Optical gain and threshold current density of strained wurtzite GaN/AlGaN quantum dot lasers

Bouchenafa, H.; Benichou, B.; Bouchenafa, H.; Benichou, B.

doi:10.31349/revmexfis.69.010503

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Revista mexicana de física

versión impresa ISSN 0035-001X

Rev. mex. fis. vol.69 no.1 México ene./feb. 2023 Epub 19-Ago-2024

https://doi.org/10.31349/revmexfis.69.010503

Condensed Matter

Optical gain and threshold current density of strained wurtzite GaN/AlGaN quantum dot lasers

H. Bouchenafa^a^b^*

B. Benichou^c

^{^a} Department of Physics, Faculty of Exact Sciences and Informatics, Hassiba Benbouali University of Chlef, 02000 Chlef, Algeria.

^{^b} Laboratory for Theoretical Physics and Material Physics, Hassiba Benbouali University, Chlef 02000, Algeria.

^{^c} Department of Electronics, Faculty of Technology, Hassiba Benbouali University of Chlef, Chlef 02000, Algeria.

Abstract

In this work, the influences of biaxial compressive and tensile strains on optical gain and threshold current density are investigated theoretically as a function of the side lengths of the quantum box in the GaN/Al_0.2Ga_0.8N structure by using a model based on the density matrix theory of semiconductor lasers with relaxation broadening. For various side lengths of the quantum box, we compare the spectra gain curves of compressive, tensile-strained, and unstrained structures of the GaN/Al_0.2Ga_0.8N cubic quantum-dot (QD) laser. The dependence of peak optical gain on carrier density and the modal gain on current density is plotted too for all cases. The results reveal that many enhancements can be made to the laser structure by introducing -0.5 % compressive strain: a higher value of optical gain of 18421 cm^-1 at L = 60 Å, a lower value of transparency of carrier density of Ntr=0.13×1019 cm^-3 and transparency current density of Jtr=26.9 A/cm² and a lower threshold current density of Jth=78.87 A/cm² at L = 100 Å.

Keywords: Quantum dot lasers; optical gain; III-N semiconductors; biaxial strain; threshold current density

1. Introduction

Modern opto-electronic devices, such as laser diodes (LDs) and light-emitting diodes (LEDs), rely heavily on semiconductor nanostructures (LEDs). The benefits of utilizing quantum dots in the active region of these devices stem from the three-dimensional confinement of charge carriers, which results in a delta function-like density of states [¹-⁴]. In order to achieve the target wavelengths of multiple applications in entertainment technologies, telecommunications, and medical engineering, various material systems must be used for the active regions.

Especially for optoelectronics, group III-nitride compounds and alloys have emerged as an essential and versatile class of semi-conductor materials. Although GaN, AlN, and InN all crystallize in the wurtzite structure [⁵], their band gaps are substantially different, ranging from 0.7 eV for InN to 6.2 eV for AlN [⁶]. At room temperature, ternary alloys with a wide bandgap range of 0.7 to 6.2 eV can be created by combining Indium and Aluminum with GaN, covering the spectral range from deep ultraviolet (UV) to infrared (IR) [⁷, ⁸].

Strain is always present in group III-nitride based devices due to the significant differences in lattice parameters and thermal expansion coefficients between the substrate and the nitride overlayers, as well as between nitride layers with various alloy compositions [⁵].

The optical properties of quantum dot structures are influenced by strain, in particular the optical gain, the energy of transitions, and the average direct band gap.

In this paper, we calculate theoretically and compare the optical gain and threshold characteristics of the GaN/Al_0.2Ga_0.8N quantum dot lasers, taking into account the influence of strain and side length boxes at an injected carrier density of Nv=3.1019 cm^-3.

2. Theoretical aspects

2.1. Optical gain and threshold current density

We determine the material gain of quantum dot lasers utilizing the method of Assada et al. [², ⁹], taking into account the intraband relaxation. In general, the optical gain may be written in the form:

g(ω)=ωnrμ0ϵ0∑lmn‍∫Eg∞<Rcv2>×gcv(fc-fv)ℏτin(Ecv-ℏω)2+ℏτin2dEcv. (1)

The conduction band (or heavy-hole band) is designated by the subscript c (or v), while fc and fv are the corresponding Fermi functions of the conduction and valence band states, respectively, Rcv is the dipole moment, Ecv is a transition energy between the conduction band and valence band, ω is the angular frequency of light, ϵ0, μ0 and nr are the dielectric constant, vacuum permeability and refractive index. The intra-band relaxation time τin is assumed to be 0.1 ps.

g _cv is the density of states for the quantum dot, given by [²]:

gcv(Ecv)=2δ(Ecv-Ecnml-Evnml-Eg)LxLyLz, (2)

where Ecnml and Evnml represent the quantized electron and hole energy levels respectively of a quantum box structure in the x, y and z directions, and δ(E) is the delta function [¹⁰]. Energy levels are given by the following equation if we assume a structure that comprises of a quantum box in the form of a cubic with dimensions of Lx, Ly and Lz [¹¹].

