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Multi-wavelength solution-processed quantum dot laser
Optics Communications 457 (2020) 124629
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Multi-wavelength solution-processed quantum dot laser Pegah Amini a , Samiye Matloub a ,∗, Ali Rostami b ,∗ a b
Quantum Photonics Research Lab., Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran Photonics and Nano-Crystals Research Lab., Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran
ARTICLE
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Keywords: Multi-wavelength Quantum dot laser Rate equations Size distribution Inhomogeneous broadening Solution Process
ABSTRACT In this paper, the multi-wavelength solution-processed quantum dot laser has been proposed by the superimposed quantum dots. To this end, the rate equation framework has been developed to model the simultaneous lasing in desired wavelengths by considering inhomogeneous broadening of energy level as a result of the size distribution of quantum dots and the homogeneous broadening due to the carries scattering inside quantum dots. Moreover, the solution process technology has been utilized in order to synthesize the quantum dots in various sizes according to the desired wavelengths. The simulation results show that simultaneous lasing has been realized by superimposition of two sizes of InGaAs quantum dots in a single cavity at two wavelengths of 1.31 μm and 1.55 μm which have special importance in telecommunication applications. Besides, the output power intensity at the desired wavelengths can be managed by controlling the density of quantum dots.
1. Introduction Nowadays, the semiconductor laser is a promise to high-speed data transmission and telecommunication applications, which are outstanding topics in technology. The multi-wavelength subject is introduced due to the importance of faster data transmission, especially at telecommunication wavelengths (i.e., 1.31 μm and 1.55 μm) [1–5]. The realization of multi-wavelength lasers is based on semiconductor quantum dot lasers (QD-lasers). The quantum confinement effect in low-dimension heterostructure like QDs modifies the semiconductors’ properties. Utilizing QDs with a delta-function-like density of states at the active region leads to improvement in QD-laser properties like lower threshold currents, higher optical gain, and high-speed operation [6–11]. The quantum dots categorization is done by their synthesize method as ‘‘epitaxial’’ (known as ‘‘self-assembled’’) or ‘‘colloidal’’. The selfassembled QDs are grown by ‘‘dry’’ method with epitaxial growth from the vapor phase, whereas the colloidal QDs are synthesized by chemical procedures in solution, which is called ‘‘wet’’ method [12]. The epitaxial method is expensive and needs a high-vacuum condition, while the cost-effective solution process is possible by a simple chemical system. Recently, solution-processed QD nanostructures have been promising candidates for optoelectronic devices [13,14]. In this procedure, the size of the QDs can be controlled by exploiting chemical synthesis methods. Generally, the speed of crystal growth can be controlled easily to obtain a precise size for synthesized QDs. Also, the size of nanoparticles can be controlled by some synthesis parameters such as concentration, temperature, pH, and the speed of rotation of solvent [15,16]. By adjusting the colloidal QDs’ size, their bandgap can
be tuned [12,13,17]. The colloidal QDs’ diameter range starts from 1 nm to 10 nm, and it is possible to synthesize nanoparticles with 0.1 nm size deviation [18,19]. Several different sizes of colloidal QDs with the solution process can be achieved by providing an appropriate experimental condition [17]. Preparing of InGaAs can be carried out from InCl3 , Ga(NO3 )3 and tris(trimethylsilyl)arsine (As[TMS]3 ) [20,21]. An inert gas like helium atmosphere and glove box or a Schlenk vacuum line is utilized to give the reaction. In order to the synthesis of InGaAs, As[TMS]3 and InCl3 can be mixed at room temperature in trioctylphosphine (TOP) and trioctylphosphine oxide (TOPO). In an inert atmosphere, the injecting of precursors should be done into hot (300 ◦ C) TOP. A common approach for dot size control is carried out by growth time and several subsequent injections after the initial nucleation. Usually, longer growth time and increased precursor injections lead to have larger dots. Normally, this synthesis method gives a broad distribution of QD sizes, but slow reaction speed and low concentration of input material, leads to obtain a narrow distribution of nanoparticles by using toluene/methanol [21]. Furthermore, to make a narrow distribution, the QDs of different sizes can be separated by the electrophoresis method. So, it is possible to obtain a narrow distribution by applying a standard synthesis method. As claimed by Hwang et al. [22], particles of 5 nm size can be separated through this method. In this area, the practical synthesis methods are progressing and obtaining a very high-resolution case. The simultaneous existence of different sizes of colloidal QDs in the active region of QD-laser, in better words, the simple condition of synthesis method of colloidal QDs is a promise in ease of access to multi-wavelength QD-laser.
∗ Corresponding authors. E-mail addresses: [email protected] (S. Matloub), [email protected] (A. Rostami).
https://doi.org/10.1016/j.optcom.2019.124629 Received 28 April 2019; Received in revised form 12 September 2019; Accepted 23 September 2019 Available online 5 October 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
P. Amini, S. Matloub and A. Rostami
Optics Communications 457 (2020) 124629
with a hole forms an exciton. We suppose that both of them are captured in pairs at GS of the QD, and the charge always holds in each QD. Also, the size of the QD is selected in a case that it has an only ground state. Based on reported researches, the rate equations are the most popular way to analyze QD-laser characteristics [8,27–29]. Here, we have developed these rate equations to achieve multi-wavelength QDlasers. As shown in Fig. 1a, the semiconductor QD-laser includes two different sizes of InGaAs QDs in the active layer, which is sandwiched by P-type and N-type AlGaAs as the carrier transition layers. Also, the P and N-type GaAs are chosen as cladding layers. Fig. 1b, illustrates the carrier transition performance in the energy band diagram. At first, the diffusion of the carriers occurs in the separated confinement heterostructure (SCH) layer, and then the carriers relax into the QD. Some of the relaxed carriers in the QDs recombine radiatively, and some recombine non-radiatively. In the radiative process, the photons are emitted from the ground state because of the stimulated emission. In Fig. 1b, the carrier number of SCH layer is 𝑁𝑠 and in the quantum dots, it is expressed by N. Also, the lifetime constants are defined as follows: 𝜏𝑐 is carrier diffusion time, 𝜏𝑒 is carrier re-excitation time, 𝜏sr and 𝜏𝑟 are carrier recombination time in the SCH layer and the QD, respectively, and 𝜏𝑝 is the photon lifetime. Here, we assume there are M groups of QDs corresponding to the M desired wavelengths, and each group is divided into (2𝑞 + 1) subgroups, due to QD size distribution during the solution process method. The ensemble of all subgroups equals with M × (2𝑞 + 1). The subgroups are selected exactly coincident with the cavity resonance modes. In other words, the subgroups depend on their resonant energy for the GS interband transition. The mode separation of each subgroup is described by energy width, which is denoted by 𝛥𝐸. Fig. 2 shows the separation of groups and subgroups.
According to reports, some of the research groups have worked on QD-lasers, and have suggested two-state lasing to achieve simultaneous lasing at two wavelengths [2,5,23]. In this case, the lasing occurs on the ground state (GS) and excited state (ES) at the same time. At first, the occurrence of lasing is on GS, and then by the increase of injected current, the ES starts to lase [1–7,23,24]. The cavity length plays a critical role in threshold current because by increasing the cavity length, the GS threshold current decreases, and the ES threshold current increases in reverse. Presence of ES in the lasing process leads to ultra-broad emission spectra [25], but when the lasing starts from ES, the output intensity of GS becomes constant and saturated [1–3,5,6,24]. This is because of the phonon bottleneck effect [23]. The main objective of this paper is designing and modeling of multi-wavelength QD-lasers by using several QDs with various radius values, in order to omit the existing reported problems of two-state QD-lasers. To this end, multi groups of QDs with various sizes are considered, each of which is related to a special wavelength. In this proposed design, due to the size distribution of QD groups during the solution process method, it is essential to presume inhomogeneous broadening (IHB) [8,9,26]. What is more, in this model, the homogeneous broadening (HB) is assumed at each subgroup of the QD groups. The homogeneous broadening is related to polarization dephasing and can be modeled by the effect of carrier-carrier and phonon-carrier scattering in QD-lasers [8,9]. As it is reported about analyzing the dynamic of carriers in [8], for the first time to our knowledge, in this paper, we provide the performance of multi-wavelength of the InGaAs/GaAs quantum dot laser, only by assuming the ground state as an energy level by solving the developed rate equations. In this model, the existence of two different sizes of the InGaAs/GaAs quantum dots in the active region leads to simultaneous lasing at both 1.31 μm and 1.55 μm wavelengths. Also, the QDs are isolated from each other with no correlation, due to the low coverage of the QDs in the active region. Achieving the different sizes of QDs to have multi-wavelength QD-laser is possible by simple and low-cost solution process method. Assuming both the HB and IHB and solving the rate equations, the characteristic of output power, gain, threshold current, and the turn-on delay time by changing QD coverage is studied, precisely.
𝛥𝐸 =
𝑐ℎ 2𝑛𝑟 𝐿𝑐𝑎𝑣
(1)
in that equation, 𝐿𝑐𝑎𝑣 is the length of the cavity, where c and h stand for the speed of light in vacuum and Planck’s constant, respectively. The 𝐿𝑐𝑎𝑣 is selected in a case that all of the main modes can resonate. The suitable cavity length is chosen according to the least common multiple between cavity lengths of wavelengths [30].
