Design and simulation to improve the structural efficiency of green light emission of GaN/InGaN/AlGaN light emitting diode

Abstract

This study considered the design of an efficient, high brightness polar InGaN/GaN light emitting diode (LED) structure with AlGaN capping layer for green light emission. The deposition of high In (>15%) composition within InGaN quantum well (QW) has limitations when providing intense green light. To design an effective model for a highly efficient InGaN green LEDs, this study considered the compositions of indium and aluminum for In_xGa_1−xN QW and Al_yGa_1−yN cap layers, along with different layer thicknesses of well, barrier and cap. These structural properties significantly affect different properties. For example, these properties affect electric fields of layers, polarization, overall elastic stress energy and lattice parameter of the structure, emission wavelength, and intensity of the emitted light. Three models with different composition and layer thicknesses are simulated and analyzed to obtain green light with in-plane equilibrium lattice parameter close to GaN (3.189 Å ) with the highest oscillator strength values. A structure model is obtained with an oscillator strength value of 1.18×10⁻¹ and least in-plane equilibrium lattice constant of 3.218 Å. This emitter can emit at a wavelength of 540 nm, which is the expected design for the fabrication of highly efficient, bright green LEDs.

Key words： green light emitting diode (LED); lattice parameter; oscillator strength; InGaN quantum well (QW); AlGaN capping layer

Introduction

Group III-Nitride-based light emittingdiodes (LEDs) are important in attaining high external quantum efficiencyin the blue wavelength range of visible spectrum [1–3]. However, the quantum efficiency of In_xGa_1−xN/GaNquantum wells (QWs) strongly decreases [2,4] forlonger wavelengths that correspond to green and yellowish green (525–565nm) colors. The degradation of optical properties is mainly attributedto the increase of In composition in In_xGa_1−xN QWs [5]. Growth temperature should be decreasedto favor large In incorporation in GaN, which can lead to an increasein extended defects or point defects density. Increased In compositionwithin QWs induces an increased stress in the structure. Risk of defectformation, such as V-defects or misfit dislocations, occurs [6] if stress is excessively high. However,growth processes for In-rich (In>20%) QWs affect the In fluctuation,carrier localization, as well as the strain relaxation processes ofthe structure; these mechanisms affect the structural and opticalproperties of the device [7–9]. The reducedoscillator strength of the fundamental transition of QW attributedto internal piezo-electric field (quantum confined Stark effect) becomesincreasingly pronounced for QWs grown along the c-plane. To compensate for the effect of strain, a numberof research groups used the growth of GaN on nonpolar or semipolarsubstrates [10,11], thick InGaN templates [12], and quantum dot structures [13]. This approach was used to demonstrategreen lasers and yellow emitting LEDs [14,15]. However,to obtain the same emission wavelength, the In composition in In_xGa_1−xN QWsgrown along a semi-polar orientation should be larger than that forthe polar orientation [16].

The optical properties of conventionalpolar growth (c-plane) systemscan be improved by using staggered InGaN structures [17,18], screening the internal electric field by dopinginto a multiple quantum well (MQW) [19], and using strain-compensated Al_yGa_1−yN inter-layersas Al_yGa_1−yN layers tensely strained on GaN [20]. Replacing part of the GaN barrierlayer by a ternary alloy can increase the magnitude of in-plane potentialfluctuations, thereby strengthening carrier localization [21]. Thus, Al_yGa_1−yN effectivelyreduces the total strain energy of MQW [22,23]. InGaN/AlGaNMQW improves the external quantum efficiency and increases the photoluminescence(PL) intensity [24]. Theuse of a thin Al_yGa_1−yN layer helps improve the PL intensity ofIn_xGa_1−xN single QW when its emission wavelength increases [25]. The most significant result isthe presence of green to red LEDs grown by metal oxide chemical vapordeposition (MOCVD) with high external quantum efficiency at this wavelengthrange [20,26–28]. The construction of efficient green to red LEDsare important in achieving monolithic white LEDs because they canact as light converters in the structure [29–31].

