Semiconductor quantum dot (QD) lasers including edge emitting and vertical cavity surface emitting lasers have more significant advantages than other lasers like bulk, quantum well, and wire in some characteristics, such as large differential gain,^[1,2] low threshold current,^[3,4] temperature insensitivity of the threshold current,^[5,6] and high output power^[7] because of their delta shape discrete density of states.

However, their carrier dynamic operations at high speed are not satisfactory enough, due to a phonon bottleneck appearing in electrons and holes capturing process^[8,9] leading to a hot-carrier problem in QD lasers. The effect of hot carrier followed by the gain compression triggers the reduction of the differential gain.^[10] Thus the resonance frequency is limited by the maximum modal gain and gain compression effects. Therefore the maximum modulation bandwidth, which is dependent on the resonance frequency and damping factor, decreases. So, the hot-carrier problem is an important reason for the disability of QD lasers to operate at high frequencies. Also, it is demonstrated that finite carrier capture, finite relaxation time, and Pauli blocking factors limit the maximum modulation bandwidth.^[11]

To solve the problem one of the solutions is to use the tunneling injection method (TI). At first, the tunneling injection method is performed in active region for quantum well semiconductor lasers. By tunneling “cold” electrons directly into the lasing states of GaAs- and InP-based quantum well lasers from an adjacent injector layer, the performance characteristics including enhanced modulation bandwidth and higher characteristic temperature ( ), the reduced Auger recombination and the reduced chirp are both demonstrated in Refs. [12]–[14]. Also, the TI scheme has been implemented in quantum well vertical-cavity surface-emitting lasers (QW-VCSELs).^[15]

The technique of TI is applied to QD lasers, too. In tunneling injection quantum dot structure, electrons are introduced directly or in the phonon-assisted way into the lasing states of dots. Thus, the probabilities of electrons existing in the wetting layer and excited state, decrease. Then, the effects of hot carrier and gain compression are reduced and the differential gain increases.

By using the tunneling injection technique, small signal modulation is measured at frequencies more than 20 GHz.^[16] In tunneling injection structure, in addition to the effect of carrier capture and relaxation, the tunneling rate also affects the resonance frequency and damping factor. So, with a smaller tunneling time, the performance of laser is increased. It is shown in Ref. [17] that with different thickness values of the barrier, the tunneling time can be adjusted.

In this paper, a numerical model is presented to simulate the behavior of tunneling injection quantum dot VCSEL by using the method described in Ref. [11]. VCSELs are proven to have some unique properties such as high reliability, high internal efficiency, low divergence angle, simple fabrication process into twodimensional arrays.^[18]

In our model, the bandwidths of modulation transfer functions associated with different levels of energy in dot and injector well are obtained. In this manner, the effect of tunnel injection on the resonance frequency and also on the bandwidth of modulation transfer function of TIQD laser is presented. Generally, the modulation transfer function components in the wetting layer (WL) and excited state (ES) limit the total bandwidth. However, in TIQD structure, modulation transfer function component related to tunneling process from injector well into QDs ground state experiences a high bandwidth leading to the increase of the entire modulation bandwidth. We base our study on the TI-QD-VCSEL model by a hetero-structure in active region and 1.3-μm QD-VCSEL which is based on experimental work in Refs. [19] and [20]. Also, in order to investigate the effect of photonic crystal (PhC) waveguide on the frequency behavior of laser, we consider a model of two-dimensional photonic crystal (PhC) waveguide embedded in entire VCSEL. The dynamic response of VCSEL varies with the structure and dimension of the laser cavity.^[21] Oxide aperture depth and diameter as well as the characteristics of photonic crystal structure such as air holes diameter and PhC lattice constant are important parameters which affect the frequency response of VCSELs due to some properties such as spatial hole burning and self-heating effect. In order to take into account all phenomena of the laser, spatial hole burning (SHB) due to stimulated recombination of carriers and thermal lensing (TL) a result of the self-heating effect and transverse mode competition needs to be considered. The coupled optical field, diffusion, and thermal conduction equations are solved by using the algorithm of the self-consistent calculation which is described in Ref. [22].

The rest of this paper is organized as follows. In Sections 2 and 3, we investigate the dynamic carrier of the laser and the factors affecting the resonance frequency and damping factor and derive a new modulation transfer function similar to that in Ref. [11] through a small signal analysis of differential rate equations. In Section 4, the numerical results of tunneling injection are presented and by comparing with the conventional quantum dot, the origin of improvement of frequency response in TIQD structure, is investigated.

The schematic structure of the oxide-confined 1.3-μm QD-VCSEL is illustrated in Fig. 1. In this device, the active region is composed of ten self-assembled QD 6-nm InAs layers. The top and bottom mirrors consist of a 22-period p-type (doped 5 × 10¹⁷cm⁻³) and a 33-period n-type (doped 1 × 10¹⁸cm⁻³) quarter-wave GaAs-Al Ga As layer distributed Bragg reflectors (DBR’s), respectively. A 12-nm-thick AlAs layer embedded in the p-DBR adjacent to the active region is used for selective oxidation with oxide apertures of 10μm–15μm in diameter. The upper metal contact is fabricated on the top of the p-doped DBR and the bottom one on the n-doped substrate. Thickness values and compositions of all layers are presented in Ref. [23]. Also, we assume that the two-dimensional PhC waveguide embedded into VCSEL cavity has air-holes which go through the active region and both mirrors. Note that increasing the etching depth results in larger photonic crystal effect. The photonic crystal is composed of hexagonally arranged air-holes with diameter and center-to center spacing A.

