# Differential Gain in InGaAsN/GaAs Double Quantum Well Structures by Numerical Simulations

Authors: M. S. Wartak; P. Weetman; P. Rusek
Journal: Research Letters in Physics
 (2008)

---

## Abstract

We performed numerical studies of differential gain in a coupled quantum well structures built from InGaAsN. Differential gain inIn0.38Ga0.62As1_yNy/GaAs quantum well structures was determined and analyzed. A 10-band k_p  Hamiltonian matrix was used in the calculations and solved self-consistently with Poisson's equation. The effect of Nitrogen composition and barrier thickness on differential gain has been determined. The influence of Nitrogen composition on differential gain is significant whereas barrier effects are modest.


---

## Body

## 1. Introduction

A new  material systemInxGa1−xAs1−yNy is currently  under intensive investigations.
Several properties of those systems have been
analyzed [1–3].  InxGa1−xAs1−yNy is a practical alternative to InP-based
materials for 1.3μm operation of quantum-well-based semiconductor
lasers. The adventages over InP systems include lower cost, higher
characteristic temperature T0,
higher output power, better heat conduction, and applicability to use in
vertical cavity surface emitting lasers.For this new material system
there is a need to analyze its properties versus different parameters. One of
those parameters is coupling between two quantum wells. Recently, we addressed
the question of the effect of well coupling inInxGa1−xAs1−yNy on the effective masses and optical gain [4, 5].
The optical gain spectra were obtained for various values of Nitrogen
composition and barrier widths. They show optimum values for particular barrier
width, although the variations are not significant. However, the magnitude of
gain peak undergoes significant changes when varying Nitrogen composition.We have also recently reported
on the results of differential gain for single quantum well [6] fabricated
using InGaAsN/GaAs material system. It was predicted that spectra of
differential gain show significant variation versus Nitrogen composition, well
width, and carrier density.In order to properly manipulate
differential gain and optical gain in achieving prescribed engineering goals,
we introduced here one more design parameter, namely barrier width of a
double-well system.  As it is known, for
aGaAs/Al0.2Ga0.8As double quantum well system [7], barrier thickness has a significant effect on differential gain. The
differential gain at the gain peak is not a monotonic function of the barrier
thickness. For relatively thick barriers inclusion of coupling reduces the
differential gain but as the barrier gets thinner the differential gain can be
significantly enhanced or suppressed compared to its uncoupled value.In the present paper, we used the theory leading to the determination
of differential gain of two wells consisting ofInxGa1−xAs1−yNy.  The details of our approach are described in
[5, 6]. The gain formulation
incorporates non-Markovian effects, Coulombic interactions, and also band-gap
renormalization. In the determination of electron and hole band structures we
also incorporated self-consistent effects by solving the relevant matrix
Schroedinger equation simultaneously with Poisson's equation.In Section2, we outline briefly our formalism and in
Section 3 present our results and conclusions.


