2 𝑖𝑗

𝑒𝑖𝑗 =

𝜕𝑢𝑖 (𝑖, 𝑗 = 1. .3) 𝜕𝑥𝑗

(13)

the vector u(x) describes the displacement due to lattice deformations and 𝑒𝑖𝑗 are deformation tensors. Pursuant to mechanical equilibrium condition

𝜕𝜎𝑘𝑙 𝜕𝑥𝑙

= 0, (𝜎𝑘𝑙

denotes the Cauchy stress tensor), elastic energy E has been minimized with finite difference method to obtain strain tensor components. 𝐸=

1 ∫ 𝐶 𝜀 𝜀 𝑑𝑉 2 𝑉 𝑖𝑗𝑘𝑙 𝑖𝑗 𝑘𝑙

(14)

where, 𝐶𝑖𝑗𝑘𝑙 is fourth-ranked elastic tensor. Details about strain term in Hamiltonian which is obtained from strain tensor can be found in Ref [28]. Accordingly, we hired eight-band k.p model [37] with EFA approximation in cubic reference to investigate the effect of QD size on optical energy emission of InAs/InP (311)B structure. Basis vectors in reference system of layer are [011̅], [2̅33] and [311]. We performed nextnano++ simulation code [38] to deal with these calculations for two types of QDs offered in the introduction with different heights. The required parameters for the Hamiltonian were extracted from Refs [24-27]. Also we have chosen truncated cone shape for capped QDs in agreement with atomic force microscopy (AFM) and cross-section scanning tunneling microscopy (X-STM) results of experimental samples [6]. Also experimental measurements show 1 nm thickness for WL so in our calculations we consider 4 (ML) wetting layer [6].

2.2. Results of k.p calculation Probability density of first three levels of electron and hole have been calculated by k.p model were illustrated in fig 1 there are similar scheme for all included samples. Actually, this reveals the first twofold degenerate level has S-like symmetry and the next two levels, with insignificant separation, can be considered as one fourfold degenerate level with P-like symmetry which is true for both electron and hole levels analogous to previous jobs [6, 20, 37].

Fig. 1. electron and hole probability density for InAs/InP quantum dot. e0, e1, and e2 stands for three first electronic states and h0, h1, and h2 stands for three first hole states.

Thus, the GS transition energy has been considered as the difference of first electron energy and first hole energy and the ES transition energy considered as the difference between mean value of second and third hole energy and mean value of second and third electron energies. Schematic of energy levels with conduction and valance band edges, were demonstrated in fig 2.

Fig. 2. Electronic structure of the strained InAs 3 nm type A dots in the InP(311)B barrier. The electronic and hole levels and bandedges are represented. GS and ES transition energies are in good agreement with experimental results. z axis is the growth direction.

The transition energies and the differences between dot levels and lowest level of wetting layer were summarized in table 1. Validation of calculations has been made according to GS transition energy as yardstick. Most experimental results like Miska et al. [38] have reported emission wavelength for dot with approximately 32 nm diameter between 0.86 and 0.73 eV at 295 K. This range is for 2-4 nm QD height which the 3 nm height gives a continuous-wave photoluminescence spectrum centered at 0.8 eV (1.55 μm). Table 1. Exciton transition energies and difference between WL, ES and GS levels in QDs with different height and diameters.

QD HEIGHT

GS (eV)

ES (eV) 0.88 0.835 0.8 0.777 0.76 0.72

E(WL-ES) (eV) 0.12 0.14 0.158 0.171 0.18 0.196

E(WL-GS) (eV) 0.14 0.163 0.182 0.198 0.21 0.244

E(ES-GS) (eV) 0.019 0.022 0.024 0.027 0.03 0.047

2 nm-A 2.5 nm-A 3 nm-A 3.5 nm-A 4 nm-A 8.5 nm-A

0.852 0.804 0.77 0.742 0.721 0.663

2 nm-B 2.5 nm-B 3 nm-B 3.5 nm-B 4.5 nm-B

0.873 0.834 0.807 0.79 0.786

0.916 0.885 0.866 0.856 0.858

0.096 0.107 0.114 0.118 0.117

0.128 0.147 0.161 0.17 0.172

0.032 0.039 0.046 0.051 0.055

On the other hand, calculation detected this emission at 2.5 nm height for type A and at 3 nm height for type B. It should be noted that, in experiment the first InP capping layer thicknesses considered as QD height which is a little bit larger (2 ML) than the effective height of the QD, including in calculations. Reduction of QD height happened during As/P exchange [6, 15, 16].

