Current-injection efficiency in semiconductor lasers with a waveguide based on quantum wells

A dynamic distributed diffusion-drift model of laser heterostructures, which takes into account carrier capture by quantum wells, is developed. The leakage currents in the lasing mode are calculated for different laser structures without wide-gap emitters: InGaAs/GaAs (lasing wavelength λ = 0.98 μm), InGaAsP/InP (λ = 1.3 μm), and InGaAs/InP (λ = 1.55 μm). It is shown that consideration of the finite carrier-capture time is of major importance for calculating structures with deep quantum wells. The ratio of the leakage currents to the total current in the structures with deep quantum wells (InGaAsP/InP and InGaAs/InP) increases with an increase in the injection current and may reach a few percent when the lasing threshold is multiply exceeded.


INTRODUCTION
Lasers with quantum wells (QWs) allow one to design a waveguide using the difference in the refrac tive indices of the QW material and the surrounding semiconductor. Although the QW width is small in comparison with the lasing wavelength, preliminary calculations showed that a small number (3)(4)(5)(6) of quantum wells are sufficient to form a waveguide [1]. In this case, the characteristic mode localization length in the direction perpendicular to the QW plane is about 1 μm (i.e., corresponds to a very wide waveguide). However, in contrast to very wide waveguides, there are no problems with mode selec tion in this design.
The lack of wide gap emitters in these structures leads to the absence of built in potential barriers in the bands, which hinder the diffusion of nonequilibrium carriers from the active region. Due to this, the leakage currents may significantly increase and the injection efficiency may fall, as occurs in separate confinement lasers upon the introduction of an extended waveguide [2]. In this case, slowed carrier capture by the QW and electron leakage from the active region into the waveguide become important factors [2,3].

THEORETICAL MODEL
2.1. Distributed Model Within the distributed model, each point in space is characterized by three variables (for example, the electrostatic potential ϕ and Fermi quasi levels for electrons (F n ) and holes (F p )) [7]. They can be found by combined numerical integration of the fundamen tal system of equations, consisting of the Poisson equation for the electrostatic potential ϕ and the con tinuity equations for the electron (j n ) and hole (j p ) cur rent densities: where n and p are the electron and hole concentra tions, respectively; N a and N d are the concentrations of ionized acceptors and donors, respectively; ε is the permittivity; ε 0 is the dielectric constant; R is the recombination rate; G is the gain; S 3D is the bulk pho ton density; v g is the group velocity; and μ n and μ p are, respectively, the electron and hole mobilities. The total which participate in charge transfer, and the concen trations of carriers localized at the QW levels (n w and p w , respectively), Here, N c (N wc ) and N v (N wv ) are the bulk (two dimen sional) effective electron and hole densities of states, respectively; T is temperature; and k is the Boltzmann constant; the z coordinate is counted from the begin ning of the quantum well layer with a thickness of d.
The hole concentration was calculated with allowance for the light and heavy hole subbands. The effective densities of states and the energy levels E ci and E vi were calculated within the effective mass approximation. The parameters ΔF wn and ΔF wp take into account the finite times of carrier capture at the QW levels. Within the approximation of fast intraband relaxation these values are zero. The energies of the bottom of the conduction band (E c ) and top of the valence band (E v ) were found using the electron affinity χ and the band gap E g of the semi conductor layers: The χ value in the QWs was chosen so as to corre late with the experimental data on the band disconti nuities [8]. To calculate the concentrations of nonlo calized carriers in the QWs, we used formulas (4), where the energies E c and E v were assumed to be equal to the corresponding values for the adjacent barrier layers.
The spontaneous recombination rate was calcu lated within the direct transition model from the approximate formula (8) Here, A cv is the Einstein coefficient for spontaneous transitions and N r is the reduced effective density of states. For QWs we applied similar formula with addi tional summation over the subbands, taking into account their occupancy. The gain G was calculated within the direct transition model. Spectral broaden ing effects were taken into consideration by formal replacement of the Heaviside function with a smoothed step function [9].