In this study, we aim to identifythe best GaN/In_xGa_1−xN/Al_yGa_1−yN structure to obtain the highestforce of oscillator or oscillator strength values, which in turn correspondsto PL, for green light (525–565 nm) emission. When searchingfor such high oscillator strength value, we calculated and soughtthe lowest overall strain energy of the structure because obtainingthe minimum strain energy value to minimize the chance of defect inthe structure is an obvious approach. Finally, we aimed to explainand justify the results obtained to propose the best structure forgreen light emission.

Theoretical background for modeling

Room temperature e1-hh1 transitionenergy for polar (c-plane) GaN/In_xGa_1−xN/Al_yGa_1−yN structureswas calculated by solving the Schrödinger equation via envelopfunction formalism. This calculation is shown in Fig. 1 as a functionof the QW thickness (L_W) and In composition [32]. The figure shows the possibility of obtaining greenlight emission from a series of In_xGa_1−xN QW thickness (L_W) and In composition(x). Examples include In composition(x) and QW thickness (L_W) of any of thefollowing combinations, x=18% and L_W = 3.5 nm (pointA), x= 25% and L_W= 2.2 nm (point B), x= 35%, and L_W= 1.5 nm (point C). We seek to determinewhich of the above combinations of the structural parameters willprovide the most efficient high-intensity green light emission.

Fig.1 Calculated wavelength asa function of In composition and QW thickness for InGaN/GaN system

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We determine the highest oscillatorstrength (F) value of the structure,which is considered proportional to the square overlap of the electron-holewave functions \left(F\propto {\left|\langle {\varphi }_e|{\varphi }_{hh}\rangle \right|}^{\mathrm{2}}\right). Radiative lifetime is inversely proportional tothe square of the overlap of the electron and hole wave function integralas follows [33]:

{\tau }_{\mathrm{rad}}^{-\mathrm{1}}=\frac{{nd}^{\mathrm{2}}{E}_{\mathrm{0}}^{\mathrm{3}}}{{\mathrm{3}\mathrm{\pi }\epsilon }_{\mathrm{0}}{h}^{\mathrm{4}}{c}^{\mathrm{3}}}{\left|\int f_{\mathrm{e}}{f}_{\mathrm{h}}\mathrm{d}z\right|}^{\mathrm{2}}=\frac{E_{\mathrm{0}}^{\mathrm{3}}}{A}{\left|\int f_{\mathrm{e}}{f}_{\mathrm{h}}\mathrm{d}z\right|}^{\mathrm{2}},

where d is the inter-band optical dipole of GaN, ε₀ is the permittivityof free space, h is Planck’sconstant, c is the speed of light, E₀ is the transitionenergy, and f_e and f_h are wave functions of electron and hole, respectively. The A value of 90 eV³·s was experimentally determined by Bretagnon et al. [34]. However, the relaxed in-planelattice parameter of GaN, InN and AlN are different from each other(a_{\mathrm{GaN}}=\mathrm{3.189} \AA , a_{\mathrm{InN}}=\mathrm{3.538} \AA \mathrm{and} a_{\mathrm{AlN}}=\mathrm{3.113} \AA) [35,36]. Thus, by pseudomorphically growing InGaN/AlGaN layerson top of GaN as substrate layer, we must consider the polarizationeffect of InGaN and AlGaN layers because of strain.

The coefficients of piezoelectricpolarization are significantly large in group-III nitrides [37,38]. Thus, the polar (c-plane) GaN/In_xGa_1−xN/Al_yGa_1−yN growth structure experiencesintense piezoelectric polarization effect because of GaN-InN latticemismatch. Overall polarization (both spontaneous and piezoelectric)effects play a major role in electric field generation in all layers,such as barrier (GaN) electric field (E_b), well (In_xGa_1−xN) electric field,or internal electric field (E_W), and cap (Al_yGa_1−yN) electric field (E_C). The effects ofinternal electric field within well regions (E_W) will deform the band structureof the GaN/InGaN/AlGaN system and reduce the effective transitionenergy between different energy levels within the QWs. The band bendingcaused by internal electric field E_W and well thickness L_W will reduce the transition energy,which means the wavelength of emitted light will get red-shifted.This effect is known as quantum confined Stark effect (QCSE). Theeffect reduces the overlap of electron-hole wave-function, which isundesirable for any targeted intense wavelength emission. Thus, toobtain the oscillator strength (F) of the structure, we must first determine the induced electricfields of the heterostructure. The electric field of any layer ofa superlattice can be determined by following expression [39]:

(1)E_j=\frac{{\Sigma }_i\left(P_i-P_j\right)\frac{L_i}{{\epsilon }_i}}{{\epsilon }_j{\Sigma }_i\frac{L_i}{{\epsilon }_i}},

where P_i and P_j are the total polarization of adjacent layer, {\epsilon }_iand {\epsilon }_jare permittivity of two adjacent layers.