Fig. 3. Variation of ground state and excited state exciton transition energy in Type A QDs as a function of QD height, solid line for GS and dash line for ES. The inset shows the results of type B.

Figure 3 demonstrates variation of transition energies with QD height for type A which is nonlinear for both GS and ES transitions. The inset graph is for type B with the same demeanor. The division between GS and ES transition has smooth variation with QD height; nevertheless, this division is larger in type B. It can be deduced that variation in height of QDs changes the depth of levels inside the QD, shallow levels for small height, while variation in diameter of QDs is responsible for separation of levels, more close levels in larger diameter.

3. Dynamic analysis

3.1. Rate Equations Model (REM)

In both type of QDs we have assumed that ensemble of QDs in active region interact through WL [41, 42]. The active region width is 5 μm and the height is about 24 nm consist of 6 layers of QDs with 4 nm spacer layer thickness in accordance with Miska et al. [43]. We have chosen optimum length of 2800 μm for type A cavity and optimum length of 300 μm for type B cavity with regards to output power diagram as function of length of cavity fig 5.

Because of charge neutrality in QDs, electron and hole levels have been considered as exiton energy stats for e-h pair [44, 45]. Owing to the fact that the time carriers transferred from barrier to WL is too short we have assumed carriers have been injected directly from contact to WL [44, 45]. After injection, carriers are captured 𝑊𝐿 from WL to ES level in the time 𝜏𝐸𝑆 and then relax from ES level to GS level in the 𝐸𝑆 𝑊𝐿 time 𝜏𝐺𝑆 . In this model direct channel from WL to GS with capture time 𝜏𝐺𝑆 have been considered. Beside this, escape of carriers from GS to ES, from ES to WL and from GS to WL due to thermal excitation have been added to equations, occurring ES 𝐺𝑆 in the times τGS ES , τWL and 𝜏𝑊𝐿 respectively, which follows Fermi distribution. Two other processes that alter the number of carriers effectively are spontaneous 𝑠𝑝𝑜𝑛 recombination from WL, ES and GS in the time 𝜏𝑊𝐿,𝐸𝑆,𝐺𝑆 and also stimulated emission from ES and GS at threshold of these levels.

Fig. 4. A schematic description of carrier dynamics model, including a direct relaxation and scape channel from WL to GS.

Considering these assumptions in the model, represented schematically in fig 4, we have five coupled differential equations (15)-(19), which have been solved with fourth order Runge-Kutta method.

dNwl dt

𝐼

𝑁𝐸𝑆

𝑒

𝐸𝑆 𝜏𝑊𝐿

= +

−

𝑁𝑊𝐿

𝑠𝑝𝑜𝑛 𝜏𝑊𝐿

− 𝑓𝐸𝑆

𝑁𝑊𝐿 𝑊𝐿 𝜏𝐸𝑆

− 𝑓𝐺𝑆

𝑁𝑊𝐿 𝑊𝐿 𝜏𝐺𝑆

+

𝑁𝐺𝑆 𝐺𝑆 𝜏𝑊𝐿

(15)