Boundary Conditions
The thicknesses of the emitter layers were chosen to be much larger than the screening length in these lay ers, which made it possible to construct the boundary conditions for the Poisson equation based on the elec troneutrality condition: (9) Here, the subscripts N and P indicate the external boundaries of the corresponding emitters. The bound ary conditions for the continuity equations were cho sen to be (10) where U is the voltage applied to the structure. In the dynamic mode, (11) where j is the total current density through the struc ture. The minority carrier transport into the bulk of the emitter layers (leakage currents) was assumed to be mainly diffusive; i.e., we used the following condi tions: where κ n (τ n ) and κ p (τ p ) are the inverse diffusion lengths (lifetimes) of minority carriers.

Lasing Dynamics
The bulk photon density near the mth QW was cal culated as (14) where Γ m is the optical confinement factor and d m is the QW width. The dynamics of the two dimensional No. 1 2014 photon density S 2D was described by the standard rate equation (15) where G m is the mean gain in the mth QW, R m is the two dimensional spontaneous recombination rate in the mth QW, ε s is the nonlinear gain factor, β is the contribution of spontaneous transitions to the laser mode, and k th is the threshold gain. Summation is per formed over all QWs.

Carrier Capture by QWs
When the processes of carrier capture and ejection to QW levels have finite rates, the quasi equilibrium between the carriers localized and nonlocalized in the QWs is violated, which limits the range of applicability of the conventional distributed model. This problem can be solved by increasing the number of variables characterizing one point in space; to this end, addi tional Fermi levels for localized electrons and holes are introduced into the QW regions. However, this proce dure increases the size of the calculated matrices and reduces the convergence rate. In addition, carrier cap ture is not local in the physical sense; i.e., carriers cap tured at any point above the QW are distributed over the entire QW.
The simplest way to solve the aforementioned problem is self consistent solution of the fundamental system of equations and the system of balance equa tions, where the rates of carrier capture at localized levels are equated with the interband recombination rates in the QWs: Here, τ cn and τ cp are, respectively, the times of electron and hole capture by an unoccupied QW; the exponen tial terms in parentheses allow for the change in the effective capture time upon QW level filling; and inte gration is performed separately for each QW. Although only two calculated parameters ΔF wn and ΔF wp are added for each QW within the model under consider ation, this approach makes it possible to introduce dif ferent Fermi quasi levels for each QW subband (as in calculations of intrasubband transition lasers [10]). The probability of free carrier capture by a QW heavily depends on the energy position of the highest bound state in the QW (it is maximum when this state is virtual [11][12][13]). The times of carrier capture by the QW in InGaAs/GaAs heterostructures were measured in [14] (based on photoluminescence analysis) to be 1 and 6.5 ps at 300 and 10 K, respectively.

Solution Algorithm
Our calculations of different quantum well struc tures showed that the coupled iterative solution to the system of equations (1)- (3) and (15)- (17) in the steady state mode is not convergent under strong inhomogeneous excitation of the QWs and for high barrier regions (ΔF wn , ΔF wp > kT). A possible reason is the existence of many steady state solutions to the nonlinear system of basic equations. This problem can be solved by the consideration of lasing dynamics.
The time sampling of all equations was performed by the implicit Euler method. The fundamental sys tem of equations (1)-(3) was solved using the finite difference scheme. Joint self consistent solution of the fundamental system of equations and Eqs. (11) and (15)- (17) was performed by the iterative method. The following algorithm, consisting of two embedded To provide solution convergence, the time integration step could be varied from several tens to several tenths of a picosecond. The obtained systems of nonlinear equations were solved by the Newton method.

