Modeling Quantum Well Lasers

Abstract

In semiconductor laser modeling, a good mathematical model gives near-reality results. Three methods of modeling solutions from the rate equations are presented and analyzed. A method based on the rate equations modeled in Simulink to describe quantum well lasers was presented. For different signal types like step function, saw tooth and sinus used as input, a good response of the used equations is obtained. Circuit model resulting from one of the rate equations models is presented and simulated in SPICE. Results show a good modeling behavior. Numerical simulation in MathCad gives satisfactory results for the study of the transitory and dynamic operation at small level of the injection current. The obtained numerical results show the specific limits of each model, according to theoretical analysis. Based on these results, software can be built that integrates circuit simulation and other modeling methods for quantum well lasers to have a tool that model and analysis these devices from all points of view.

1. Introduction

Laser semiconductor diodes are key components in modern optical communications, storage, printing, medicine, and information processing. These types of devices were developed and evolved constantly in the direction of size reduction and integration. A quantum well (QW) laser improves the functioning characteristics of laser diodes in the direction of low threshold current and narrow emission band as well as emitted wavelength dependence on nanostructure dimension (quantum size effect) [1–8].

A quantum well laser is a structure in which the active region of the device is so narrow that quantum confinement occurs, according to quantum mechanics. The wavelength of the light emitted by a quantum well laser is determined by the width of the active region rather than just the band gap of the material from which the device is realized. Consequently, much shorter wavelengths can be obtained from quantum well lasers than from conventional laser diodes using a particular semiconductor material. The efficiency of a quantum well laser is also greater than a conventional laser diode due to the stepwise form of its density of states function. In the optoelectronic integrated circuits (OEICs), the quantum well lasers will interact optically and electrically with other devices as integrated modulators, optical fibers and coherent amplifiers, detectors, and optical integrated guides [9–19]. A fully analytical method to describe these kind of complex systems cannot be realistic without questionable simplifying assumptions so that the implementations of refined mathematical methods and algorithms as well as circuit simulators become essentially in the applications development [20–24].

2. General Modeling Methods and Techniques

This paper aims at presenting and analyzing modeling techniques for quantum well lasers starting from the rate equations. The issue of modeling of quantum well semiconductor lasers has been addressed by several authors [1–3, 7, 13]. In operation of a laser, effects are involved that can impact a variety of performance parameters of the device. In some cases is crucial to obtain a clear picture of how a laser works in various conditions or parameters. A good laser model which allows calculation and testing of certain parameters of the device is an important result. The main models of laser with quantum wells are based on the description of the semiconductor lasers using the rate equations formalism.

While the first models were based on one pair of equations to describe the density of photons and carriers in the active region, recent approaches include additional rate equations, to take into account, and carriers transport between active region and adjacent layers of the structure.

Note that in most cases the rate equations lead to multiple solutions, although only one solution is correct. Javro and Kang [25] showed that incorrect solutions or without physical sense can be eliminated or avoided through a change of variables in the rate equations.

However, the transformations used are available under certain conditions and for some cases give unrealistic solutions. These shortcomings are caused mainly due to the linear character of the gain-saturation coefficient. More general expression of the gain-saturation coefficient, proposed by Channin can be used to obtain models for operation regimes having a unique solution. Agrawal suggests another expression for this coefficient, which is also suitable. As it is shown, any of these two forms of the gain-saturation coefficient can be used to obtain models with a solution unique of the operation regime.

In this paper, based on known rate equations models we try, firstly, the circuit simulation for two different modeling systems: Simulink and SPICE. The Simulink modeling technique is very simple. It is based on standard rate equations with the sets of parameters given directly in Simulink. The second model is based on the standard rate equations that use a nonlinear gain-saturation term proposed by Channin. In this second model we also describe the circuit level in SPICE with simulation results.

A numerical experiment in MathCad which gives satisfactory results for the study of the transitory and dynamic operation at small level of the injection signal is shortly presented, also.