Ecnml=ℏ22mc*nπLx2+mπLy2+lπLz2, (3a)

Evnml=ℏ22mv*nπLx2+mπLy2+lπLz2, (3b)

where the effective masses of an electron and a hole are mc*= and mv*, respectively, while the labels of the quantized energy levels in the box are n, m and l.

In Eq. (1), the electron and hole in the quantum box are assumed to be in equilibrium, as determined by the quasi-Fermi levels Efc and Efv.

It is been assumed that the transition from the first conduction band to the first valence band (heavy hole band) is made because the density of states in the light hole band is lower than that of the heavy hole band, and its probability of occurrence is greater than that of the other transitions.

The modal gain is another main parameter characteristic of lasing action. To calculate it, multiply the optical gain by the confinement factor. The lasing action occurs when the modal gain exceeds the total loss [⁹, ¹²].

It is written as gm=Γ.g, where Γ is the optical confinement factor.

The threshold current density is written as [², ¹³, ¹⁴]:

Jth=kηqLzNthτs, (4)

where q indicates electron charge, η is the rate of quantum box surface area included in the total area, k defines the number of layers in the quantum dot array, and τs represents carrier life time.

The threshold carrier density Nth is given as follow [¹⁵]:

Nth=Ntr+1Γgdαi+12Lcln1R, (5)

where internal loss is αi, cavity length and reflectivity are denoted by Lc and R, respectively, differential gain is gd, and Ntr represents transparency carrier density.

2.2. Effects of strain

Consider a layer of wurtzite crystal of GaN grown along the c axis (z-axis) on a thick Al_x Ga_1-x N layer with a compressive or tensile strain. The strain effects are calculated in the following manner. First, the strain in the plane ϵxx of the epitaxial growth assumed to be positive for tension and negative for compression is [¹⁶-¹⁹]:

ϵxx=ϵyy=a0-aa, (6)

where a0 and a are the lattice constants of the Al_x Ga_1-x N barrier and the GaN dot layers, respectively.

The strain in the perpendicular direction can be expressed as:

ϵzz=-2C13C33.ϵxx, (7)

with ϵxy=ϵyz=ϵzx=0, where C13 and C33 are the stiffness constants of the GaN dots (well) layer.

It is widely known that the primary distinction between III-nitride wurtzite-type and other direct band gap semi-conductors is that the conduction band does not degenerate at the Γ point. However, the valence band is split into three sub-bands: heavy hole (hh), light hole (lh), and crystal field split-off band (ch) [²⁰].

The strain induced an energy shift in the conduction and the three valence bands at the Γ point [⁵]:

δEc=acz.ϵzz+act.(ϵxx+ϵyy), (8a)

δEhh/lh=(D1+D3).ϵzz+(D2+D4).(ϵxx+ϵyy), (8b)

δEch=D1.ϵzz+D2.(ϵxx+ϵyy), (8c)

where, the strain tensor component is ϵkl and the conduction-band deformation potentials along the x axis and perpendicular to the c axis are acz and act respectively. Dj(j=1,...4) is the potential deformation.

Following the convention for the wurtzite structure, we note that the energy origin is selected at the maximum of the unstrained valence band.

3. Results and discussions

We suppose the quantum dot structure under analysis contains a GaN active layer with L side length of (L=Lx=Ly=Lz) sandwiched by Al_x Ga_1-x N barriers. Taking into account the strain effect, we compute quantum dot quantized energy levels for conduction and valence bands, as well as strain band-gap energy, which are then used to determine optical gain and threshold current density in the model stated above. Table I summarizes the values of the various physical parameters utilized in the calculation.

Table I The parameters used in the calculations. (m ₀ is the free electron mass) [⁵, ¹⁷, ¹⁹, ²¹].