## 2. Formalism

The formalism used in our paper has been recently described
by us [5]. We use a plane-wave expansion method to calculate the electron and
hole bandstructures of the quantum well and a 10×10 Hamiltonian as presented by Tomić et al. [8],
where the effect of adding nitrogen to the structure is modeled perturbatively.
The Hamiltonian has been also analyzed by O’Reilly et al. [9] and Hader et al.
[10]. It can be considered as a good model of InGaAsN system. The selection of
parameters has been discussed extensively by Choulis et al. [11].  Specific form of this Hamiltonian is (1)H=(ENVNc00000000EC−3T+2U−U000−T−−2T−EHH2S−S000−R−2RELHQ0T+∗R03SESO02T+∗2R−3S0ENVNc000EC−3T−2U−UEHH2S∗−S∗ELHQESO). Here the subscripts N, C, HH, LH, and SO stand for
nitrogen, conduction, heavy-hole, light-hole, and split-off bands,
respectively.The self-consistent method is similar to the one which we
used before [12] which consists in a self-consistent solution of the Poisson
equation with the matrix Schroedinger equation described by Hamiltonian (1).
The Poisson equation is (2)ddz[ε(z)ddzφ(z)]=−e[ρHH(z)+ρLH(z)−ρC(z)], where e is the fundamental charge, ε(z) is the position-dependent permittivity, ρC(z), ρHH(z) and ρLH(z) are the position-dependent electron and
hole band density distributions,
respectively. The function φ(z) is the electrostatic potential. The density
distributions are [12] (3)ρα(z)=kBTπℏ2{∑nm¯α∗|Fn,αp(z)|2ln⁡[1+exp⁡(Eαf−En,αpkBT)]}. The symbol α represents the conduction (C), heavy-hole (HH)
and light-hole (LH) bands and n is an index over the subbands. Ecf is the conduction band Fermi level and EHHf=ELHf is the valence band Fermi level. Their values
are determined by the standard methods [13]. The symbol kB is Boltzmann’s constant, T is temperature, m¯α∗ is the average effective mass for the
particular band which is approximated as the effective mass in the well since
most of the carriers are confined there. Here, Fn,αp(z) and En,αp are the respective envelope eigenfunctions and
eigenvalues of the various subbands.We use our previous relation for optical gain quantity
[5, 6], which incorporates band-gap renormalization, Coulombic, and
non-Markovian effects [14–19]. The TE material gain is given by (4)g(N,ω)=ωμcnrV∑vlm∑k∥|ε^⋅Mlmv(k∥)|2×(flc(N,k∥)−fmv(N,k∥))ℜΞlmv(0,Δlmv(k∥))1−ℜq1lmv(N,k∥). The details of this equation can be found in [19]. We have
added the functional dependence on carrier density N for clarity. The term Mlmv(k∥) represents the dipole matrix elements for
which the calculation of electron and hole wavefunctions was required. For this
case, the holes are solved in the LK approximation. The superscript v on the variables represents the heavy and
light hole bands and l and m represent the subband indices of the
conduction and hole subbands.The calculations are performed in a self-consistent way as
described before [12]. Optical gain is determined using (4). Finally, the
differential gain is determined by numerically differentiating expression for
TE gain as (5)g′(E,N)≈g(E,N+δN)−g(E,N)δN, where δN is a small increment in the average carrier
density in the wells.


## 3. Results and Conclusions

We performed our analysis for a double quantum well system
containingIn0.38Ga0.62As1−yNy wells and In0.05Ga0.95As0.985N0.15 barriers. In
our calculations, the well widths are 7 nm and composition of nitrogen in the
wells varies from 0.5% to 1.5%. The barrier was varied between 2 nm  and 6 nm.Figure1 shows differential gain calculated at optical gain
peak versus nitrogen composition for several values of barrier thickness. One
observes significant dependence of the differential gain on nitrogen
composition. For the considered cases, it attains the largest value of 4.45×10−18cm2 for 1.5% of nitrogen, which fits well in the 1.3μm telecommunication window. At the same time,
the dependence on the barrier width is relatively weak.Figure 1
Differential gain calculated at optical gain peak versus
nitrogen composition for several values of barrier thickness.To analyze barrier dependence, in Figure2 we showed
differential gain calculated at optical gain peak versus barrier thickness for
several values of nitrogen composition. The strongest dependence on barrier
thickness is for 0.5% of nitrogen composition. For 1.5% of nitrogen the value
of differential gain for 2 nm to 6 nm barrier's thicknesses is almost flat.Figure 2
Differential gain calculated at optical gain peak versus
barrier thickness for several values of Nitrogen composition.In Figure3 we analyze
spectra of differential gain for the case of 0.5% of nitrogen. The peak of
differential gain is around 1080 nm range and shifts towards smaller
wavelengths when barrier width increases. It is far from the peak value of
optical gain which for 0.5% of nitrogen is around 1190 nm wavelength, see Figure 4.
A similar situation can be observed for 1.0% and 1.5% of nitrogen where also
peak of differential gain lies in a different range of wavelengths than peak of
optical gain.Figure 3
Differential gain spectra for several values of
barrier thickness and nitrogen composition of 0.5%.Figure 4
Wavelength of maximum gain and differential gain versus
nitrogen composition for 2 nm barrier thickness.Finally, in Figure4 we showed the results of the wavelength
dependence of maximum gain and differential gain versus nitrogen composition
for 2 nm barrier thickness. One can observe linear dependence of nitrogen
composition in both cases. The differential gain spectrum is a direct implication of the conduction
band nonparabolicity of GaInNAs alloys which results in the high electron
effective mass.In conclusion,
we have analyzed the effects of nitrogen composition and barrier width in a
double quantum well system consisting of InGaAsN/GaAs. A 10-bandk⋅p Hamiltonian
matrix was used in the calculations and solved self-consistently with Poisson's
equation. The effect of nitrogen composition and barrier thickness on differential
gain has been determined. The influence of nitrogen composition on differential
gain is significant whereas barrier effects are modest.



---

*Source: 786939-2008-07-03.xml*