3. The Model with Linear Gain Saturation, in Simulink

One of the prevailing laser diode models is based on a set of rate equations. The rigorous derivation of these equations originates from Maxwell equations with a quantum mechanical approach for the induced polarization. However, the rate equation could also be derived by considering physical phenomena [1, 3, 5]: 𝑑𝑁=𝐼𝑑𝑡𝑞𝑉_act−𝑔₀𝑁−𝑁₀()𝑁1−𝜀𝑆𝑆−𝜏_𝑝+𝑁_𝑒𝜏_𝑛,𝑑𝑆𝑑𝑡=Γ𝑔₀𝑁−𝑁₀(1−𝜀𝑆)𝑆+Γ𝛽𝑁𝜏_𝑛−𝑆𝜏_𝑝𝑆,𝑃_𝑓=Γ𝜏_𝑝𝜆₀𝑉_act𝜂ℎ𝑐=𝜗. (3.1)(3.2)(3.3) Equation (3.1) relates the rate of change in carrier concentration 𝑁 to the injection current 𝐼, the carrier recombination rate and the stimulated emission rate. Equation (3.2) relates the rate of change in photon density 𝑆 to the photons loss, the rate of coupled recombination into the lasing mode, and the stimulated emission rate. The ratio between photon density 𝑆 to the output power 𝑃_𝑓 is described in (3.3). The other parameters used have well-known significance [2]. This simple model can be directly implemented with Simulink without any problems.

A direct implementation looks like that in Figure 1.

Figure 1 

Construction of rate equation block.

Figures 1 and 2 show how the rate equations model is constructed in Simulink. with 𝐼 as input parameter from a signal generator and 𝑆, 𝑁, and 𝑃_𝑓 as output parameters. All the parameters in the rate equations can be modified before the simulation starts.

Figure 2 

Rate equation in Simulink.

For the simulation we used the following parameters found in specialized papers [1–5]: 𝜆₀=1.502×10⁻⁴ cm (laser wavelength), 𝑉_act=9×10⁻¹¹ cm³ (active region volume), Γ=0.44 (optical confinement factor), 𝐵=4×10⁻⁴ (spontaneous emission factor), 𝑔₀=3×10⁻⁶ cm³/s (gain coefficient), 𝑁₀=1.2×10¹⁸ cm⁻³ (optical transparency density), 𝜏_𝑛=3 ns (carrier lifetime), 𝜏_𝑝=1 ps (photon lifetime), 𝜂=0.1 (quantum efficiency), 𝑁_𝑒=5.41×10¹⁰ cm⁻³ (equilibrium carrier density), and 𝜀=3.4×10⁻¹⁷ cm³ (gain saturation factor).

With the above model different signal forms can be used as input for the quantum well laser. They show that theoretical response of the equations is good in comparison with real results that are obtained in applications. Signals like step function, saw tooth, and sine types are used as input. The results are shown in Figures 3 and 4, which illustrate a very fast response at low current level. Low threshold current is the main feature of quantum well lasers, and it is directly shown for the basic form of rate equations. The simulation is not perfect, and this is because of negative solutions for 𝑁 and 𝑆 and high power solution of the equations.

Figure 3 

Output for step input signal of 10 mA amplitude and frequency of 5 MHz.

(a)

(b)

Figure 4 

Output for saw tooth input signal of 10 mA amplitude and frequency of 5 MHz (a) and for sinusoidal input signal (b).

4. The Model with Nonlinear Gain Saturation, in SPICE

To obtain nonnegative solutions for the photon density and the density of carriers in terms of a nonnegative current injection, a different version of the standard rate equations must be formulated. In the corresponding generalized equations (3.1) and (3.2), the linear gain-saturation term is replaced by a nonlinear term proposed by Channin or Agrawal as follows [1–3]: 𝑑𝑁=𝜂𝑑𝑡^𝑖𝐼𝑞𝑁_𝑤𝑉_act−𝑅_𝑤(𝑁)−Γ_𝑐𝜈_gr𝛼(𝑁)Φ(𝑆)𝑆,𝑑𝑆𝑆𝑑𝑡=−𝜏_𝑝+𝑁_𝑤𝑅_𝑤𝛽(𝑁)+𝑁_𝑤Γ_𝑐𝜈_gr𝛼(𝑁)Φ(𝑆)𝑆𝑆,𝑃_𝑓=𝜆𝜏_𝑝𝜂_𝑐𝑉_actℎ𝑐=𝜗. (4.1)(4.2)(4.3) Equation (4.1) describes the carrier concentration rate dependence on the injection current, the carrier recombination rate 𝑅_𝑤(𝑁) and the stimulated emission rate. To take into account all mechanisms of recombination, we consider: 𝑅_𝑤(𝑁)=𝐴𝑁+𝐵𝑁²+𝐶𝑁³, where 𝐴, 𝐵, and 𝐶 are, respectively, the unimolecular, radiative, and Auger recombination coefficients.