	GaN	AlN
Energy parameters (eV)
Eg at 300 K	3.43	6.2
Δ₁ (eV)	0.01	-0.0227
Δ₂ = Δ₃ (eV)	0.00567	0.012
Effective masses
m_e	0.2 m ₀	0.3 m ₀
m _hh	0.8 m ₀	1.14 m ₀
refractive index n _r	2.67	2.03
Deformation potentials (eV)
a _cz	4.6	4.5
a _ct	4.6	4.5
D ₁	-3.7	-17.1
D ₂	4.5	7.9
D ₃	8.2	8.8
D ₄	-4.1	-3.9
Elastic stiffness constants (GPa)
C ₁₃	106	108
C ₃₃	398	373

In Fig. 1, at an injected carrier density of Nv=3.1019 cm^-3, the optical gain is presented as a function of the transition energy for various side lengths of a cubic quantum box (L=60,80,100 Å) for compressive strain, unstrained, and tensile strain structures, respectively. It is intriguing to observe a significant increase in optical gain for compressive strain GaN quantum dots compared to unstrained and tensile strain GaN quantum dots, because compressive strain raises the average direct band-gap and makes it easier to populate inversions for given carrier densities.

Figure 1 Optical gain spectra for GaN/Al_0.2Ga_0.8N quantum dot lasers with -0.5% compressive strain, unstrained and 0.5% tensile strain for various sizes of quantum box at Nv=3.1019 cm^-3.

It is also noted that optical gain values decrease with increasing box sizes. Due to the increase in carrier density for population inversion in small-sized quantum boxes, the optical gain is higher in both cases for a small-sized box with L = 60 Å.

On the other hand, when the quantum dot’s size becomes larger, the carriers in the box are dispersed throughout useless levels, and the separation between energy levels is insufficient to achieve a high gain.

The variation of optical gain as a function of the transition energy under a compressive strained and a tensile strained for different magnitudes |ϵxx|=0.25, 0.5 and 0.75% are presented in Fig. 2, where we observe that the compressive strain presents a higher optical gain with similar values for the three cases (-0.25,-0.5,-0.75%) compared to that of tensile strain. It can also be seen that, in the case of tensile strain, the value of optical gain increases as the magnitude |ϵxx| decreases.

Figure 2 Optical gain spectra for a strained GaN/Al_x Ga_1-x N (|ϵxx|=0.25, 0.5 and 0.75%) quantum dot lasers at L = 60 Å, Nv=3.1019 cm^-3: a) compressive strain, b) tensile strain.

Table II summarizes the maximum optical gain values and the peak photon energy obtained from Fig. 2 for an injected carrier density Nv=3.1019 cm^-3 at L = 60 Å.

Table II Comparative table of value of the maximum gain and peak transition energy for GaN/Al_x Ga_1-x N compressive and tensile strain.

L = 60 Å	Compressive strain	Tensile strain
	-0.25% -0.5% -0.75%	0.25% 0.5% 0.75%
Gain max (cm^-1)	17198 18074 18370	12505 8780 5092
Transition energy (eV)	3.665 3.668 3.67	3.615 3.611 3.609

Figure 3 shows a comparison of peak optical gain as a function of carrier density for compressive strain, unstrained and tensile strain structures with varied quantum box sizes. This diagram contains two zones (on the positive side) as well as absorption (negative side). We can observe that the optical gain increases rapidly as carrier density increases above N_tr , the point at which the material begins to amplify the photon whose energy fulfills the Bernard-Duraffourg conduction (Eg<hv< E_fc - E_fv ) for each box size (where hv represents photon energy).

Figure 3 Peak optical gain as a function of carrier density for a GaN/Al_0.2Ga_0.8N quantum dot laser with -0.5% compressive strain, unstrained and 0.5% tensile strain for various sizes of quantum box.

The gain coefficient of a smaller sized quantum dot has a higher transparent carrier density value, than that of a larger sized quantum dot. In comparison to the other two structures, we can see that the compressively strained quantum dot laser has a lower transparent carrier density. On the other hand, while having the biggest differential gain, the tensile strained quantum dot laser has the highest transparent carrier density. Figure 4 shows the relationship between peak modal gain and current density values for three distinct quantum box sizes with: compressive strain, unstrained and tensile strain structures. This curve is significant because it depicts the relationship between the three primary parameters: gain modal, current density and side of quantum box length, and enables quick comparison of various quantum dots.

Figure 4 Peak modal gain for GaN/Al_0.2Ga_0.8N quantum box as a function of current density, Nv=3.1019 cm^-3, for various box sizes for -0.5% compressive strain, unstrained, and 0.5% tensile strain.

It is apparent from this figure that the modal gain increases as current density increases and remains almost constant at higher values of current density.

We also notice that as the side length of the quantum box increases, the transparency current density Jtr (intercept at gain = 0) decreases. At this value, the active layer cannot absorb nor amplify the light of the lasing wavelength. Furthermore, when compared to the other two structures, the tensile strained quantum dot laser has the highest transparent current density, while the compressively strained structure has the lowest.