Fig. 1.

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	Fig. 1. Schematic diagram of the TIQD-VCSEL model.

In the active region, we consider a model similar to the tunnel injection quantum dot active region structure in Ref. [16] whose sketch is shown in Fig. 2 consisting of the In Ga As injector well, an Al As tunnel barrier, and three coupled In As QD layers. In order to efficiently realize the phonon assisted tunneling from the injector layer to the dots, the conduction band states of the injector layer with the bound states in QDs are aligned.^[16] In our model, it is assumed that the active region consists of only one QD ensemble including two energy levels: a ground state (GS) with a two-fold degenerate and an excited state (ES) with a four-fold degeneration. The system is assumed to consist of excitonic energy states, while QDs are always neutral, and electrons and holes are treated as electron–hole pairs.

Fig. 2.

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	Fig. 2. (color online) Transport processes in the active region of TIQD laser.

According to the carrier dynamics in the conduction band shown in Fig. 2, the external carriers are first injected into SCH and then into the injector well. Some of the carriers are then either introduced into the QD ground state directly or by phonon-assisted tunneling, and some of them participate in spontaneous emission processes, with a rate time of . Once in the WL, carriers are captured into the ES within time and directly into the GS within time or participate in spontaneous emission processes with a rate time of . Carriers on the excited state are relaxed into the GS within time and on the other hand, they are also thermally reemitted from the ES to the WL with a which is governed by the Fermi distribution for the quasi-thermal equilibrium without external excitation.^[24] Carriers on the ground state level have a similar dynamic behavior to ES, while some of them recombine spontaneously with a spontaneous time . Most carriers reside in the injector well and in the wetting layer at room temperature, so the Auger recombination has not been explicitly considered in ES and GS levels.^[16] In our model, the stimulated emission from the GS is just taken into account and occurs when the threshold is reached. Following the sketch in Fig. 2, the carrier dynamics defined by a set of rate equations given in Ref. [16] is as follows:

(2)

(3)

(5)

(6)

Now, we define the relaxation resonance frequency and damping factor for the lasing level versus coefficients of matrix as

(13)

(14)

(15)

For GaAs,

(16)

(17)

At first, we numerically solve the steady state of the coupled equations (13)–(17) by using the algorithm of the self-consistent calculation. Diffusion equations of TIQD structure are created from the rate equation with adding ∇² N _w and ∇² N _WL terms in steady state and is solved with the nonlinear Newton method for determining the radial profile of carrier concentration. Thermal equation is solved by the finite difference method with Dirichlet boundary conditions. Field equation is solved by the finite difference method with absorbing boundary conditions accordingly to . Then, the frequency behavior of laser is studied by using an analytical approximation of the model which has been discussed in the previous section.

The components of approximate solution presented in Eqs. (10a)–(10d) are shown in Fig. 3. The inset presents the precise modulation transfer function given by Eq. (6) and the approximate solution which shows that the and curves are approximately close to each other.

Fig. 3.

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	Fig. 3. (color online) Components of approximate solution . The inset shows precise modulation transfer function and the approximate solution for TIQD-VCSEL.

It can be seen that the behavior of response of to frequency is similar to that of quantum well structure.^[11] While, among the components of , is the underlying limitation to the enhancement of the total modulation bandwidth, because the modulation bandwidth of is much smaller than others’. Of course, in TIQD structure, the carrier dynamics in the inter-band of dots has no significant influence on , while in CQD, the finite carrier capture time and finite carrier relaxation time are the effective physical limitations to the enhancement of modulation bandwidth.^[11] This is the evidence for the advantage of using a tunnel injection quantum dot compared with using the conventional QD. In addition, has the highest bandwidth, which leads to the increase of the total modulation bandwidth. Given that this component is related to the tunneling time, the behavior of modulation will be improved for shorter tunneling time.

Generally, due to tunnelling injection, total frequency behaviour of laser has less dependence on inter-band dynamic of QD. However, the back tunnelling time from GS to the injector well is a more important parameter for determining the 3-dB bandwidth of total transfer function by its direct influence on the bandwidth of . Figure 4 shows frequency responses of the laser for four different values of back tunnelling time from GS to the injector well ( ) at , while the inset shows variations of with for differentvalues of carrier tunnelling time ( ). By comparing these results, it can be seen that by increasing the , of total transfer function is increased due to high increment of of as shown in the inset. As depicted, in the case of low back tunnelling time, the changes of 3-dB bandwidth of is intense, but in the case of high value of back tunnelling time, 3-dB bandwidth of changes slower so that, forshort tunneling time, it is approximately independent of . Also, with small back tunnelling time, has an indirect influence on the bandwidth of lasing transfer function component due to generated photon. While, in the case of large back tunnelling time, the effect of on the bandwidth of is very weak. Because, by increasing , there will be no back tunneling process approximately toward injector well. So, carrier densities at the ES and the GS levels of dots as well as the Pauli blocking factors have small changes due to photon density created in the cavity and will almost be less dependent on the value of . According to the expressions of and Γ in and their dependence on photon density, 3-dB bandwidth of is less impressible to for large value of .