2. Proposed multi-wavelength QD-laser
𝐿𝑐𝑎𝑣 = 𝐿𝑐𝑎𝑣1 𝐿𝑐𝑎𝑣2 =
In the following, the theoretical model of multi-wavelength QDlasers is provided. As mentioned earlier, it is possible to achieve simultaneous two-state lasing from both GS and ES at two different wavelengths, but while starting the lasing from ES, the output intensity of GS decreases and then becomes saturated. To overcome saturating of the GS’s output intensity, as in prior works, in this presented model, the PIN heterostructure QD-laser with the QD active layer in the role of intrinsic layer, including various sizes of QDs is suggested which leads to multi-wavelength lasing. Here, the coverage of the QDs is chosen in a case that they are isolated and have no correlation between each other. The active layer is sandwiched between the P and N-type layers as the carrier transition layers. Also, due to the importance of the resonance of all the main modes, the cavity length is selected relevant to the appropriate case. Besides, the implementation of the different sizes of QDs in the active region of QD-laser is possible by solution process method. In this method, a cost-effective and straightforward chemical system provides controllable experimental condition to synthesize different sizes of QDs. However, the exact size of QDs cannot be attained because there are some insignificant limitations of synthesizing condition; due to this size distribution, the IHB is considered. In addition, the existence of carriercarrier and phonon-carrier scattering in QD-laser leads to HB of energy levels [8,9]. Primarily the performance of the QD-lasers is based on carrier and photon dynamics like capture and relaxation in the QD. An electron
𝑎1 𝜆1 𝑎2 𝜆2 𝑎𝜆 𝜆 = ( 1 )2 2 2𝑛𝑟 2𝑛𝑟 2𝑛𝑟
(2)
where 𝑛𝑟 is refractive index and 𝐿𝑐𝑎𝑣1 , 𝐿𝑐𝑎𝑣2 corresponds to the cavity length of each wavelength. The cavity length should be selected in a way that the coefficients a, 𝑎1 and 𝑎2 become integer. It is good to be mentioned that in this equation, only the value of each wavelength is considered and the calculated 𝐿𝑐𝑎𝑣 is expressed in μm. Due to solution process technology, the size of each group of QDs which corresponds to the desired wavelengths can be deviated from the central value of the radius, leading to the distribution of energy levels in each energy group. The inhomogeneous broadening of energy levels can be modeled by the Gaussian function [8] which is defined as ⎛ − (𝐸 − 𝐸 )2 ⎞ ( ) 𝑖,𝑗 𝑖 ⎟ 1 𝐺𝑖 𝐸𝑖,𝑗 = √ exp ⎜ ⎜ ⎟ 2𝜉02 2𝜋𝜉0 ⎝ ⎠
(3)
𝑗 = 1, 2, … , 2𝑞 + 1 According to the Gaussian function, the energy of the ith group and jth subgroup of the QD is given by 𝐸𝑖,𝑗 = 𝐸𝑖 − (𝑞 − 𝑗) 𝛥𝐸
(4)
The main mode of each group is defined by 𝐸𝑖 = 𝐸𝑖,𝑞 . The FWHM of IHB is 𝛤0 , and it is given by 𝛤0 = 2.35𝜉0 . The multi-wavelength QD-laser can be realized by superimposition of several groups of QDs. Hence, the 2
P. Amini, S. Matloub and A. Rostami
Optics Communications 457 (2020) 124629
Fig. 1. (A) The structure of QD-laser with two different sizes of InGaAs QDs in active region (R2 < R1 ), (B) The energy band diagram of the QD-laser-active region and the carrier relaxation process in the ground state of QD.
subgroup gives to the nth mode photons is given as 𝑔𝑚,𝑛,𝑖,𝑗 =
2 ) ( ) ( ) 2𝜋𝑒2 ℏ𝑁𝐷𝑖 ||𝑃𝑐𝑣 || ( 2𝑃𝑖,𝑗 − 1 × 𝐺 𝐸𝑖,𝑗 B𝑚,𝑛,𝑖,𝑗 𝐸𝑚,𝑛 − 𝐸𝑖,𝑗 𝐸𝑖 𝑐𝑛𝑟 𝜀0 𝑚2
(7)
0
where 𝑛𝑟 is the refractive index, e is the electron charge and 𝑁𝐷𝑖 is the density of QD at the ith group which is given by 𝜉𝑖 = 𝑁𝐷𝑖 𝑉𝐷𝑖 that 𝜉𝑖 is QD group’s coverage, and 𝑉𝐷𝑖 = 34 𝜋𝑅3𝑖 is the volume of a QD with a radius of 𝑅𝑖 . Determination of lasing wavelength comes possible by a change in the size of 𝑅𝑖 . It is good to be mentioned that different size of QDs with no correlation have been spatially isolated from each other when the HB is small in comparison with IHB of energy levels. So, the optical gain of lasing modes is independent of other modes. In other words, when the HB is comparable to IHB, not only the resonant QDs but also non-resonant QDs whose energy levels lie within the amount of HB are considered in the optical gain of lasing modes. The occupation probability in the ground state of the ith group and jth subgroup of the QD is in accordance with Pauli’s exclusion and is represented by
Fig. 2. Groups and subgroups of QD superimposition. The black solid indicates superimposition of multiple Gaussian profiles for various QD groups; also the HB is included in the diagram (red line).. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
𝑃𝑖,𝑗 =
𝑀 𝑀 ⎛ − (𝐸 − 𝐸 )2 ⎞ ∑ ( ) ∑ ( ) 𝑖,𝑗 𝑖 ⎟ 1 𝐺 𝐸𝑖,𝑗 = 𝐺𝑖 𝐸𝑖,𝑗 = √ exp ⎜ ⎜ ⎟ 2𝜉02 2𝜋𝜉0 𝑖=1 𝑖=1 ⎝ ⎠
|𝑃𝑐𝑣 |2 = |𝐼𝑐𝑣 |2 𝑀 2 | | | | 𝑏
(9)
The 𝐼𝑐𝑣 is the overlap integral between the envelope functions of a hole and an electron inside each group of QDs. 𝑀𝑏 is given by k.p theory with the first-order interaction between the conduction and valence bands. The transition matrix element is defined as follows [8] ( ) 𝑚20 𝐸𝑔 𝐸𝑔 + 𝛥 𝑀𝑏 = (10) 2𝛥 12𝑚∗𝑒 𝐸𝑔 + 3 here, 𝐸𝑔 and 𝛥 refer to the bandgap and spin–orbit interaction energy of QD material, respectively; also 𝑚∗𝑒 is the electron effective mass. According to defined relations, as mentioned above, the rate equations of multi-wavelength QD-lasers are expressed by
(5)
Moreover, in this model, the assumed HB has a Lorentz shape [8,31] as defined by ( ) ℏ𝛤𝐵 ∕𝜋 B𝑚,𝑛,𝑖,𝑗 𝐸𝑚,𝑛 − 𝐸𝑖,𝑗 = ( )2 ( )2 𝐸𝑚,𝑛 − 𝐸𝑖,𝑗 + ℏ𝛤𝐵 𝑖 = 1, 2, … , 𝑀, 𝑗 = 1, 2, … , 2𝑞 + 1
(8)
where 𝑁𝑖,𝑗 is the carrier number in a QD that belongs to the ith group and jth subgroup, 𝑉𝐴 is the active region’s volume. In Eq. (7), ||𝑃𝑐𝑣 || is the transition matrix element that is given as
modeling is developed by superimposition of Gaussian function which can be written as follows
𝑗 = 1, 2, … , 2𝑞 + 1
𝑁𝑖,𝑗 ( ) 2𝑁𝐷𝑖 𝑉𝐴 𝐺 𝐸𝑖,𝑗
(6)
According to B𝑚,𝑛,𝑖,𝑗 the FWHM is 2ℏ𝛤𝐵 in which 𝛤𝐵 is the polarization dephasing or scattering rate. In this equation, the homogeneous broadenings of all subgroups are assumed and as mentioned, the HB comes from carrier-carrier and phonon-carrier scattering. Here, 𝐸𝑚,𝑛 refers to the energy at mth mode of the group, and nth mode of the subgroup. Based on density matrix theory, the linear optical gain of the QD-laser which the ith QD group gives to the mth mode photons, and the jth QD
𝑀 2𝑞+1 ∑ ∑ 𝑁𝑖,𝑗 𝑁 𝑁 𝑑𝑁𝑠 𝐼 = − 𝑠 − 𝑠 + 𝑑𝑡 𝑒 𝜏𝑠𝑟 𝜏𝑒 𝜏𝑐 𝑖=1 𝑗=1
𝑑𝑁𝑖,𝑗 𝑑𝑡
=
(11)
𝑀 2𝑞+1 𝑁𝑖,𝑗 )( ) 𝑁𝑖,𝑗 𝑁𝑠 ( 𝑐𝛤 ∑ ∑ 𝐺 𝐸𝑖,𝑗 1 − 𝑃𝑖,𝑗 − − − 𝑔 𝑆 𝜏0 𝜏𝑟 𝜏𝑒 𝑛𝑟 𝑚=1 𝑛=1 𝑚,𝑛,𝑖,𝑗 𝑚,𝑛
(12) 3
P. Amini, S. Matloub and A. Rostami
Optics Communications 457 (2020) 124629
Fig. 3. (A) The potential energy along 𝑥-direction at constant y-position (black line) and the Eigen-energy of QD-array respect to conduction band edge. The blue and pink solid lines correspond to big and small QDs, respectively. (B) The normalized wave-function corresponding to Eigen-energy of big QDs and (C) small QDs. These schematics indicate the electron Eigen-energy of both small and big QD which is summed with hole Eigen-energy and band gap of InGaAs to achieve the resonant energies of 0.8 eV and 0.946 eV corresponding to the emission wavelength 1.55 μm and 1.31 μm, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 Parameters of two-wavelength QD-Laser. Description
Parameter
Value
Quantum dot radius Quantum dot radius Volume of a QD Optical confinement factor Mirrors reflectivity [8] Internal loss [8] Stripe width [8] The volume of the active region Cavity length Spontaneous emission coupling efficiency [8] Refractive index [8] The bulk semiconductor’s bandgap The effective mass of the electron Recombination lifetime of the carrier in quantum dot [8] Initial carrier relaxation lifetime [8] Central interband transition energy Central interband transition energy
𝑅1 (nm) 𝑅2 (nm) 𝑉𝐷𝑖 (m3 ) 𝛤 𝑟1 , 𝑟2 𝛼𝑖 (cm−1 ) 𝑑(μm) 𝑉𝐴 (m3 ) 𝐿𝑐𝑎𝑣 (μm) 𝛽 𝑛𝑟 𝐸𝑔 (eV) 𝑚∗𝑒 𝜏𝑟 (ns) 𝜏0 (ps) 𝐸𝑖,1 (eV) 𝐸𝑖,2 (eV)
2.76 1.96 4 𝜋𝑅3𝑖 3 6% 0.3, 0.9 600 10 1.62 × 10−16 812.2 10−4 3.5 0.42 0.26𝑚0 2.8 10 0.8 0.946
𝑑𝑆𝑚,𝑛 𝑑𝑡
=
𝛽𝑁𝑚,𝑛 𝜏𝑟
+
𝑀 2𝑞+1 𝑆𝑚,𝑛 𝑐𝛤 ∑ ∑ 𝑔 𝑆 − 𝑛𝑟 𝑖=1 𝑗=1 𝑚,𝑛,𝑖,𝑗 𝑚,𝑛 𝜏𝑝
Besides, the 𝜏𝑝 is the photon lifetime of the cavity, which is defined by [8] ( ) 1 1 𝑐 𝛼𝑖 + ln (15) 𝜏𝑝−1 = 𝑛𝑟 2𝐿𝑐𝑎𝑣 𝑟1 𝑟2 where 𝛼𝑖 is the internal loss and 𝑟1 , 𝑟2 are the cavity mirrors reflectivity. The laser’s output power from one of the mirrors is given by 𝑃𝑚,𝑛 =
𝑀 2𝑞+1 ∑ ∑ 1 ( ) ( ) 1 − 𝑃𝑖,𝑗 𝐺𝑖 𝐸𝑖,𝑗 𝜏 𝑖=1 𝑗=1 0
2𝑛𝑟 𝐿𝑐𝑎𝑣
ln
1 𝑟
(16)
where 𝐸𝑚,𝑛 is the emitted photon energy, and r is the mirror reflectivity, which can be substituted with 𝑟1 or 𝑟2 . 3. Simulations results 3.1. Two-wavelength QD-laser In this model, it is presumed that the active layer of the twowavelength laser contains two different sizes of QDs with no correlation. In order to acknowledge the isolation of QDs assumption, according to Fig. 3, the modal analysis is carried out to determinate the Eigen-energy and Eigen-function of the two different sizes of QDs array by solving the Schrödinger equation. According to calculations, the transition probability based on matrix element between both ground states of big and small QDs, are less than the other transitions rates. Fig. 3a demonstrates the GS energy levels of each QD for electrons in which the GS energy of big QDs equals to 0.27 eV while the GS energy of the small ones equals to 0.38 eV; the corresponding Eigen-functions of big and small QDs are illustrated in Fig. 3b and c, respectively. It is obvious that there is no correlation between them.