To reach laser oscillation, it is necessary that the modal gain must be equal the total losses αtotal. The laser oscillation condition is given as [²²-²⁶]:

Gmod=Γgth=αi+12Lcln1R=αtotal. (9)

In Fig 5, we show how threshold current density varies with inverse cavity length for the three GaN/Al_0.2Ga_0.8N quantum dot lasers: compressive strain, unstrained, and tensile strain at L = 100 Å and Nv=3.1019 cm^-3. Assuming that αi=5 cm^-1, R = 0.3, n = 1, Lc1=2.4 mm and Lc2=4.8 mm. As can be seen from the graph, the threshold current density rises as the reciprocal cavity length grows due to a corresponding increase in mirror loss. As a result, a long cavity length is required to obtain a low threshold current density.

Figure 5 Threshold current density versus reciprocal cavity length for GaN/Al_0.2Ga_0.8N QD structure of -0.5% compressive strain, unstrained and 0.5% tensile strain at L = 100 Å and Nv=3.1019 cm^-3.

Table III shows a summary of the findings from this study. Depending on the above results and Table III, we can deduce that:

The value of optical gain increases as the quantum dot laser size reduces. On the other hand, the value of the threshold current decreases as the size of the box increases.
Compressively strained quantum dot lasers are more effective as a result of increased optical gain, while tensile strained has a smaller impact.
In comparison to the unstrained and tensile strained structures, the compressively strained quantum dot laser has a lower transparent carrier density (Ntr) and transparency current density (Jtr). While the tensile straine has the highest transparent carrier density and transparency current density (for the same size), what is important is the threshold current density that can be defined by the total losses due to the facet transmission and to the intrinsic absorption. For example, we can see that the compressively strained quantum dot laser, for a smaller threshold gain Γgth1=αT1=7.5 cm^-1 (long cavity length), will get a lower threshold current density indicated by the intersection between the threshold gain line and the curve of peak modal gain as a function of current density.
The quantum box’s optimal size for the minimum threshold is shifted to a longer length of the side because low gain can be attained without carrier injection for the ineffective levels. Which means that the compressively strained quantum dot is the optimal structure with L = 100 Å, a gain of 5921 cm^-1 and a minimum threshold current density of 78.87 A/cm² for a cavity length of Lc=4.8 mm.

Table III Performance characteristics of GaN/Al_0.2Ga_0.8N quantum dot structures for various sizes of cubic boxes of -0.5% compressive strain, unstrained and 0.5% tensile strain at Nv=3.1019 cm^-3.

	Compressive strain	unstrained	Tensile strain
Side length (Å)	60 80 100	60 80 100	60 80 100
Gain max (cm^-1)	18421 10362 5921	15415 9688 5666	9079 7697 5053
Transition energy (eV)	3.669 3.56 3.507	3.626 3.54 3.501	3.612 3.525 3.48
Peak wavelenght (nm)	338 348 353	342 350 354	343 352 356
Emission spectrum	UV UV UV	UV UV UV	UV UV UV
N _tr (10¹⁹ cm^-3)	0.58 0.25 0.13	0.926 0.39 0.2	1.48 0.63 0.32
Transparency current J _tr (A/cm²)	78.06 41.13 26.9	148.37 69.6 33.4	163.37 88.58 57.24
J _th (A/cm²) α _T1 = 7.5	137.06 100.6 78.87	240.5 183 127	320 208 194
α _T2 = 10	173 138.8 121.56	295.8 242 203	413 280.7 337.8

4. Conclusion

In conclusion, we have presented the background theory for the calculation of the material gain and threshold current density for GaN/Al_0.2Ga_0.8N quantum dot lasers. We also studied the impact of quantum dot strain and size on its performance. The peak modal gain has been plotted as a function of current density with the influence of strain and various quantum box sizes, as well as the threshold current density with reciprocal cavity length. It is found that adequately compressive strain improves the optical gain, and reduces the threshold current. For a larger cavity length, a quantum dot with a compressive strain has the minimum Jth. So, better performance can be achieved with a compressively strained GaN/Al_0.2Ga_0.8N quantum dot laser with L = 100 Å and a long cavity length compared to unstrained and tensile strain structures.

Acknowledgments

This research was supported by the Algerian General Directorate for Scientific Research and Technological Development (DGRSDT). https://www.mesrs.dz/en/dgrsdt.

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Received: April 16, 2022; Accepted: June 13, 2022

^* E-mail: bouchenafa_halima@yahoo.fr

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