Equation (4.2) shows the photon density rate dependence on photon loss, the rate of coupled recombination into the lasing mode, and the stimulated emission rate. Equation (4.3) relates the output power 𝑃_𝑓 to the photon density 𝑆. The other parameters used have well-known significance [2]. In (4.1) and (4.2), the correlation between the material gain and the carrier density is given by the logarithmic carrier-dependent gain term: 𝛼(𝑁)=𝐺₀𝑅ln^𝑤(𝑁)𝑅_𝑤𝑁₀, (4.4) where 𝐺₀ is the one quantum well gain coefficient and 𝑁₀ is the optical transparency density while the gain-saturation function can take on one of the following two forms: 𝜙⁻¹(𝑆)=11+𝜀Γ_𝑐𝑆(Channin)or𝜙⁻¹(𝑆)=11+𝜀Γ_𝑐𝑆Agrawal. (4.5) An equivalent circuit model based on (4.1)–(4.3) can be implemented in SPICE as in Figure 5. Unlike models based on the rate equations that use a linear gain-saturation term, this circuit model is applicable for all nonnegative values of injection current.

Figure 5 

The circuit implementation of the model.

Figure 6 shows the circuit implementation of the model, using suitable variables transformations [25]: 𝑁=𝑁_𝑒exp(𝑞𝑉/𝑛𝑘𝑇) and 𝑃_𝑓=(𝑚+𝛿)², which impose nonnegative solutions for 𝑁 and 𝑃_𝑓. The linear recombination and charge storage in the device are described by diodes 𝐷1 and 𝐷2 and current sources 𝐼_𝑐1 and 𝐼_𝑐2. The effects of additional recombination mechanisms and stimulated emission, respectively, on the carrier density are modeled by 𝐵_𝑟1 and 𝐵_𝑠1, according to (4.11) and (4.13).

(a)

(b)

Figure 6 

Answer of optical power with variation of the input bias current between 0 and 25 mA (a) and between 0 and 50 mA (b).

The effects of spontaneous and stimulated emission, respectively, are accounted in the circuit by 𝐵_𝑟2 and 𝐵_𝑠2 while 𝑅_ph and 𝐶_ph complets the model for the time variation of the photon density. The voltage 𝐵_𝑝𝑓 describe the optical output power of the laser.

The equations for the elements of the circuit in Figure 8 according to [2] are as follows: 𝐼=𝐼_𝑇1+𝐼_𝐷1+𝐼_𝐶1+𝐵_𝑟1+𝐵_𝑠1, where 𝐼_𝑇1=𝐼_𝐷1+𝐼_𝐶1,2𝜏_𝑝𝑑𝑚𝑑𝑡+𝑚=𝐵_𝑟2+𝐵_𝑠2,𝐵_𝑝𝑓=(𝑚+𝛿)²𝐼_𝐷1,=𝑞𝑁_𝑤𝑉_act𝑁_𝑒2𝜂_𝑖𝜏_𝑛exp𝑞𝑉𝐼𝑛𝑘𝑇−1,_𝐷2=𝑞𝑁_𝑤𝑉_act𝑁_𝑒2𝜂_𝑖𝜏_𝑛exp𝑞𝑉𝑛𝑘𝑇−1+2𝑞𝜏_𝑝𝑛𝑘𝑇exp𝑞𝑉𝑛𝑘𝑇𝑑𝑉𝐼𝑑𝑡_𝑐1,=𝐼_𝑐2=𝑞𝑁_𝑤𝑉_act𝑁_𝑒2𝜂_𝑖𝜏_𝑛𝐵_𝑟1,=𝑞𝑁_𝑤𝑉_act𝜂_𝑖𝑅_𝜔2Θ𝐼_𝑇1𝐵_𝑠1,=𝜆𝜏_𝑝𝑁_𝑤Γ_𝑐𝜐_gr𝜂𝑖𝜂_𝑐𝛼ℎ𝑐Θ𝐼_𝑇1𝜙𝜗(𝑚+𝛿)²𝜗(𝑚+𝛿)²𝐵_𝑟2,=𝑁_𝑤𝜂_𝑐𝑉_actℎ𝑐()𝑅𝜆𝜗𝑚+𝛿^𝜔2Θ𝐼_𝑇1𝐵_𝑠2,=𝜏_𝑝𝑁_𝑤Γ_𝑐𝜐_gr𝛼Θ𝐼_𝑇1𝜙𝜗(𝑚+𝛿)²(𝑚+𝛿)−𝛿. (4.6)(4.7)(4.8)(4.9)(4.10)(4.11)(4.12)(4.13)(4.14) The SPICE netlist was implemented in AIM-SPICE. Calculating the parameters in the netlist is time consuming, and if a calculation is wrong, the netlist will fail in SPICE or the result will not be good. Figures 6, 7, and 8 resulting from simulation with simple DC sweep and transitory sweep illustrate the following:(i)answer of optical power with variation of the input bias current between 0 and 25 mA (a) and between 0 and 50 mA (b) (Figure 6);(ii)transient output power in response between 0 and 100 ps (Figure 7);(iii)transient output power in response between 0 and 5 ns (Figure 8).