(13)
where I is the injected current, 𝑁𝑠 is the carrier number of SCH layer, and 𝑆𝑚,𝑛 is the photon number of mth mode of the group, and nth mode of the subgroup. The spontaneous emission coupling efficiency is defined by 𝛽. When the GS of QD is unoccupied, the initial carrier relaxation lifetime is 𝜏0 and 𝜏 𝑐 is the average carrier relaxation lifetime which is given as 𝜏 −1 𝑐 =
𝑐𝐸𝑚,𝑛 𝑆𝑚,𝑛
(14)
4
P. Amini, S. Matloub and A. Rostami
Optics Communications 457 (2020) 124629
Fig. 4. (A) The output power versus injected current for the central lasing mode at two wavelengths of 1.55 μm and 1.31 μm when the QD coverage of big and small QDs are 0.05 and the FWHM of HB and IHB are set to 2ℏ𝛤𝐵 = 20 meV and 𝛤0 = 20 meV, respectively. (B) The photon numbers characteristics of two wavelengths. (C) The light emission spectrum. (d) The optical gain spectrum. The injected current in (B), (C) and (D) is equal to I = 88 mA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Selecting the specific size of QDs and modal analysis of their array determines the Eigen-energies which allow choosing the supposed wavelength. So, as it is depicted in Fig. 2, the size distribution of the QDs group is defined by the Gaussian function. Here, the energy of the desired wavelengths is shown by Ei ; also in this paper, the two wavelengths 1.55 μm and 1.31 μm are assumed, each of which relates to the energies E1 and E2 that are described previously. Based on the developed model described in Section 2, the photon numbers, light emission spectra, gain, and output power have been calculated. In this direction, the rate equations are numerically solved based on fourth-order Rung-Kutta method [8] to achieve simultaneous lasing at 1.31 μm and 1.55 μm. In the proposed model, the re-excitation of carriers from the ground state to SCH layer is ignored, i.e., 𝜏e = ∞; also it is assumed that all of the carriers are injected into the SCH layer, i.e., 𝜏sr = ∞. Furthermore, two types of spherical shape InGaAs QD with the radii of 1.96 nm and 2.76 nm has been exploited. The solution process permits to achieve tunable QD sizes, and it offers to control the size and surface of QDs. The solution process
is a low-cost method to provide high-performance devices, which is a goal for improvement in telecommunication applications [14,32]. In this process, the size of QDs may have 4%–6% tolerances, which cause the broadening of origin energy. For this reason, the assumed inhomogeneous broadening is about 20 meV. Also, the HB is assumed 20 meV at room temperature [8]. The parameters of two-wavelength QD-laser, 1.31 μm and 1.55 μm, in simulation are given in Table 1. In this paper, the multi-wavelength model is investigated by assuming the two wavelengths of 1.31 μm and 1.55 μm. These wavelengths have special importance in telecommunication applications. The distance of about 240 nm, is an appropriate magnitude to the independent lasing for these wavelengths to prevent overlapping, while as it is reported in [23], at the presence of both GS and ES, the two wavelengths are chosen near to each other and their distance is about 100 nm due to the limitation of energy difference of GS and ES. Based on the rate equations described in Section 2, the output power intensity depends on the photon numbers, which can be controlled by injection current and the QD coverage at each wavelength. 5
P. Amini, S. Matloub and A. Rostami
Optics Communications 457 (2020) 124629
Fig. 5. (A) The output power as a function of injected current for big QDs relating to 1.55 μm radiation wavelength when its QD coverage is 𝜉1 = 0.05, 0.1 and 0.2 corresponding to blue, pink and green solid lines, respectively. The output power versus injected current of small QD (1.31 μm radiation wavelength) with 𝜉2 = 0.1 is shown as black solid line. (B) The output power as a function of injected current for small QDs relating to 1.31 μm radiation wavelength when its QD coverage is 𝜉2 = 0.05, 0.1 and 0.2 corresponding to blue, pink and green solid lines, respectively. The output power versus injected current of big QD with 𝜉1 = 0.1 is shown as black solid line. (C) The output power spectrum for different QD coverage of both big and small QDs which relates to 1.55 μm and 1.31 μm at different injected currents. The FWHM of HB and IHB are set to 2ℏ𝛤𝐵 = 20 meV and 𝛤0 = 20 meV, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Consequently, the same intensity of output power for two different wavelengths is attained at specific injection current and QD density. It is worthy of mention that single-mode lasing at injected current larger than threshold current, can be achieved at each wavelength when the FWHM of HB is comparable or equal to IHB [9]. Therefore, the output power as a function of injected currents for two-wavelength laser has been shown in Fig. 4a when the FWHM of HB and IHB are considered 20 meV. Also, the coverage of each QD group equals to 𝜉1 = 𝜉2= 5%. The blue solid curve relates to 1.55 μm and the pink solid one to 1.31 μm. Although, the output power corresponding to each wavelength is enhanced linearly by increasing the injected current, the threshold current and slope efficiency of them are different. The injected electrons in the active region tend to relaxing in big QDs whose density is smaller than small ones (see Eq. (14)). So, the population inversion is occurred in big QDs. For this reason, as shown in Fig. 4a, the threshold current for longer wavelength is smaller and for 1.55 μm and 1.31 μm is about 10 mA and 25 mA, respectively. Moreover, the slope efficiency of longer wavelength is decreased when the shorter one start lasing. Hence, at special injected current, the output powers of the two-wavelength laser are the same.