dNES dt

dNGS dt

dSES dt

= 𝑓𝐸𝑆

𝑁𝑊𝐿

= 𝑓𝐺𝑆

𝑁𝐸𝑆

𝑊𝐿 𝜏𝐸𝑆

𝐸𝑆 𝜏𝐺𝑆

= 𝛤𝑣𝑔 𝐾𝐸𝑆

+ 𝑓𝐸𝑆

𝑁𝐺𝑆

+ 𝑓𝐺𝑆

𝑁𝑊𝐿

𝐺𝑆 𝜏𝐸𝑆

𝑊𝐿 𝜏𝐺𝑆

−

1+𝜀𝐸𝑆 𝑆𝐸𝑆

− 𝑓𝐺𝑆

𝐸𝑆 𝜏𝑊𝐿

− 𝑓𝐸𝑆

2𝑁𝐸𝑆 −1) 𝐸𝑆 𝑁𝐷

𝑆𝐸𝑆 (𝜇

𝑁𝐸𝑆

−

𝑁𝐺𝑆 𝐺𝑆 𝜏𝐸𝑆

𝑆𝐸𝑆 𝜏𝑝

−

𝑁𝐸𝑆 𝐸𝑆 𝜏𝐺𝑆

−

𝑁𝐺𝑆

𝑠𝑝𝑜𝑛 𝜏𝑆𝑆

+ 𝛽𝑠𝑝

𝑁𝐸𝑆

𝑠𝑝𝑜𝑛

𝜏𝐸𝑆

− 𝛤𝑣𝑔 𝐾𝐸𝑆

− 𝛤𝑣𝑔 𝐾𝐸𝑆

𝑆𝐸𝑆 (

2𝑁𝐸𝑆 −1) 𝜇𝐸𝑆 𝑁𝐷

1+𝜀𝐸𝑆 𝑆𝐸𝑆

2𝑁𝐺𝑆 −1) 𝐺𝑆 𝑁𝐷

𝑆𝐺𝑆 (𝜇

1+𝜀𝐺𝑆 𝑆𝐺𝑆

−

𝑁𝐺𝑆

(16)

(17)

𝐺𝑆 𝜏𝑊𝐿

𝑁𝐸𝑆

(18)

𝑠𝑝𝑜𝑛

𝜏𝐸𝑆

2N SGS (μ GS − 1) 𝑆 dSGS 𝑁𝐺𝑆 𝐺𝑆 GS ND = 𝛤𝑣𝑔 𝐾𝐸𝑆 − + 𝛽𝑠𝑝 𝑠𝑝𝑜𝑛 dt 1 + εGS SGS 𝜏𝑝 𝜏𝐺𝑆

(19)

𝐾𝐸𝑆,𝐺𝑆 are gain parameter for GS and ES obtained through (20)-(21) 𝐾𝐸𝑆

2 2𝜋𝑒 2 ℏ × 𝜇𝐸𝑆 × 𝜉(𝑝𝑐𝑣 ) = 𝑐𝑛𝑟 𝜀0 𝑚02vd Γinhom EES

(20)

𝐾𝐺𝑆

2 2𝜋𝑒 2ℏ × 𝜇𝐺𝑆 × 𝜉(𝑝𝑐𝑣 ) = 2 𝑐𝑛𝑟 𝜀0 𝑚0 vd Γinhom EGS

(21)

Also, we have considered effect of gain compression in our simulations. The equations for GS and ES gain compression are 𝜀𝐺𝑆,𝐸𝑆 =

𝜖𝑚𝐺𝑆,𝐸𝑆 𝛤 𝑉𝑎

that 𝜖𝑚𝐺𝑆,𝐸𝑆 are

calculated from following relation 𝜖𝑚𝐺𝑆,𝐸𝑆

2 𝑒 2 𝑝𝑐𝑣 𝜏𝑝 = 4ℏ𝑛𝑟2 𝑚02𝜀0 𝐸𝐺𝑆,𝐸𝑆 𝛤ℎ𝑜𝑚

(22)

The probability occupations for ES and GS are obtained from 𝑓𝐺𝑆 =

𝑁𝐺𝑆 −1 𝜇𝐺𝑆 𝑁𝐷

𝑓𝐸𝑆 =

𝑁𝐸𝑆 −1 𝜇𝐸𝑆 𝑁𝐷

(23)

𝑁𝑊𝐿,𝐸𝑆,𝐺𝑆 are carrier numbers in wetting layer, ES level and GS level respectively. 𝑆𝐸𝑆,𝐺𝑆 are photon numbers of stimulated recombination from ES level and GS level. The value of capture and relaxation time have been taken from [46, 47] for type A 𝐸𝑆 𝑊𝐿 𝑊𝐿 𝑊𝐿 𝜏𝐺𝑆 =11 ps, 𝜏𝐸𝑆 =25 ps, and also 𝜏𝐺𝑆 = 𝜏𝐸𝑆 [45]. Due to change in the occupied area of WL surface by QDs for type B these times were recalculated according to τWL ES,GS = (σn vn