The induced elastic strain energyper unit surface of the structure should also be considered by usingthe following expression:

(2)E_{\mathrm{e}\mathrm{l}}\left(a_{\mathrm{e}\mathrm{q}}\right)={\Sigma }_{i=\mathrm{1}}^{\mathrm{3}}{M}_i{L}_i{\Delta }_i^{\mathrm{2}},

where M_i is the biaxial modulus, L_i is the layer thickness, {\Delta }_i=\left(a_{\mathrm{e}\mathrm{q}}-a_i\right)/a_i is thestrain in the growth plane, a_i is the relaxed lattice parameter of each i layer ( i corresponds to i=1= InGaN, i= 2=AlGaN and i= 3=GaN layer) and a_{\mathrm{e}\mathrm{q}} is the overall in-plane equilibrium lattice parameter of the entireGaN/InGaN/AlGaN heterostructure. This elastic strain energy valueshould be at the minimum to reduce the chance of defect formationwithin the structure because defects act as a non-radiative recombinationcenter.

Biaxial modulus M_i and relaxed lattice parameter a_i of epilayers depend on the particular compositionvalues of the layer. The biaxial modulus of In_xGa_1−xN layershould be calculated based on the following equation:

(3)M_{{\ln }_x{\mathrm{Ga}}_{\mathrm{1}-x}\mathrm{N}}=\left(c_{\mathrm{11}}\left(x\right)+c_{\mathrm{12}}\left(x\right)-\frac{{\mathrm{2}c}_{\mathrm{13}}{\left(x\right)}^{\mathrm{2}}}{c_{\mathrm{33}}\left(x\right)}\right),

where c_{jk} are the elastic stiffness constants of In_xGa_1−xN layerand their values can be obtained using Vegard’s law as

(4)c_{jk}\left(x\right)={xc}_{jk}\left(\mathrm{InN}\right)+\left(\mathrm{1}-x\right)c_{jk}\left(\mathrm{GaN}\right).

Here, c_{jk}\left(\mathrm{InN}\right) and c_{jk}\left(\mathrm{GaN}\right) values are consideredfrom Nikolaev et al. [40]. Similar equations should be considered for Al_yGa_1−yN layer.In this work, we did not consider the bowing parameter (b =0) when calculating the band gap energyto avoid further complexity. We may also consider that E_{\mathrm{e}\mathrm{l}} is the minimum when \frac{{\mathrm{d}E}_{\mathrm{e}\mathrm{l}}}{{\mathrm{d}a}_{\mathrm{e}\mathrm{q}}}=\mathrm{0} , and the in-planeequilibrium lattice parameter of the whole structure can be expressedas

(5)a_{\mathrm{e}\mathrm{q}}=\frac{{\Sigma }_{i=\mathrm{1}}^{\mathrm{3}}\left(M_i{L}_i\prod { }_{j\ne i}{a}_j\right)}{{\Sigma }_{i=\mathrm{1}}^{\mathrm{3}}\left(M_i{L}_i\prod {}_{j\ne i}{a_j}^{\mathrm{2}}\right)}{\prod }_{i-\mathrm{1}}^{\mathrm{3}}{a}_i.

This study considered different parametersof the structure, such as the In composition (x) and layer thickness (L_W) of In_xGa_1−xN QW region, Al composition(y) and layer thickness (L_C) of Al_yGa_1−yN caplayer while keeping GaN barrier layer thickness (L_b) at 12 nm. The electric fieldof different layers is then calculated using Eq. (1). Then, usingthe different layers’ electric field values along with the structuralparameters, we attempted to simulate and identify the transition energy(emission wavelength), oscillator strength, elastic strain energyper surface, and in-plane equilibrium lattice parameter a_{\mathrm{e}\mathrm{q}} values of the structure.