At injected current I = 88 mA, the photon numbers’ characteristics are illustrated in Fig. 4b for both small QDs (solid pink line) and big QDs (solid blue line) corresponding to the wavelengths of 1.31 μm and 1.55 μm, respectively. As shown in this figure, the turn-on delay time for big QDs is smaller than the small ones as expected. So, the population of photons in big QD groups reaches steady-state earlier than small ones. Additionally, Fig. 4c shows the output power spectrum for each QD group at I = 88 mA in both wavelengths, each of which has the same intensity peak and equals to 16 mW. When the HB is comparable with IHB, for specific injected currents, there are more QD subgroups lying within the HB of central subgroups (corresponds to the resonant mode of each QD groups). Consequently, the carriers emitting into the central mode of each group increase in comparison to the other modes, and the central modes become prominent. So, the single-like lasing mode can be observed. Finally, the optical gain spectrum at I = 88 mA is illustrated in Fig. 4d. The output power intensity corresponding to the wavelengths of 1.55 μm and 1.31 μm as a function of injected current for various QD coverages are demonstrated in Fig. 5a and b, respectively. At constant injected current, due to lack of carriers, the number of occupied big 6
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Fig. 6. (A) The turn-on delay time of small QDs relating to wavelength 1.31 μm versus the density of big QDs for different values of small QDs coverage. (B) The turn-on delay time of big QDs relating to wavelength 1.55 μm versus the density of small QDs for different value of big QDs coverage. (C) The output power of big QDs (black line) as function of big QDs coverage and the output power of small QDs (colorful lines) as function of big QDs coverage for different values of small QDs coverage. (D) The output power of small QDs (black line) as function of small QDs coverage and the output power of big QDs (colorful lines) as function of small QDs coverage for different values of big QDs coverage. The injected current is set to I = 200 mA and the FWHM of HB and IHB are set to 2ℏ𝛤𝐵 = 20 meV and 𝛤0 = 20 meV, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(small) QDs lying within the scope of central mode of big (small) QDs is declined when the density of big (small) QDs is increased; therefore, the corresponding output power intensity is decreased, and the threshold current is increased as illustrated in Fig. 5a (5b). In other words, by increasing the density of big (small) QDs, higher injection current is needed to achieve the threshold gain and start lasing. In all cases, the output power increases linearly. In each diagram of Fig. 5a and b, when the density of each group of QDs doubles, the threshold current doubles likewise. Also, at the same current, for different densities of QDs, the small density has high output intensity. Furthermore, by comparing Fig. 5a and b, it is revealed that the threshold current of small QDs is increased in comparison with big ones at the same QD coverage (corresponding to the higher density of small QDs), and the output power intensity is decreased according to the mentioned description. As Fig. 5a depicts, the slope efficiency for big and small QDs is different, in which the same value of output powers can be obtained at specific injection current for various QD coverage of big QDs. While
the density of the small QDs is constant, by increasing the density of big QDs, at lower injected current, the output powers in both wavelengths equal to each other. In other words, the output power curve of small QDs for 𝜉2 = 0.1 (the black curve), intersects the output power curve of big QDs by assuming the different densities (i.e. 𝜉1 = 0.1 and 𝜉1 = 0.2 corresponding the pink and green curves), at I = 201 mA and 127 mA, leading to observation of the same output power like the blue and pink curves in Fig. 5c. Similarly, the comparison of curves in Fig. 5b demonstrate while the density of big QDs is constant, increasing the coverage of small ones, leads to the equality in output powers of both wavelengths at higher injected current. In continuation to the explanation, Fig. 5c is in a good agreement with Fig. 5a and b. Herein figure, the output power is illustrated for the different density of QDs at different currents. As it is seen in this figure, by changing the QD coverage, the intensity of the output powers for different densities is changed. Also, at the specific current, for different densities of both QDs groups, 7
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Optics Communications 457 (2020) 124629
Fig. 7. (A) The output power versus injected current for the central lasing mode at three wavelengths of 1.3 μm, 1.4 μm and 1.55 μm when the QD coverage for all groups has been set to 0.05 and the FWHM of HB and IHB are set to 2ℏ𝛤𝐵 = 20 meV and 𝛤0 = 20 meV, respectively. (B) The photon numbers of all three wavelengths (C) Light emission spectrum (D) The optical gain spectrum. The injected current in (B), (C) and (D) is equal to 160 mA.
can be controlled by QD coverage which can be easily provided by the synthesis method. The output power as a function of the different QD coverages for each group is depicted in Fig. 6c and d. As shown in Fig. 6c (6d), the output power of big (small) QDs (the black curve) is independent of the density of small (big) ones, while it is varying by increasing the density of big (small) QDs, and the maximum output power is observed for the specific big (small) QD coverage. For example, at 𝜉1 = 0.05 the maximum output power for the wavelength of 1.55 μm is observed (Fig. 6c). On the other hand, the output power of small (big) QD is decreased by increasing the density of small (big) QD groups as a result of decline in the number of QDs which satisfied the threshold condition at constant injected current. Moreover, the output power of small (big) QDs is decreased by increasing the density of the big (small) QDs when the big (small) QDs coverage is small. However, it is saturated in high QDs coverage because of constant injected current, as above explanation. It should be mentioned that the same intensity of output power at both wavelengths can be achieved by choosing an appropriate QD coverage for small and big QDs. As an example, while the small and
the output intensities are equal with each other. In better words, by controlling QDs coverage and injected current, the same output power at two-wavelength can be observed. At certain injected currents, the output power and turn-on delay time for each wavelength are depended on the QD coverage of groups. In this direction, the turn-on delay time of small (big) QDs relating to the wavelength of 1.31 μm (1.55 μm) versus the density of big (small) QDs for the different density of small (big) QDs is illustrated in Fig. 6a (6b). As shown in Fig. 6a (6b), the turn-on delay time of small (big) QDs is slightly increased by variation of the big (small) QDs as a result of weak coupling between the subgroups of big (small) QD groups lying within the HB of subgroups inside small (big) QD groups. However, the turn-on delay time corresponding to small (big) QDs lasing mode is increased by rising the density of small (big) QD. As explained earlier, because of increasing the QDs density, the injected current cannot satisfy all the QDs threshold condition, and the lasing is started later. The solution process is a low-cost method to control the density of colloidal QDs during the synthesis process. Consequently, based on the results observed in Fig. 6c and d, the intensity of the output power 8
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big QD coverage are equal to 0.12 and 0.03, respectively (𝜉2 = 0.12, 𝜉1 = 0.03), the intensity of the output radiation for both wavelengths will be approximately 38 mW.
[5] V.V. Korenev, A.V. Savelyev, A.E. Zhukov, A.V. Omelchenko, M.V. Maximov, Effect of carrier dynamics and temperature on two-state lasing in semiconductor quantum dot lasers, Semiconductors 47 (2013) 1397–1404, http://dx.doi.org/10. 1134/S1063782613100151. [6] Z.R. Lv, H.M. Ji, S. Luo, F. Gao, F. Xu, D.H. Xiao, T. Yang, Dynamic characteristics of two-state lasing quantum dot lasers under large signal modulation, AIP Adv. 5 (2015) http://dx.doi.org/10.1063/1.4933194. [7] A. Röhm, B. Lingnau, K. Lüdge, Ground-state modulation-enhancement by twostate lasing in quantum-dot laser devices, Appl. Phys. Lett. 106 (2015) 1–6, http://dx.doi.org/10.1063/1.4921173. [8] M. Sugawara, K. Mukai, Y. Nakata, H. Ishikawa, A. Sakamoto, Effect of homogeneous linewidth of optical gain on lasing spectra in self-assembled 𝐼𝑛𝑥 𝐺𝑎1−𝑥 𝐴𝑠∕𝐺𝑎𝐴𝑠 quatum dot lasers, Phys. Rev. B 61 (2000) 7595. [9] D. 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3.2. Design of multi-wavelength QD-laser As mentioned earlier, the realization of multi-wavelength QD-laser is possible by the superimposition of various sizes of QDs at the active region of QD-laser. Each group of QD sizes leads to lasing at a specific wavelength. In order to have an improvement in our multi-wavelength QDlaser model, the results of three-wavelength QD-laser are investigated. The cavity length of this three-wavelength QD-laser is 1128.4 μm. The output power versus injected current for three wavelengths has been illustrated in Fig. 7a, while the FWHM of HB and IHB are considered 2ℏ𝛤𝐵 = 20 meV and 𝛤0 = 20 meV, respectively. Also, the QD coverage for all three groups of QDs is 𝜉 = 5%. The more injected current, the more output power intensity for all radiation wavelengths. As explained for two-wavelength QD-laser, the threshold current of longer wavelength is smaller than the others. Due to different slope efficiency for the three radiation wavelength, at I = 160 mA, the power of 1.3 μm and 1.4 μm radiations are equal. When the injected current is set to I = 160 mA, the photon number characteristics of three wavelengths of 1.3 μm, 1.4 μm, and 1.55 μm have been calculated based on the developed rate equations that is demonstrated in Fig. 7b. The photon number of longer wavelength is more than the shorter one at steadystate. In addition, the turn-on delay time of longer wavelength is less than the others as expected. Besides, the output power and the optical gain spectra are illustrated in Fig. 7c and d, respectively. It is noticed that the intensity of output power from shorter wavelength is small; however, the peak value of the optical gain for all wavelengths is somewhat larger than threshold gain which is equal for all of them. 4. Conclusion In brief, considering the solution process technology as a simple and cost-effective method to implement size distribution of QDs, the realization of the multi-wavelength QD-laser was investigated by superimposition of different QD sizes. Furthermore, simultaneous lasing of proposed multi-wavelength QD-laser was modeled based on the modified rate equations in which the inhomogeneous broadening as a consequence of size distribution of QDs and homogeneous broadening due to polarization dephasing were included. It was shown that twowavelength lasing of the proposed InGaAs/GaAs QD-laser at 1.31 μm and 1.55 μm was realized. Also, the effect of changing the QD’s density on the threshold current, turn-on delay time, and the output power intensity of each lasing wavelength was discussed in detail. Last but not least, in this work the design procedure for multi-wavelength QD-laser was proposed, and the solution process technology as the realization method was introduced. References [1] S. Meinecke, B. Lingnau, K. Lüdge, Increasing stability by two-state lasing in quantum-dot lasers with optical injection, Proc. SPIE 10098 (2017) 100980J, http://dx.doi.org/10.1117/12.2251791. [2] A. Markus, J.X. Chen, O. Gauthier-Lafaye, J.G. Provost, C. Paranthoën, A. Fiore, Impact of intraband relaxation on the performance of a quantum-dot laser, IEEE J. Sel. Top. Quantum Electron. 9 (2003) 1308–1314, http://dx.doi.org/10.1109/ JSTQE.2003.819494. [3] K. Lüdge, E. Schöll, Temperature dependent two-state lasing in quantum dot lasers, in: 2011 5th Rio La Plata Work. Laser Dyn. Nonlinear Photonics, LDNP 2011, 2011, http://dx.doi.org/10.1109/LDNP.2011.6162081. [4] M.V. Maximov, Y.M. Shernyakov, F.I. Zubov, A.E. Zhukov, N.Y. Gordeev, V.V. Korenev, A.V. Savelyev, D.A. Livshits, The influence of p-doping on two-state lasing in InAs/InGaAs quantum dot lasers, Semicond. Sci. Technol. 28 (2013) http://dx.doi.org/10.1088/0268-1242/28/10/105016. 9
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Optics Communications 457 (2020) 124629 [31] M. Sugawara, T. Akiyama, N. Hatori, Y. Nakata, H. Ebe, H. Ishikawa, Quantumdot semiconductor optical amplifiers for high-bit-rate signal processing up to 160 Gb s-1and a new scheme of 3R regenerators, Meas. Sci. Technol. 13 (2002) 1683–1691, http://dx.doi.org/10.1088/0957-0233/13/11/304. [32] V. Sukhovatkin, S. Musikhin, I. Gorelikov, S. Cauchi, L. Bakueva, E. Kumacheva, E.H. Sargent, Room-temperature amplified spontaneous emission at 1300 nm in solution-processed PbS quantum-dot films, Opt. Lett. 30 (2005) 171, http: //dx.doi.org/10.1364/OL.30.000171.