Figure 7 

Transient output power in response between 0 and 100 ps.

Figure 8 

Transient output power in response between 0 and 5 ns. Note: Output power is in 𝑉 because SPICE cannot output variables in 𝑊.

5. Numerical Experiment in MathCad

Study of functional characteristics of QWL is possible based on the models described previosly, by implementing MathCad programs to integrate these equations using appropriate algorithms corresponding to different types of pumping signals.

The model given by the equations with linear gain-saturation term (3.1)–(3.3) was integrated for a constant current injection 𝐼_𝑖=0.05 A. The waveforms corresponding to 𝐼_𝑖 and to output optical power 𝑃 are obtained [13].

The model given by the equations with nonlinear gain saturation (4.1)–(4.3) was integrated for a constant current injection 𝐼_𝑖(𝑡)=0.025 A (Figure 9), sinus current injection (Figure 10):𝐼_𝑖(𝑡)=1+sin5⋅10⁸𝑡, (5.1) and rectangular current injection (Figure 11), which for several periods can be written as: 𝐼_𝑖(𝑡)=10⁻⁸+0.01+0.25𝐼_𝑖𝑡−2⋅10⁻⁹+0.25𝐼_𝑖𝑡−7⋅10⁻⁹+0.25𝐼_𝑖𝑡−12⋅10⁻⁹. (5.2) Simulations results for these large-signal models have shown excellent agreement with experimental data. A corresponding small-signal model has been derived for the quantum well lasers, to study the modulation properties in the frequency domain [13].

Figure 9 

The waveforms corresponding to 𝐼𝑖 and to output optical power 𝑃.

Figure 10 

The waveforms corresponding to 𝐼𝑖 and to output optical power 𝑃.

Figure 11 

The waveforms corresponding to 𝐼𝑖 and to output optical power 𝑃.

6. Conclusions

(1)In the paper, some modeling methods and techniques for the quantum well lasers have been implemented and validated. A suitable modeling technique is very important for technology designing and analyzing of the integrated optoelectronic circuits.(2) A method based on the rate equations for using Simulink to describe quantum well lasers was presented. For different signal types like step function, saw tooth, and sinus used as input, the results show a good theoretical response of the equations in comparison with the real results obtained in applications. This method is useful to determine the various regions of operation of the laser diode also.(3) The SPICE simulation of the circuits shows that modifying one parameter will result in new calculations and new SPICE netlist, which is a rather complicated procedure. Simulation is not limited to SPICE; any all-purpose circuit simulator can be used to get similar results.(4) Generally, all studied models give satisfactory results for the study of the transitory and dynamic operation at small level of the injection signals. At large injection signal levels the obtained numerical results show the specific limits of each model, according to theoretical analysis.(5) As a future development, software can be built that integrates circuit simulations and other modeling methods for quantum well lasers to have a tool that models these devices from all points of view.

Acknowledgments

The author A. R. Sterian acknowledges the financial support from CNCSIS-UEFISCSU, project number PNII-IDEI ID_954/2007 and ID_123/2008.