Simulation process

To obtain green light (525–565nm) emission, we initially chose three models that consider the sample(as shown schematically in Fig. 2) parameters shown in Table 1. QWthickness and In composition values were taken to match those of thepoints from Fig. 1. The Al composition of each model are initiallytaken arbitrarily to 50% and AlGaN cap layer thickness (L_C) values to makethe one period length in between 14.5 and 16.5 nm.

Fig.2 Schematic diagram of thesimulated structure

Full size|PPT slide

Tab.1 Initial structuralparameters of the models for green light emission

	barrier length, Lb/nm	In composition (x)/%	InxGa1−xN QW thickness, LW/nm	Al composition (y)/%	cap layer (AlyGa1−yN) thickness, LC/nm
Model 1	12	25	2.2	50	1.2
Model 2	12	18	3.5	50	1.5
Model 3	12	35	1.5	50	1.0

Oscillator strength (F) and in-plane equilibrium lattice parameter \left(a_{\mathrm{e}\mathrm{q}}\right) should be identified todetermine the best structural parameters for high-intensity greenemission. To identify the highest possible oscillator strength andthe lowest elastic strain energy values, we analyzed the structureby varying each parameter individually while keeping all other parametersfixed. The following steps are then considered.

Step 1: The Al composition (y) of Al_yGa_1−yN cap layer is variedwhile keeping other parameters fixed to obtain the best value of Alcomposition.

Step 2: The In composition (x) of In_xGa_1−xN QW layer is variedwhile keeping the best value of Al composition obtained in Step 1and other parameters fixed.

Step 3: The In_xGa_1−xN QW thickness(L_W) valueis varied while keeping the best values of Al and In compositionsobtained in Step 1 and 2 and other parameters fixed.

Step 4: The Al_yGa_1−yN caplayer thickness (L_C) value is varied, while keeping all other best values of parametersobtained from the previous steps.

Result and discussion

Model 1

The structural parameters for Model1 are provided in Table 2 where Al compositions are varied. The resultsobtained for Model 1 of the simulation process are discussed. First,we calculated different layer electric field values for each Al compositionusing Eq. (1). Then, we simulated the band diagram of the structurefor each change. Figure 3 shows the band diagram of Model 1, whereinstructural parameters are taken from Table 1 and electric field valuesare given as: barrier electric field, E_b = 227.9 kV/cm, InGaN QW electric field, E_W = 4158.1 kV/cmand AlGaN cap layer electric field, E_C = 5343.7 kV/cm. We can easily identify theband bending phenomenon in the figure because of different valuesof layers’ field and the green line in the figure that indicatesthe simulated electron wave function within the QW region of the structure.

Tab.2 Structural parametersfor Model 1 as in Step 1 of the simulation process

type	barrier length, Lb/nm	In composition (x)/%	QW thickness, LW/nm	Al composition (y)/%	cap layer (AlyGa1−yN) thickness, LC/nm
Model 1	12	25	2.2	variable	1.2

Fig.3 Energy band diagram of Model1, wherein the structural parameters are taken from Table 1. The redlines show the valance band (VB) and conduction band (CB) and thegreen line is the electron wave function of the GaN/In₂₅Ga₇₅N/Al₅₀Ga₅₀N/GaN structure

Full size|PPT slide

The band diagram allows us to determinethe transition energy and oscillator strength values. We calculatedthe elastic strain energy (E_{\mathrm{e}\mathrm{l}}) and overall in-plane equilibrium lattice parameter (a_{\mathrm{e}\mathrm{q}}) using Eqs. (2) and (5), respectively. The results are presentedin Table 3.

Tab.3 Variation of in-planeequilibrium lattice parameter, oscillator strength, and emission wavelengthas a function of Al composition

Al composition(y)/%	in-plane equilibrium lattice parameter, {{\displaystyle a}}_{\mathrm{e}\mathrm{q}}/Å	oscillator strength, F	wavelength/nm
50	3.226	1.54×10−1	527
60	3.222	1.53×10−1	528
70	3.219	1.49×10−1	529
78	3.216	1.46×10−1	530