[29] Y. Kaptan, A. Röhm, B. Herzog, B. Lingnau, H. Schmeckebier, D. Arsenijević, V. Mikhelashvili, O. Schöps, M. Kolarczik, G. Eisenstein, D. Bimberg, U. Woggon, N. Owschimikow, K. Lüdge, Stability of quantum-dot excited-state laser emission under simultaneous ground-state perturbation, Appl. Phys. Lett. 105 (2014) 191105, http://dx.doi.org/10.1063/1.4901051. [30] G.P. Agrawal, N.K.S. Dutta, Semiconductor Lasers, Springer, US, 1993, http: //dx.doi.org/10.1007/978-1-4613-0481-4.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Multi-wavelength solution-processed quantum dot laser Pegah Amini a , Samiye Matloub a ,∗, Ali Rostami b ,∗ a b
Quantum Photonics Research Lab., Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran Photonics and Nano-Crystals Research Lab., Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran
ARTICLE
INFO
Keywords: Multi-wavelength Quantum dot laser Rate equations Size distribution Inhomogeneous broadening Solution Process
ABSTRACT In this paper, the multi-wavelength solution-processed quantum dot laser has been proposed by the superimposed quantum dots. To this end, the rate equation framework has been developed to model the simultaneous lasing in desired wavelengths by considering inhomogeneous broadening of energy level as a result of the size distribution of quantum dots and the homogeneous broadening due to the carries scattering inside quantum dots. Moreover, the solution process technology has been utilized in order to synthesize the quantum dots in various sizes according to the desired wavelengths. The simulation results show that simultaneous lasing has been realized by superimposition of two sizes of InGaAs quantum dots in a single cavity at two wavelengths of 1.31 μm and 1.55 μm which have special importance in telecommunication applications. Besides, the output power intensity at the desired wavelengths can be managed by controlling the density of quantum dots.
1. Introduction Nowadays, the semiconductor laser is a promise to high-speed data transmission and telecommunication applications, which are outstanding topics in technology. The multi-wavelength subject is introduced due to the importance of faster data transmission, especially at telecommunication wavelengths (i.e., 1.31 μm and 1.55 μm) [1–5]. The realization of multi-wavelength lasers is based on semiconductor quantum dot lasers (QD-lasers). The quantum confinement effect in low-dimension heterostructure like QDs modifies the semiconductors’ properties. Utilizing QDs with a delta-function-like density of states at the active region leads to improvement in QD-laser properties like lower threshold currents, higher optical gain, and high-speed operation [6–11]. The quantum dots categorization is done by their synthesize method as ‘‘epitaxial’’ (known as ‘‘self-assembled’’) or ‘‘colloidal’’. The selfassembled QDs are grown by ‘‘dry’’ method with epitaxial growth from the vapor phase, whereas the colloidal QDs are synthesized by chemical procedures in solution, which is called ‘‘wet’’ method [12]. The epitaxial method is expensive and needs a high-vacuum condition, while the cost-effective solution process is possible by a simple chemical system. Recently, solution-processed QD nanostructures have been promising candidates for optoelectronic devices [13,14]. In this procedure, the size of the QDs can be controlled by exploiting chemical synthesis methods. Generally, the speed of crystal growth can be controlled easily to obtain a precise size for synthesized QDs. Also, the size of nanoparticles can be controlled by some synthesis parameters such as concentration, temperature, pH, and the speed of rotation of solvent [15,16]. By adjusting the colloidal QDs’ size, their bandgap can
be tuned [12,13,17]. The colloidal QDs’ diameter range starts from 1 nm to 10 nm, and it is possible to synthesize nanoparticles with 0.1 nm size deviation [18,19]. Several different sizes of colloidal QDs with the solution process can be achieved by providing an appropriate experimental condition [17]. Preparing of InGaAs can be carried out from InCl3 , Ga(NO3 )3 and tris(trimethylsilyl)arsine (As[TMS]3 ) [20,21]. An inert gas like helium atmosphere and glove box or a Schlenk vacuum line is utilized to give the reaction. In order to the synthesis of InGaAs, As[TMS]3 and InCl3 can be mixed at room temperature in trioctylphosphine (TOP) and trioctylphosphine oxide (TOPO). In an inert atmosphere, the injecting of precursors should be done into hot (300 ◦ C) TOP. A common approach for dot size control is carried out by growth time and several subsequent injections after the initial nucleation. Usually, longer growth time and increased precursor injections lead to have larger dots. Normally, this synthesis method gives a broad distribution of QD sizes, but slow reaction speed and low concentration of input material, leads to obtain a narrow distribution of nanoparticles by using toluene/methanol [21]. Furthermore, to make a narrow distribution, the QDs of different sizes can be separated by the electrophoresis method. So, it is possible to obtain a narrow distribution by applying a standard synthesis method. As claimed by Hwang et al. [22], particles of 5 nm size can be separated through this method. In this area, the practical synthesis methods are progressing and obtaining a very high-resolution case. The simultaneous existence of different sizes of colloidal QDs in the active region of QD-laser, in better words, the simple condition of synthesis method of colloidal QDs is a promise in ease of access to multi-wavelength QD-laser.
∗ Corresponding authors. E-mail addresses: [email protected] (S. Matloub), [email protected] (A. Rostami).
https://doi.org/10.1016/j.optcom.2019.124629 Received 28 April 2019; Received in revised form 12 September 2019; Accepted 23 September 2019 Available online 5 October 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
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Optics Communications 457 (2020) 124629
with a hole forms an exciton. We suppose that both of them are captured in pairs at GS of the QD, and the charge always holds in each QD. Also, the size of the QD is selected in a case that it has an only ground state. Based on reported researches, the rate equations are the most popular way to analyze QD-laser characteristics [8,27–29]. Here, we have developed these rate equations to achieve multi-wavelength QDlasers. As shown in Fig. 1a, the semiconductor QD-laser includes two different sizes of InGaAs QDs in the active layer, which is sandwiched by P-type and N-type AlGaAs as the carrier transition layers. Also, the P and N-type GaAs are chosen as cladding layers. Fig. 1b, illustrates the carrier transition performance in the energy band diagram. At first, the diffusion of the carriers occurs in the separated confinement heterostructure (SCH) layer, and then the carriers relax into the QD. Some of the relaxed carriers in the QDs recombine radiatively, and some recombine non-radiatively. In the radiative process, the photons are emitted from the ground state because of the stimulated emission. In Fig. 1b, the carrier number of SCH layer is 𝑁𝑠 and in the quantum dots, it is expressed by N. Also, the lifetime constants are defined as follows: 𝜏𝑐 is carrier diffusion time, 𝜏𝑒 is carrier re-excitation time, 𝜏sr and 𝜏𝑟 are carrier recombination time in the SCH layer and the QD, respectively, and 𝜏𝑝 is the photon lifetime. Here, we assume there are M groups of QDs corresponding to the M desired wavelengths, and each group is divided into (2𝑞 + 1) subgroups, due to QD size distribution during the solution process method. The ensemble of all subgroups equals with M × (2𝑞 + 1). The subgroups are selected exactly coincident with the cavity resonance modes. In other words, the subgroups depend on their resonant energy for the GS interband transition. The mode separation of each subgroup is described by energy width, which is denoted by 𝛥𝐸. Fig. 2 shows the separation of groups and subgroups.
According to reports, some of the research groups have worked on QD-lasers, and have suggested two-state lasing to achieve simultaneous lasing at two wavelengths [2,5,23]. In this case, the lasing occurs on the ground state (GS) and excited state (ES) at the same time. At first, the occurrence of lasing is on GS, and then by the increase of injected current, the ES starts to lase [1–7,23,24]. The cavity length plays a critical role in threshold current because by increasing the cavity length, the GS threshold current decreases, and the ES threshold current increases in reverse. Presence of ES in the lasing process leads to ultra-broad emission spectra [25], but when the lasing starts from ES, the output intensity of GS becomes constant and saturated [1–3,5,6,24]. This is because of the phonon bottleneck effect [23]. The main objective of this paper is designing and modeling of multi-wavelength QD-lasers by using several QDs with various radius values, in order to omit the existing reported problems of two-state QD-lasers. To this end, multi groups of QDs with various sizes are considered, each of which is related to a special wavelength. In this proposed design, due to the size distribution of QD groups during the solution process method, it is essential to presume inhomogeneous broadening (IHB) [8,9,26]. What is more, in this model, the homogeneous broadening (HB) is assumed at each subgroup of the QD groups. The homogeneous broadening is related to polarization dephasing and can be modeled by the effect of carrier-carrier and phonon-carrier scattering in QD-lasers [8,9]. As it is reported about analyzing the dynamic of carriers in [8], for the first time to our knowledge, in this paper, we provide the performance of multi-wavelength of the InGaAs/GaAs quantum dot laser, only by assuming the ground state as an energy level by solving the developed rate equations. In this model, the existence of two different sizes of the InGaAs/GaAs quantum dots in the active region leads to simultaneous lasing at both 1.31 μm and 1.55 μm wavelengths. Also, the QDs are isolated from each other with no correlation, due to the low coverage of the QDs in the active region. Achieving the different sizes of QDs to have multi-wavelength QD-laser is possible by simple and low-cost solution process method. Assuming both the HB and IHB and solving the rate equations, the characteristic of output power, gain, threshold current, and the turn-on delay time by changing QD coverage is studied, precisely.