Nb −1 b

−1 ) and τES [13, 14]. Here b is the barrier GS = (σn vn n1 )

thickness, Nb is the surface density of QDs, σn is the cross-section that carriers are captured into a QD have been taken equal to cross-section of QDs, vn is the carrier thermal velocity and n1 the quantity characterizes the carrier thermal escape from a QD to the WL. By comparison, for type B the capture times are considered about 0.5 of value used in type A and the relaxation time is considered about 1.7 times of corresponding value in type A. Escape times from ES to WL, from GS to WL and from GS to ES are given by following relations 𝐸𝑊𝐿 −𝐸𝐸𝑆 4𝑁𝑏 = ×( )𝑒 𝐾𝑏 𝑇 2 𝑚𝑒 𝐾𝑏 𝑇𝜋ℏ 𝐸𝑊𝐿 −𝐸𝐺𝑆 2𝑁𝑏 𝑊𝐿 𝐺𝑆 𝜏𝑊𝐿 = 𝜏𝐺𝑆 × ( )𝑒 𝐾𝑏 𝑇 𝑚𝑒 𝐾𝑏 𝑇𝜋ℏ2 𝐺𝑆 1 𝐸𝐸𝑆𝐾−𝐸 𝐺𝑆 𝐸𝑆 𝜏𝐸𝑆 = 𝜏𝐺𝑆 × 𝑒 𝑏 𝑇 2 𝐸𝑆 𝜏𝑊𝐿

𝜏𝑝

𝑊𝐿 𝜏𝐸𝑆

(24)

(25) (26)

is the photon life time with following relation

𝜏𝑝

=

1 𝑐 1 1 𝑛𝑟 (𝛼𝑖 + 2𝐿𝑐𝑎 ln 𝑅1𝑅2)

(27)

c is light velocity, 𝑛𝑟 is refractive index, 𝛼𝑖 is internal modal loss, 𝑅1,2 are mirror reflectivity and 𝐿𝑐𝑎 is cavity length. Other parameters in rate equations were summarized in table 2.

Table 2. Values of laser parameters.

Symbols

𝐬𝐩𝐨𝐧 𝛕𝐖𝐋 𝐬𝐩𝐨𝐧 𝛕𝐄𝐒 𝐬𝐩𝐨𝐧 𝛕𝐆𝐒

𝐧𝐫 𝚪 𝛃𝐬𝐩 𝛂𝐢 𝐑𝟏 = 𝐑𝟐 𝛍𝐆𝐒 𝛍𝐄𝐒

Parameters spontaneous time of WL spontaneous time of ES spontaneous time of GS Refractive index Optical confinement factor

Values 500 ps 500 ps 1200 ps 3.27 0.06

Spontaneous emission factor

Internal modal loss Mirror reflectivity degeneracy of the GS level degeneracy of the ES level

6 cm−1 0.3 2 4

1 × 10−4

By this model of rate equations output power has been calculated verses cavity length for four different injected current which is illustrated in fig 5. It figure out that 2.8 mm length for type A and 0.3 mm length for type B, are minimum lengths for each type in which both GS and ES lasing could occur.

Fig. 5. Output power at several injected currents for 3 nm QD height array, a) for GS and ES transition in type A, b) for GS and ES transition in type B. It represents optimum cavity length for each type.

Modulation response can be obtained from a sinusoidal current modulation around threshold current therefore relations for injection current, carrier numbers and photon numbers are

𝐼 = 𝐼0 + 𝐼1 𝑒 𝑗𝑤𝑡 𝑁𝑊𝐿,𝐸𝑆,𝐺𝑆 = 𝑁𝑊𝐿,𝐸𝑆,𝐺𝑆 0 + 𝑁𝑊𝐿,𝐸𝑆,𝐺𝑆 1 𝑒 𝑗𝑤𝑡 𝑆𝐸𝑆,𝐺𝑆 = 𝑆𝐸𝑆,𝐺𝑆 0 + 𝑆𝐸𝑆,𝐺𝑆 1 𝑒 𝑗𝑤𝑡

(28) (29) (30)

which the zero indexes are values for constant current. Through inserting these relations in the Rate Equations, modulation response function (31) has been solved numerically [48] according to the altered rate equation 𝐻 (𝑤 ) =

𝑆(𝑤)/𝐼(𝑤) 𝑆(0)/𝐼(0)

(31)

3.2. Results

At first the model has been validated with properties of system such as gain and photon number, which were calculated numerically with constant dc current in rate equations. For this purpose gain as function of current were illustrated in fig 6. For GS level, comparison between inset fig 6(a) and inset fig 6(b) for type A and B respectively, shows threshold gain starts at smaller current when the diameter is smaller as expected because of lower size dispersion in type B. Same result can be concluded for ES level from fig 6(a) and fig 6(b). For both type A and B the GS threshold gain starts at grater current for smaller height however, ES threshold gain starts at almost same current for different height. On the other hand, saturation gain in type B is higher than type A due to smaller linewidth which is because of smaller size dispersion.