Table 3 shows that the best resultis found when Al composition (y) is 70% because it has high oscillator strength (F=\mathrm{1.49}\times {\mathrm{10}}^{-\mathrm{1}}) at wavelength of 529 nm and low in-plane equilibriumlattice parameter \left(a_{\mathrm{e}\mathrm{q}}\cong \mathrm{3.219}\AA \right) value,which is slightly mismatched to GaN lattice parameter \left(a_{\mathrm{GaN}}\cong \mathrm{3.189}\AA \right) of 0.94%. For Al compositions (y) higher than 70%, the observed oscillator strength valuebecomes low, whereas lower Al composition (y<70%) increases oscillator strength and in-plane equilibriumlattice parameter. However, increased value of in-plane equilibriumlattice parameter is undesirable. This finding is attributed to thefacts that the higher in-plane equilibrium lattice parameter valuerelates to increased chance of defect formation within the structure;these defects act as non-radiating recombination centers for electron-holepairs. Thus, emitted light intensity will become degraded for structureswith lower (<70%) Al composition cap layers.

The variation of In composition (x) in In_xGa_1−xN QW was examined.In composition (x) was varied toa maximum of ±3% to that of initial considered value (25%)to determine its effect on in-plane equilibrium lattice parameter,oscillator strength, and emission wavelength values. This method isadopted to keep other parameters fixed and Al composition is fixedat 70%, as obtained Step 1. The obtained results are shown in Table4.

Tab.4 Variation of in-planeequilibrium lattice parameter, oscillator strength and emission wavelengthas a function of In composition

In composition(x)/%	in-plane equilibrium lattice parameter, {{\displaystyle a}}_{\mathrm{e}\mathrm{q}}/Å	oscillator strength, F	wavelength/nm
23	3.215	1.67×10−1	509
24	3.217	1.58×10−1	519
25	3.219	1.49×10−1	529
26	3.221	1.41×10−1	539
27	3.223	1.34×10−1	550
28	3.225	1.27×10−1	561

Table 4 indicates a slight decreaseof In composition (x) at 25% willcause the emission wavelength to blue shift from 529 nm to 509 nm,which is undesirable. However, a slight increase of In composition(x) red shifts the emission tofacilitate green light emission. Oscillator strength (F) decreases and in-plane equilibrium latticeparameter (a_{\mathrm{e}\mathrm{q}}) increases, which increase the chance of dislocation, thereby decreasinglight intensity. Thus, In composition of 25% is the best parameteras it provides low in-plane equilibrium lattice parameter \left(a_{\mathrm{e}\mathrm{q}}\cong \mathrm{3.219}\AA \right) andhigh oscillator strength (F=\mathrm{1.49}\times {\mathrm{10}}^{-\mathrm{1}}) in the wavelength of green light emission (525–565nm).

The thickness variation of InGaNQW was also examined by simulating the structure and maintaining thecompositions of Al and In compositions fixed at 70% and 25%. The resultobtained from such simulation for emission wavelength, in-plane equilibriumlattice parameter, and oscillator strength is given in Table 5.

Tab.5 Variation of in-planeequilibrium lattice parameter, oscillator strength and emission wavelengthas a function of QW thickness

QW thickness, LW/nm	in-plane equilibrium lattice parameter, {{\displaystyle a}}_{\mathrm{e}\mathrm{q}}/Å	oscillator strength, F	wavelength/nm
2	3.216	2.19×10−1	511
2.1	3.217	1.82×10−1	520
2.2	3.219	1.49×10−1	529
2.3	3.220	1.21×10−1	539
2.35	3.221	8.35×10−2	548

The result in Table 5 shows thatdecreased QW thickness improved the parameters, but emission wavelengthblue shifts from green emission. Thus, we cannot excessively decreaseQW thickness. However, a slight increase in QW thickness value (from2.2 to 2.3 nm) facilitates the central emission peak (539 nm) of greenlight spectrum (525–565 nm), but the lattice parameter of in-planeequilibrium and oscillation strength was slightly degraded. Thus,we selected QW thickness of 2.3 nm to obtain an emission wavelengthof 539 nm, in-plane equilibrium lattice parameter \left(a_{\mathrm{e}\mathrm{q}}\right), and oscillator strength(F) of about 3.220 Å and1.21×10⁻¹.