Fig. 4.

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	Fig. 4. (color) Frequency responses of TIQD-VCSEL for four values of . The inset shows variations of with for different values of .

The frequency response of the laser is plotted in Fig. 5 for two different values of back tunnelling time at and . It can be concluded that for a large value of , the dependence of 3-dB bandwidth on the bias current is not so strong. The inset shows laser 3-dB bandwidth changes of with bias current for three different values of . It is observed that by increasing , the dependence of on the bias current decreases with the increase of current. It should be noted that the bandwidth of is not dependent on , nor . But, the tunnelling time affects the bandwidth of strongly.

Fig. 5.

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	Fig. 5. (color online) Frequency responses of TIQD-VCSEL for two values of at and . The inset shows variations of with bias current for three values of .

Now, in order to investigate the effect of photonic crystal (PhC) waveguide on the frequency behavior of laser, the 3-dB bandwidth of transfer function of TIQD-PhC-VCSEL and its components are plotted in Figs. 6(a)–6(c) as a function of injected current for various values of air hole diameter ( ) with a fixed value of PhC lattice constant (A = 2 μm). Oxide aperture diameter (s) and spreading current (r) are 10 μm and 1 μm, respectively. The threshold currents are obtained to be 3.2 mA, 4.6 mA, and 59 mA. The 3-dB bandwidth of transfer function of TIQD-VCSEL with the same structure is shown in the inset of Fig. 6(a). As can be seen from Fig. 6(a) the bandwidth of increases with increasing bias current and is saturated for high values. The occurrence of rollover of the modulation response at the high bias current is due to both self-heating effect (with more effectiveness) and the emergence of high-order modes (with less effectiveness). The rollovers are observed at GHz for , GHz for , and GHz for , respectively. While in TIQD-VCSEL without photonic crystal structure the rollover is at GHz at 3.4-mA threshold current. Relaxation rate in QD will decrease as a result of phonon bottleneck caused by increasing temperature in the active region. For larger air hole diameter, the bandwidth increases more intensely at low bias currents, and is saturated later. Since the air holes diameter has a significant influence on the reduction of thermal lensing and spatial hole burning effect. So, with its increase, the temperature decreases and the gain increases in the center of the active region. In TIQD structures, this rollover is less than that in conventional quantum dot structure by solving the phonon bottleneck problem. In order to further analyse the variations of 3-dB bandwidth of and versus bias current are shown in Figs. 6(b)–6(c). The presented results point out that 3-dB bandwidth of increases with the increase of current, while of is reduced, slightly. The effect of air-hole diameter on the 3-dB bandwidth of is almost similar to that of . The reduction of 3-dB bandwidth of is more intense for smaller air-hole diameter. Based on the variations of and as well as , in the case of low bias current the effect of increase of 3-dB bandwidth of is dominant over the effect of reduction of 3-dB bandwidth of , while frequency behavior of will be predominant in the case of high bias current. On the other hand, carrier dynamic behaviors in wetting layer and excited state levels are more influenced by spatial hole burning and thermal effects. In the TIQD structures, current dependence of and Γ are negligible despite CQD decreasing linearly with the increase of current.^[11] Thus, the limitation to the bandwidth of does not restrict the total modulation bandwidth with the increase of current. This is attributed to the current being not dependent on Pauli blocking factor nor . The of is approximately independent of the current (not shown here). Based on Eqs. (5) and (10d), the results indicate that the constant Pauli blocking factor in ground state is effective to small reduction of 3-dB bandwidth of in TIQD when increasing the current.

Fig. 6.

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	Fig. 6. (color online) Plots of 3-dB frequency of approximation modulation transfer function with its components versus for various values of air-hole diameter with A = 2 μm in TIQD-PhC-VCSEL.

Through the analysis of modulation transfer function, the characteristics of the photonic crystal tunneling injection quantum dot vertical cavity surface emitting lasers (PhC-TIQD-VCSEL) are studied. Based on the rate equations describing the carrier dynamics, the modulation transfer function is obtained and divided into the components related to the energy levels in the dot and the injector well. Numerical investigation clarifies the effect of the tunneling injection on the enhancement of the modulation response. The results show that the carrier back-tunneling time from the dot ground state into the injector well is the most effective parameter to determine the frequency response of TIQD-VCSEL. The frequency behaviors of WL-ES and GS levels are also affected by the variations of the structure and dimension of laser cavity as well as air-hole diameter and lattice constant of photonic crystal structure due to some nonlinear effects such as spatial hole burning and thermal lensing effects. However the component of tunneling process is almost independent of them.