Fig. 6. Gain versus injected current for different QD height. a) ES gain of type A and the inset is GS gain. b) ES gain of type B and the inset is GS gain. It represents threshold gain at lower current and higher saturation gain in type B.

In Fig 7 variation of photon number with time for three deferent injected currents, represents the increase of oscillation frequency and steady-state photon number beside decrease of turn-on delay for higher currents, which is true for both types of QDs and both GS and ES levels. These results are in consonant with modulation response function fig 8, representing increase in resonance frequency and modulation band width with increase in current.

Fig. 7. Oscillation frequency properties at various injected currents. a) Photon number vs time for ES transition in type A, the inset is for GS transition. b) Photon number vs time for ES transition in type B, the inset is for GS transition.

Fig. 8. Modulation responses at various injected currents, a) for GS transition in type A, b) for GS transition in type B, c) for ES transition in type A, d) for ES transition in type B.

To deal with effect of size, photon number verses time has been plotted for different QD height in fig 9, which reveals the increase in oscillation frequency and photon number of steady state for GS level with increasing dot height. This behavior is observed for both type A and B in the inset fig 9(a) and the inset fig 9(b) respectively. However compare between these two diagrams displays higher steady-state photon number for greater diameter (type A) according to longer cavity and higher frequency for smaller diameters (type B) according to smaller capture time. Same diagram have been plotted for ES level fig 9(a) and fig 9(b) for type A and B respectively, which illustrates higher steady-state photon number for smaller dot height. However comparison between type A and B in this diagram is

same as GS level, longer cavity leads to higher steady-state photon number for type A and smaller capture time leads to higher frequency for type B.

Fig. 9. Oscillation frequency and steady-state photon number properties for different QD height. a) Photon number vs time for ES transition in type A, the inset is for GS transition. b) Photon number vs time for ES transition in type B, the inset is for GS transition.

Finally to compare modulation response function of both types of QDs, results of these calculations for 2.5 nm, 3 nm and 3.5 nm QD heights of each type were represented in fig 10 which is in good conformity with fig 9.

Fig. 10. Modulation responses for different QD height, a) for GS transition in type A and type B, b) for ES transition in type A and type B. It represents higher modulation bandwidth in type B.

The higher resonance frequency and modulation bandwidth for more height QDs are in conformity with higher relaxation oscillation frequency and higher steady-state photon number respectively, which is valid for GS level of both type fig 10 a. Inversely, for ES level, lower modulation band width for more height QDs approved by lower steady-state photon number were illustrated in fig 10 b. On the other hand for GS level, faster capture time in type B increments oscillation frequency that leads to higher resonance frequency than type A. indeed, greater modulation bandwidth observed in type B despite type A is subjection to higher steady-state photon number has been justified according to higher gain in type B fig 6. Similarly for ES level, greater modulation bandwidth in type B than type A has been justified with higher gain and faster captures time in type B.

4. Conclusion

Through this paper, we have investigated the effect of variations in size, as an experimental parameter in engineering QD lasers, on dynamic parameters and also the contribution of each variation. Therefore, energy levels for electron and hole inside InAs/InP (311)B QD have been calculated with 8×8 k.p method. Calculations have been run for QDs with 32 nm diameter and 24 nm diameter, and for each diameter different height 2, 2.5, 3 and 3.5 nm have been considered. Results show increase in transition energy with reduction of size and also 1.55 μm wavelength suitable for optic fiber, equivalent to 0.8 eV, is achievable for GS level of QD with 2.5 nm height and 32 nm diameter and QD with 3 nm height and 24 nm diameter. Likewise, ES level in QD with 3 nm height and 32 nm diameter has 0.8 eV transition energy either. Furthermore, dynamic analysis of carriers in the QD structure have been done with coupled rate equations through which laser properties like variation of gain with current, variation of photon number with time and small signal modulation response function have been calculated for different QD size. However it reveals the significant effect of diameter decrement which leads to decrease in size dispersion of QDs and increase in surface density that respectively results in decrease of inhomogeneous broadening and shorter capture time. Accordingly, QD arrays with smaller diameter have larger modulation bandwidth while threshold injection current is smaller. However modulation bandwidth of GS level is larger for more height QDs but for ES level it is larger for less height QDs.

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