In Step 4, the variation of Al_0.7Ga_0.3N cap layer thickness(L_C) wasexamined. The results are shown in Table 6. Cap layer thickness L_C of 1.3 nm providesthe best results, but a slight increase in L_C lowers the oscillator strengthfrom 1.18×10⁻¹ to 1.14×10⁻¹. A slight decrease in L_C (see Table 6) increases oscillatorstrength (F) value and in-planeequilibrium lattice parameter (a_{\mathrm{e}\mathrm{q}}), which will increase the chance of generating dislocation in thestructure.

Tab.6 Variation of in-planeequilibrium lattice parameter, oscillator strength and emission wavelengthas a function of AlGaN cap layer thickness

cap layer thickness, LC/nm	in-plane equilibrium lattice parameter, {{\displaystyle a}}_{\mathrm{e}\mathrm{q}}/Å	oscillator strength, F	wavelength/nm
1.2	3.220	1.21×10−1	539
1.3	3.218	1.18×10−1	540
1.4	3.215	1.14×10−1	541

Comparison

The best parameters for Models 2and 3 (as described in Table 1) were also identified and summarizedin Table 7. A comparison of the three models shows that each modelwill provide green light, but the intensity of Model 2 will be lowbecause oscillator strength (F)is the lowest (F=\mathrm{1.05}\times {\mathrm{10}}^{-\mathrm{2}}); however, the in-plane equilibrium lattice parameteris lowest \left(a_{\mathrm{e}\mathrm{q}}\cong \mathrm{3.217}\AA \right), whichsuggest the least possible defects in the structure. A comparisonbetween Model 1 and 3 suggests that Model 1 is slightly better becausethe in-plane equilibrium lattice parameter \left(a_{\mathrm{e}\mathrm{q}}\right) is slightly lower in valuethan that of Model 3 (Table 7). This finding suggests that the changein defect generation is low for Model 1 because the strain value islower than that of Model 3 (for Model 1, strain is \Delta =\left(\mathrm{3.2182}-\mathrm{3.189}\right)/\mathrm{3.189}=\mathrm{9.18}\times {\mathrm{10}}^{-\mathrm{3}} and that forModel 3 is \Delta =\left(\mathrm{3.22498}-\mathrm{3.189}\right)/\mathrm{3.189}=\mathrm{11.28}\times {\mathrm{10}}^{-\mathrm{3}}). However, oscillatorstrength is slightly low.

Tab.7 Summary of thethree models

type	barrier length, Lb/nm	In composition (x)/%	QW thickness, LW/nm	Al composition (y)/%	cap layer (AlyGa1−yN) thickness, LC/nm	emission wavelength, λ/nm	oscillator strength, F	in-plane equilibrium lattice parameter, {{\displaystyle a}}_{\mathrm{e}\mathrm{q}}/Å
Model 1	12	25	2.3	70	1.3	540	1.18×10−1	3.218
Model 2	12	18	3.6	40	1.8	542	1.05×10−2	3.217
Model 3	12	35	1.6	67	1.2	548	3.18×10−1	3.224

Conclusion

We simulated and analyzed GaN/In_xGa_1−xN/Al_yGa_1−yN heterostructuresto identify the best structural parameters for high-intensity greenand yellowish green light (525–565 nm) emissions. We determinethe appropriate structure that provides the highest oscillator strength(F) and lowest elastic strain energyvalues. A comparison of the data of three models (Table 7) shows thestructure of Model 1 for In_xGa_1−xN QW, In composition (x = 25%) and thickness (L_W = 2.3 nm), Al_yGa_1−yN caplayer for Al composition (y = 70%)with a cap layer thickness of L_C = 1.3 nm. This value provides a relativelyhigh oscillator strength (F) valueof 1.18×10⁻¹ and low in-planeequilibrium lattice parameter \left(a_{\mathrm{e}\mathrm{q}}\right) of 3.218 Å. Thus,the structure of Model 1 will have the least structural defects andwill provide green light with higher intensity than the other twomodels (i.e., Models 2 and 3).

Acknowledgements

The authors thank Dr. Benjamin Damilano andMr. Philippe Vennéguès, Centre de Recherche sur l’H´et´ero-Epitaxieet ses Applications (CRHEA), Centre National de la Recherche Scientifique(CNRS), Valbonne 06560, France for their support.

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