NUMERICAL CALCULATION AND RESULTS
The parameters of the numerical calculation (band gap E g and effective electron (m c ) and hole (m vl , m vh ,  m vlt , and m vht ) masses) for quaternary (A x B 1 -x C y D 1 -y ), ternary (A x B 1 -x C), and binary (AB) compounds were approximated based on experimental and theoretical data according to [15]. The electron (μ n ) and hole (μ p ) mobilities were taken from the database [16]. When calculating the band structure, the potential well depths in the conduction (ΔE c ) and valence (ΔE v ) bands were calculated according to [8]. The interpola tion data for Ga 0.30 In 0.70 As 0.65 P 0.35 , In 0.2 Ga 0.8 As, In 0.53 Ga 0.47 As, GaAs, and InP compounds are listed in Table 1. The mobilities in the emitter layers are given in Table 2. Figure 1 shows the calculated energy band dia grams and positions of the Fermi quasi levels for elec trons (F n ) and holes (F p ). The emitters were assumed to be doped with acceptors to the concentration N a = 2 × 10 18 cm -3 and with donors to the concentration N d = 5 × 10 17 cm -3 . The weakly doped (N d = 5 × 10 16 cm -3 ) layers 100 nm thick are adjacent to the QWs. In a structure with relatively shallow QWs (Fig. 1a), the ΔF wn and ΔF wp values, which character ize the inhomogeneous excitation of the QWs and bar rier regions, do not exceed the thermal energy kT, and the excitation levels of all QWs are almost identical. In structures with lasing wavelengths of 1.31 and 1.55 μm (Figs. 1b and 1c), the ΔF wn and ΔF wp values exceed kT; in addition, these structures are characterized by sig nificant inhomogeneous QW excitation and short cap ture times (~1 ps or shorter). Calculation of the latter structure under similar conditions but with carrier capture disregarded indicates a significant voltage drop in the QW region (Fig. 1d) due to the high ohmic resistance of the barrier regions with low carrier con centrations.
The results presented in Fig. 2 were obtained in the dynamic mode with the application of a pump current linearly increasing from 0 to 10 kA/cm 2 in 100 ns. The time scale in Fig. 2 is replaced with the scale of the corresponding injection current density. A short tran sient process occurred after the onset of lasing. The relaxation oscillations strongly affected only the leak age currents calculated with the processes of carrier capture by QWs neglected. Afterwards, the lasing became quasi steady state. The InGaAs/GaAs structure has a low injection efficiency near the threshold (Fig. 2a); however, the fraction of leakage currents j leak may decrease after the onset of lasing. The reason is that the carrier concen trations in the QWs in the lasing mode are fairly stable near the threshold level, the population of the barrier regions (which determines the leakage currents) barely changes for short capture times, and injection into the QWs increases proportionally to the induced recom bination rate. Beginning with a certain current value, the fraction of the leakage currents increases again. The leakage currents in the conduction band are higher than those in the valence band because of a higher electron mobility. The leakage currents can be reduced to an acceptable level by introducing blocking layers of wide gap materials [17,18]. These layers must be as thin as possible to exclude elimination of the waveguide mode as a result of their antiwaveguide effect.
The InGaAsP/InP and InGaAs/InP structures exhibit similar dependences of the leakage currents, which increase with an increase in the injection cur rent (Figs. 2b and 2c). After the onset of lasing, this increase in only slowed (the jump in the dependence at the threshold is caused by the lasing onset delay). When the threshold is several times exceeded, the leakage currents may reach several percent. In the InGaAs/InP structure with deeper QWs, the leakage currents are lower than those in the InGaAsP/InP structure by a factor of about 2. The current depen dences of the leakage coefficient, which are calculated for zero capture times, differ significantly from the set of curves obtained at different τ cn and τ cp values. The reason is that calculation of the energy band diagrams without regard for carrier capture by QWs leads to sig nificant overestimation of the voltage drop on the ohmic resistance of the active region (Figs. 1c, 1d).
The multiple excess of the thermal energy by the homojunction potential barrier can be considered as a qualitative criterion of smallness of the leakage cur rents. For the heterostructures under study, the poten tial barriers U = E g -(F e -F h ) ≈ E g -hc/λ at 300 K were found to be 6, 15, and 21kT, respectively; this series correlates with the successive decrease (from the first to the third structure) in the leakage currents.

CONCLUSIONS
We developed a dynamic distributed diffusiondrift model of laser heterostructures, which takes into account the processes of carrier capture by QWs. The leakage currents were calculated for different QW waveguide laser structures with lasing wave lengths of 0.98, 1.31, and 1.55 μm. It was shown that significant inhomogeneous excitation of the QWs in the InGaAsP/InP (1.31 μm) and InGaAs/InP (1.55 μm) structures occurs at capture times less than 1 ps. At the lasing threshold the leakage currents were estimated to be 35-45% for InGaAs/GaAs (0.98 μm), less than 5% for InGaAsP/InP (1.31 μm), and less than 1% for InGaAs/InP (1.55 μm). The leakage currents in the structures with deep QWs increase with an increase in the injection current and may reach several percent at a multiply exceeded threshold.