𝛥𝐸 =
𝑐ℎ 2𝑛𝑟 𝐿𝑐𝑎𝑣
(1)
in that equation, 𝐿𝑐𝑎𝑣 is the length of the cavity, where c and h stand for the speed of light in vacuum and Planck’s constant, respectively. The 𝐿𝑐𝑎𝑣 is selected in a case that all of the main modes can resonate. The suitable cavity length is chosen according to the least common multiple between cavity lengths of wavelengths [30].
2. Proposed multi-wavelength QD-laser
𝐿𝑐𝑎𝑣 = 𝐿𝑐𝑎𝑣1 𝐿𝑐𝑎𝑣2 =
In the following, the theoretical model of multi-wavelength QDlasers is provided. As mentioned earlier, it is possible to achieve simultaneous two-state lasing from both GS and ES at two different wavelengths, but while starting the lasing from ES, the output intensity of GS decreases and then becomes saturated. To overcome saturating of the GS’s output intensity, as in prior works, in this presented model, the PIN heterostructure QD-laser with the QD active layer in the role of intrinsic layer, including various sizes of QDs is suggested which leads to multi-wavelength lasing. Here, the coverage of the QDs is chosen in a case that they are isolated and have no correlation between each other. The active layer is sandwiched between the P and N-type layers as the carrier transition layers. Also, due to the importance of the resonance of all the main modes, the cavity length is selected relevant to the appropriate case. Besides, the implementation of the different sizes of QDs in the active region of QD-laser is possible by solution process method. In this method, a cost-effective and straightforward chemical system provides controllable experimental condition to synthesize different sizes of QDs. However, the exact size of QDs cannot be attained because there are some insignificant limitations of synthesizing condition; due to this size distribution, the IHB is considered. In addition, the existence of carriercarrier and phonon-carrier scattering in QD-laser leads to HB of energy levels [8,9]. Primarily the performance of the QD-lasers is based on carrier and photon dynamics like capture and relaxation in the QD. An electron
𝑎1 𝜆1 𝑎2 𝜆2 𝑎𝜆 𝜆 = ( 1 )2 2 2𝑛𝑟 2𝑛𝑟 2𝑛𝑟
(2)
where 𝑛𝑟 is refractive index and 𝐿𝑐𝑎𝑣1 , 𝐿𝑐𝑎𝑣2 corresponds to the cavity length of each wavelength. The cavity length should be selected in a way that the coefficients a, 𝑎1 and 𝑎2 become integer. It is good to be mentioned that in this equation, only the value of each wavelength is considered and the calculated 𝐿𝑐𝑎𝑣 is expressed in μm. Due to solution process technology, the size of each group of QDs which corresponds to the desired wavelengths can be deviated from the central value of the radius, leading to the distribution of energy levels in each energy group. The inhomogeneous broadening of energy levels can be modeled by the Gaussian function [8] which is defined as ⎛ − (𝐸 − 𝐸 )2 ⎞ ( ) 𝑖,𝑗 𝑖 ⎟ 1 𝐺𝑖 𝐸𝑖,𝑗 = √ exp ⎜ ⎜ ⎟ 2𝜉02 2𝜋𝜉0 ⎝ ⎠
(3)
𝑗 = 1, 2, … , 2𝑞 + 1 According to the Gaussian function, the energy of the ith group and jth subgroup of the QD is given by 𝐸𝑖,𝑗 = 𝐸𝑖 − (𝑞 − 𝑗) 𝛥𝐸
(4)
The main mode of each group is defined by 𝐸𝑖 = 𝐸𝑖,𝑞 . The FWHM of IHB is 𝛤0 , and it is given by 𝛤0 = 2.35𝜉0 . The multi-wavelength QD-laser can be realized by superimposition of several groups of QDs. Hence, the 2
P. Amini, S. Matloub and A. Rostami
Optics Communications 457 (2020) 124629
Fig. 1. (A) The structure of QD-laser with two different sizes of InGaAs QDs in active region (R2 < R1 ), (B) The energy band diagram of the QD-laser-active region and the carrier relaxation process in the ground state of QD.
subgroup gives to the nth mode photons is given as 𝑔𝑚,𝑛,𝑖,𝑗 =
2 ) ( ) ( ) 2𝜋𝑒2 ℏ𝑁𝐷𝑖 ||𝑃𝑐𝑣 || ( 2𝑃𝑖,𝑗 − 1 × 𝐺 𝐸𝑖,𝑗 B𝑚,𝑛,𝑖,𝑗 𝐸𝑚,𝑛 − 𝐸𝑖,𝑗 𝐸𝑖 𝑐𝑛𝑟 𝜀0 𝑚2
(7)
0
where 𝑛𝑟 is the refractive index, e is the electron charge and 𝑁𝐷𝑖 is the density of QD at the ith group which is given by 𝜉𝑖 = 𝑁𝐷𝑖 𝑉𝐷𝑖 that 𝜉𝑖 is QD group’s coverage, and 𝑉𝐷𝑖 = 34 𝜋𝑅3𝑖 is the volume of a QD with a radius of 𝑅𝑖 . Determination of lasing wavelength comes possible by a change in the size of 𝑅𝑖 . It is good to be mentioned that different size of QDs with no correlation have been spatially isolated from each other when the HB is small in comparison with IHB of energy levels. So, the optical gain of lasing modes is independent of other modes. In other words, when the HB is comparable to IHB, not only the resonant QDs but also non-resonant QDs whose energy levels lie within the amount of HB are considered in the optical gain of lasing modes. The occupation probability in the ground state of the ith group and jth subgroup of the QD is in accordance with Pauli’s exclusion and is represented by
Fig. 2. Groups and subgroups of QD superimposition. The black solid indicates superimposition of multiple Gaussian profiles for various QD groups; also the HB is included in the diagram (red line).. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
𝑃𝑖,𝑗 =
𝑀 𝑀 ⎛ − (𝐸 − 𝐸 )2 ⎞ ∑ ( ) ∑ ( ) 𝑖,𝑗 𝑖 ⎟ 1 𝐺 𝐸𝑖,𝑗 = 𝐺𝑖 𝐸𝑖,𝑗 = √ exp ⎜ ⎜ ⎟ 2𝜉02 2𝜋𝜉0 𝑖=1 𝑖=1 ⎝ ⎠
|𝑃𝑐𝑣 |2 = |𝐼𝑐𝑣 |2 𝑀 2 | | | | 𝑏
(9)
The 𝐼𝑐𝑣 is the overlap integral between the envelope functions of a hole and an electron inside each group of QDs. 𝑀𝑏 is given by k.p theory with the first-order interaction between the conduction and valence bands. The transition matrix element is defined as follows [8] ( ) 𝑚20 𝐸𝑔 𝐸𝑔 + 𝛥 𝑀𝑏 = (10) 2𝛥 12𝑚∗𝑒 𝐸𝑔 + 3 here, 𝐸𝑔 and 𝛥 refer to the bandgap and spin–orbit interaction energy of QD material, respectively; also 𝑚∗𝑒 is the electron effective mass. According to defined relations, as mentioned above, the rate equations of multi-wavelength QD-lasers are expressed by
(5)
Moreover, in this model, the assumed HB has a Lorentz shape [8,31] as defined by ( ) ℏ𝛤𝐵 ∕𝜋 B𝑚,𝑛,𝑖,𝑗 𝐸𝑚,𝑛 − 𝐸𝑖,𝑗 = ( )2 ( )2 𝐸𝑚,𝑛 − 𝐸𝑖,𝑗 + ℏ𝛤𝐵 𝑖 = 1, 2, … , 𝑀, 𝑗 = 1, 2, … , 2𝑞 + 1
(8)
where 𝑁𝑖,𝑗 is the carrier number in a QD that belongs to the ith group and jth subgroup, 𝑉𝐴 is the active region’s volume. In Eq. (7), ||𝑃𝑐𝑣 || is the transition matrix element that is given as
modeling is developed by superimposition of Gaussian function which can be written as follows
𝑗 = 1, 2, … , 2𝑞 + 1
𝑁𝑖,𝑗 ( ) 2𝑁𝐷𝑖 𝑉𝐴 𝐺 𝐸𝑖,𝑗
(6)
According to B𝑚,𝑛,𝑖,𝑗 the FWHM is 2ℏ𝛤𝐵 in which 𝛤𝐵 is the polarization dephasing or scattering rate. In this equation, the homogeneous broadenings of all subgroups are assumed and as mentioned, the HB comes from carrier-carrier and phonon-carrier scattering. Here, 𝐸𝑚,𝑛 refers to the energy at mth mode of the group, and nth mode of the subgroup. Based on density matrix theory, the linear optical gain of the QD-laser which the ith QD group gives to the mth mode photons, and the jth QD
𝑀 2𝑞+1 ∑ ∑ 𝑁𝑖,𝑗 𝑁 𝑁 𝑑𝑁𝑠 𝐼 = − 𝑠 − 𝑠 + 𝑑𝑡 𝑒 𝜏𝑠𝑟 𝜏𝑒 𝜏𝑐 𝑖=1 𝑗=1
𝑑𝑁𝑖,𝑗 𝑑𝑡
=
(11)
𝑀 2𝑞+1 𝑁𝑖,𝑗 )( ) 𝑁𝑖,𝑗 𝑁𝑠 ( 𝑐𝛤 ∑ ∑ 𝐺 𝐸𝑖,𝑗 1 − 𝑃𝑖,𝑗 − − − 𝑔 𝑆 𝜏0 𝜏𝑟 𝜏𝑒 𝑛𝑟 𝑚=1 𝑛=1 𝑚,𝑛,𝑖,𝑗 𝑚,𝑛
(12) 3
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Optics Communications 457 (2020) 124629
Fig. 3. (A) The potential energy along 𝑥-direction at constant y-position (black line) and the Eigen-energy of QD-array respect to conduction band edge. The blue and pink solid lines correspond to big and small QDs, respectively. (B) The normalized wave-function corresponding to Eigen-energy of big QDs and (C) small QDs. These schematics indicate the electron Eigen-energy of both small and big QD which is summed with hole Eigen-energy and band gap of InGaAs to achieve the resonant energies of 0.8 eV and 0.946 eV corresponding to the emission wavelength 1.55 μm and 1.31 μm, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 Parameters of two-wavelength QD-Laser. Description
Parameter
Value
Quantum dot radius Quantum dot radius Volume of a QD Optical confinement factor Mirrors reflectivity [8] Internal loss [8] Stripe width [8] The volume of the active region Cavity length Spontaneous emission coupling efficiency [8] Refractive index [8] The bulk semiconductor’s bandgap The effective mass of the electron Recombination lifetime of the carrier in quantum dot [8] Initial carrier relaxation lifetime [8] Central interband transition energy Central interband transition energy
𝑅1 (nm) 𝑅2 (nm) 𝑉𝐷𝑖 (m3 ) 𝛤 𝑟1 , 𝑟2 𝛼𝑖 (cm−1 ) 𝑑(μm) 𝑉𝐴 (m3 ) 𝐿𝑐𝑎𝑣 (μm) 𝛽 𝑛𝑟 𝐸𝑔 (eV) 𝑚∗𝑒 𝜏𝑟 (ns) 𝜏0 (ps) 𝐸𝑖,1 (eV) 𝐸𝑖,2 (eV)
2.76 1.96 4 𝜋𝑅3𝑖 3 6% 0.3, 0.9 600 10 1.62 × 10−16 812.2 10−4 3.5 0.42 0.26𝑚0 2.8 10 0.8 0.946
𝑑𝑆𝑚,𝑛 𝑑𝑡
=
𝛽𝑁𝑚,𝑛 𝜏𝑟
+
𝑀 2𝑞+1 𝑆𝑚,𝑛 𝑐𝛤 ∑ ∑ 𝑔 𝑆 − 𝑛𝑟 𝑖=1 𝑗=1 𝑚,𝑛,𝑖,𝑗 𝑚,𝑛 𝜏𝑝
Besides, the 𝜏𝑝 is the photon lifetime of the cavity, which is defined by [8] ( ) 1 1 𝑐 𝛼𝑖 + ln (15) 𝜏𝑝−1 = 𝑛𝑟 2𝐿𝑐𝑎𝑣 𝑟1 𝑟2 where 𝛼𝑖 is the internal loss and 𝑟1 , 𝑟2 are the cavity mirrors reflectivity. The laser’s output power from one of the mirrors is given by 𝑃𝑚,𝑛 =
𝑀 2𝑞+1 ∑ ∑ 1 ( ) ( ) 1 − 𝑃𝑖,𝑗 𝐺𝑖 𝐸𝑖,𝑗 𝜏 𝑖=1 𝑗=1 0
2𝑛𝑟 𝐿𝑐𝑎𝑣
ln
1 𝑟
(16)
where 𝐸𝑚,𝑛 is the emitted photon energy, and r is the mirror reflectivity, which can be substituted with 𝑟1 or 𝑟2 . 3. Simulations results 3.1. Two-wavelength QD-laser In this model, it is presumed that the active layer of the twowavelength laser contains two different sizes of QDs with no correlation. In order to acknowledge the isolation of QDs assumption, according to Fig. 3, the modal analysis is carried out to determinate the Eigen-energy and Eigen-function of the two different sizes of QDs array by solving the Schrödinger equation. According to calculations, the transition probability based on matrix element between both ground states of big and small QDs, are less than the other transitions rates. Fig. 3a demonstrates the GS energy levels of each QD for electrons in which the GS energy of big QDs equals to 0.27 eV while the GS energy of the small ones equals to 0.38 eV; the corresponding Eigen-functions of big and small QDs are illustrated in Fig. 3b and c, respectively. It is obvious that there is no correlation between them.
(13)
where I is the injected current, 𝑁𝑠 is the carrier number of SCH layer, and 𝑆𝑚,𝑛 is the photon number of mth mode of the group, and nth mode of the subgroup. The spontaneous emission coupling efficiency is defined by 𝛽. When the GS of QD is unoccupied, the initial carrier relaxation lifetime is 𝜏0 and 𝜏 𝑐 is the average carrier relaxation lifetime which is given as 𝜏 −1 𝑐 =
𝑐𝐸𝑚,𝑛 𝑆𝑚,𝑛
(14)
4
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Optics Communications 457 (2020) 124629
Fig. 4. (A) The output power versus injected current for the central lasing mode at two wavelengths of 1.55 μm and 1.31 μm when the QD coverage of big and small QDs are 0.05 and the FWHM of HB and IHB are set to 2ℏ𝛤𝐵 = 20 meV and 𝛤0 = 20 meV, respectively. (B) The photon numbers characteristics of two wavelengths. (C) The light emission spectrum. (d) The optical gain spectrum. The injected current in (B), (C) and (D) is equal to I = 88 mA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Selecting the specific size of QDs and modal analysis of their array determines the Eigen-energies which allow choosing the supposed wavelength. So, as it is depicted in Fig. 2, the size distribution of the QDs group is defined by the Gaussian function. Here, the energy of the desired wavelengths is shown by Ei ; also in this paper, the two wavelengths 1.55 μm and 1.31 μm are assumed, each of which relates to the energies E1 and E2 that are described previously. Based on the developed model described in Section 2, the photon numbers, light emission spectra, gain, and output power have been calculated. In this direction, the rate equations are numerically solved based on fourth-order Rung-Kutta method [8] to achieve simultaneous lasing at 1.31 μm and 1.55 μm. In the proposed model, the re-excitation of carriers from the ground state to SCH layer is ignored, i.e., 𝜏e = ∞; also it is assumed that all of the carriers are injected into the SCH layer, i.e., 𝜏sr = ∞. Furthermore, two types of spherical shape InGaAs QD with the radii of 1.96 nm and 2.76 nm has been exploited. The solution process permits to achieve tunable QD sizes, and it offers to control the size and surface of QDs. The solution process
is a low-cost method to provide high-performance devices, which is a goal for improvement in telecommunication applications [14,32]. In this process, the size of QDs may have 4%–6% tolerances, which cause the broadening of origin energy. For this reason, the assumed inhomogeneous broadening is about 20 meV. Also, the HB is assumed 20 meV at room temperature [8]. The parameters of two-wavelength QD-laser, 1.31 μm and 1.55 μm, in simulation are given in Table 1. In this paper, the multi-wavelength model is investigated by assuming the two wavelengths of 1.31 μm and 1.55 μm. These wavelengths have special importance in telecommunication applications. The distance of about 240 nm, is an appropriate magnitude to the independent lasing for these wavelengths to prevent overlapping, while as it is reported in [23], at the presence of both GS and ES, the two wavelengths are chosen near to each other and their distance is about 100 nm due to the limitation of energy difference of GS and ES. Based on the rate equations described in Section 2, the output power intensity depends on the photon numbers, which can be controlled by injection current and the QD coverage at each wavelength. 5
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Optics Communications 457 (2020) 124629
Fig. 5. (A) The output power as a function of injected current for big QDs relating to 1.55 μm radiation wavelength when its QD coverage is 𝜉1 = 0.05, 0.1 and 0.2 corresponding to blue, pink and green solid lines, respectively. The output power versus injected current of small QD (1.31 μm radiation wavelength) with 𝜉2 = 0.1 is shown as black solid line. (B) The output power as a function of injected current for small QDs relating to 1.31 μm radiation wavelength when its QD coverage is 𝜉2 = 0.05, 0.1 and 0.2 corresponding to blue, pink and green solid lines, respectively. The output power versus injected current of big QD with 𝜉1 = 0.1 is shown as black solid line. (C) The output power spectrum for different QD coverage of both big and small QDs which relates to 1.55 μm and 1.31 μm at different injected currents. The FWHM of HB and IHB are set to 2ℏ𝛤𝐵 = 20 meV and 𝛤0 = 20 meV, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Consequently, the same intensity of output power for two different wavelengths is attained at specific injection current and QD density. It is worthy of mention that single-mode lasing at injected current larger than threshold current, can be achieved at each wavelength when the FWHM of HB is comparable or equal to IHB [9]. Therefore, the output power as a function of injected currents for two-wavelength laser has been shown in Fig. 4a when the FWHM of HB and IHB are considered 20 meV. Also, the coverage of each QD group equals to 𝜉1 = 𝜉2= 5%. The blue solid curve relates to 1.55 μm and the pink solid one to 1.31 μm. Although, the output power corresponding to each wavelength is enhanced linearly by increasing the injected current, the threshold current and slope efficiency of them are different. The injected electrons in the active region tend to relaxing in big QDs whose density is smaller than small ones (see Eq. (14)). So, the population inversion is occurred in big QDs. For this reason, as shown in Fig. 4a, the threshold current for longer wavelength is smaller and for 1.55 μm and 1.31 μm is about 10 mA and 25 mA, respectively. Moreover, the slope efficiency of longer wavelength is decreased when the shorter one start lasing. Hence, at special injected current, the output powers of the two-wavelength laser are the same.
At injected current I = 88 mA, the photon numbers’ characteristics are illustrated in Fig. 4b for both small QDs (solid pink line) and big QDs (solid blue line) corresponding to the wavelengths of 1.31 μm and 1.55 μm, respectively. As shown in this figure, the turn-on delay time for big QDs is smaller than the small ones as expected. So, the population of photons in big QD groups reaches steady-state earlier than small ones. Additionally, Fig. 4c shows the output power spectrum for each QD group at I = 88 mA in both wavelengths, each of which has the same intensity peak and equals to 16 mW. When the HB is comparable with IHB, for specific injected currents, there are more QD subgroups lying within the HB of central subgroups (corresponds to the resonant mode of each QD groups). Consequently, the carriers emitting into the central mode of each group increase in comparison to the other modes, and the central modes become prominent. So, the single-like lasing mode can be observed. Finally, the optical gain spectrum at I = 88 mA is illustrated in Fig. 4d. The output power intensity corresponding to the wavelengths of 1.55 μm and 1.31 μm as a function of injected current for various QD coverages are demonstrated in Fig. 5a and b, respectively. At constant injected current, due to lack of carriers, the number of occupied big 6
P. Amini, S. Matloub and A. Rostami
Optics Communications 457 (2020) 124629
Fig. 6. (A) The turn-on delay time of small QDs relating to wavelength 1.31 μm versus the density of big QDs for different values of small QDs coverage. (B) The turn-on delay time of big QDs relating to wavelength 1.55 μm versus the density of small QDs for different value of big QDs coverage. (C) The output power of big QDs (black line) as function of big QDs coverage and the output power of small QDs (colorful lines) as function of big QDs coverage for different values of small QDs coverage. (D) The output power of small QDs (black line) as function of small QDs coverage and the output power of big QDs (colorful lines) as function of small QDs coverage for different values of big QDs coverage. The injected current is set to I = 200 mA and the FWHM of HB and IHB are set to 2ℏ𝛤𝐵 = 20 meV and 𝛤0 = 20 meV, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(small) QDs lying within the scope of central mode of big (small) QDs is declined when the density of big (small) QDs is increased; therefore, the corresponding output power intensity is decreased, and the threshold current is increased as illustrated in Fig. 5a (5b). In other words, by increasing the density of big (small) QDs, higher injection current is needed to achieve the threshold gain and start lasing. In all cases, the output power increases linearly. In each diagram of Fig. 5a and b, when the density of each group of QDs doubles, the threshold current doubles likewise. Also, at the same current, for different densities of QDs, the small density has high output intensity. Furthermore, by comparing Fig. 5a and b, it is revealed that the threshold current of small QDs is increased in comparison with big ones at the same QD coverage (corresponding to the higher density of small QDs), and the output power intensity is decreased according to the mentioned description. As Fig. 5a depicts, the slope efficiency for big and small QDs is different, in which the same value of output powers can be obtained at specific injection current for various QD coverage of big QDs. While
the density of the small QDs is constant, by increasing the density of big QDs, at lower injected current, the output powers in both wavelengths equal to each other. In other words, the output power curve of small QDs for 𝜉2 = 0.1 (the black curve), intersects the output power curve of big QDs by assuming the different densities (i.e. 𝜉1 = 0.1 and 𝜉1 = 0.2 corresponding the pink and green curves), at I = 201 mA and 127 mA, leading to observation of the same output power like the blue and pink curves in Fig. 5c. Similarly, the comparison of curves in Fig. 5b demonstrate while the density of big QDs is constant, increasing the coverage of small ones, leads to the equality in output powers of both wavelengths at higher injected current. In continuation to the explanation, Fig. 5c is in a good agreement with Fig. 5a and b. Herein figure, the output power is illustrated for the different density of QDs at different currents. As it is seen in this figure, by changing the QD coverage, the intensity of the output powers for different densities is changed. Also, at the specific current, for different densities of both QDs groups, 7
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Optics Communications 457 (2020) 124629
Fig. 7. (A) The output power versus injected current for the central lasing mode at three wavelengths of 1.3 μm, 1.4 μm and 1.55 μm when the QD coverage for all groups has been set to 0.05 and the FWHM of HB and IHB are set to 2ℏ𝛤𝐵 = 20 meV and 𝛤0 = 20 meV, respectively. (B) The photon numbers of all three wavelengths (C) Light emission spectrum (D) The optical gain spectrum. The injected current in (B), (C) and (D) is equal to 160 mA.
can be controlled by QD coverage which can be easily provided by the synthesis method. The output power as a function of the different QD coverages for each group is depicted in Fig. 6c and d. As shown in Fig. 6c (6d), the output power of big (small) QDs (the black curve) is independent of the density of small (big) ones, while it is varying by increasing the density of big (small) QDs, and the maximum output power is observed for the specific big (small) QD coverage. For example, at 𝜉1 = 0.05 the maximum output power for the wavelength of 1.55 μm is observed (Fig. 6c). On the other hand, the output power of small (big) QD is decreased by increasing the density of small (big) QD groups as a result of decline in the number of QDs which satisfied the threshold condition at constant injected current. Moreover, the output power of small (big) QDs is decreased by increasing the density of the big (small) QDs when the big (small) QDs coverage is small. However, it is saturated in high QDs coverage because of constant injected current, as above explanation. It should be mentioned that the same intensity of output power at both wavelengths can be achieved by choosing an appropriate QD coverage for small and big QDs. As an example, while the small and
the output intensities are equal with each other. In better words, by controlling QDs coverage and injected current, the same output power at two-wavelength can be observed. At certain injected currents, the output power and turn-on delay time for each wavelength are depended on the QD coverage of groups. In this direction, the turn-on delay time of small (big) QDs relating to the wavelength of 1.31 μm (1.55 μm) versus the density of big (small) QDs for the different density of small (big) QDs is illustrated in Fig. 6a (6b). As shown in Fig. 6a (6b), the turn-on delay time of small (big) QDs is slightly increased by variation of the big (small) QDs as a result of weak coupling between the subgroups of big (small) QD groups lying within the HB of subgroups inside small (big) QD groups. However, the turn-on delay time corresponding to small (big) QDs lasing mode is increased by rising the density of small (big) QD. As explained earlier, because of increasing the QDs density, the injected current cannot satisfy all the QDs threshold condition, and the lasing is started later. The solution process is a low-cost method to control the density of colloidal QDs during the synthesis process. Consequently, based on the results observed in Fig. 6c and d, the intensity of the output power 8
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Optics Communications 457 (2020) 124629
big QD coverage are equal to 0.12 and 0.03, respectively (𝜉2 = 0.12, 𝜉1 = 0.03), the intensity of the output radiation for both wavelengths will be approximately 38 mW.
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3.2. Design of multi-wavelength QD-laser As mentioned earlier, the realization of multi-wavelength QD-laser is possible by the superimposition of various sizes of QDs at the active region of QD-laser. Each group of QD sizes leads to lasing at a specific wavelength. In order to have an improvement in our multi-wavelength QDlaser model, the results of three-wavelength QD-laser are investigated. The cavity length of this three-wavelength QD-laser is 1128.4 μm. The output power versus injected current for three wavelengths has been illustrated in Fig. 7a, while the FWHM of HB and IHB are considered 2ℏ𝛤𝐵 = 20 meV and 𝛤0 = 20 meV, respectively. Also, the QD coverage for all three groups of QDs is 𝜉 = 5%. The more injected current, the more output power intensity for all radiation wavelengths. As explained for two-wavelength QD-laser, the threshold current of longer wavelength is smaller than the others. Due to different slope efficiency for the three radiation wavelength, at I = 160 mA, the power of 1.3 μm and 1.4 μm radiations are equal. When the injected current is set to I = 160 mA, the photon number characteristics of three wavelengths of 1.3 μm, 1.4 μm, and 1.55 μm have been calculated based on the developed rate equations that is demonstrated in Fig. 7b. The photon number of longer wavelength is more than the shorter one at steadystate. In addition, the turn-on delay time of longer wavelength is less than the others as expected. Besides, the output power and the optical gain spectra are illustrated in Fig. 7c and d, respectively. It is noticed that the intensity of output power from shorter wavelength is small; however, the peak value of the optical gain for all wavelengths is somewhat larger than threshold gain which is equal for all of them. 4. Conclusion In brief, considering the solution process technology as a simple and cost-effective method to implement size distribution of QDs, the realization of the multi-wavelength QD-laser was investigated by superimposition of different QD sizes. Furthermore, simultaneous lasing of proposed multi-wavelength QD-laser was modeled based on the modified rate equations in which the inhomogeneous broadening as a consequence of size distribution of QDs and homogeneous broadening due to polarization dephasing were included. It was shown that twowavelength lasing of the proposed InGaAs/GaAs QD-laser at 1.31 μm and 1.55 μm was realized. Also, the effect of changing the QD’s density on the threshold current, turn-on delay time, and the output power intensity of each lasing wavelength was discussed in detail. Last but not least, in this work the design procedure for multi-wavelength QD-laser was proposed, and the solution process technology as the realization method was introduced. References [1] S. Meinecke, B. Lingnau, K. Lüdge, Increasing stability by two-state lasing in quantum-dot lasers with optical injection, Proc. SPIE 10098 (2017) 100980J, http://dx.doi.org/10.1117/12.2251791. [2] A. Markus, J.X. Chen, O. Gauthier-Lafaye, J.G. Provost, C. Paranthoën, A. Fiore, Impact of intraband relaxation on the performance of a quantum-dot laser, IEEE J. Sel. Top. Quantum Electron. 9 (2003) 1308–1314, http://dx.doi.org/10.1109/ JSTQE.2003.819494. [3] K. Lüdge, E. Schöll, Temperature dependent two-state lasing in quantum dot lasers, in: 2011 5th Rio La Plata Work. Laser Dyn. Nonlinear Photonics, LDNP 2011, 2011, http://dx.doi.org/10.1109/LDNP.2011.6162081. [4] M.V. Maximov, Y.M. Shernyakov, F.I. Zubov, A.E. Zhukov, N.Y. Gordeev, V.V. Korenev, A.V. Savelyev, D.A. Livshits, The influence of p-doping on two-state lasing in InAs/InGaAs quantum dot lasers, Semicond. Sci. Technol. 28 (2013) http://dx.doi.org/10.1088/0268-1242/28/10/105016. 9
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