Review of the Properties of GaN, InN, and Their Alloys Obtained in Cubic Phase on MgO Substrates by Plasma-Enhanced Molecular Beam Epitaxy

Abstract

Gallium nitride (GaN) semiconductors and their broadband InGaN alloys in their hexagonal phase have been extensively studied over the past 30 years and have allowed the development of blue-ray lasers, which are essential disruptive developments. In addition to high-efficiency white light-emitting diodes, which have revolutionized lighting technologies and generated a great industry around these semiconductors, several transistors have been developed that take advantage of the characteristics of these semiconductors. These include power transistors for high-frequency applications and high-power transistors for power electronics, among other devices, which have far superior achievements. However, less effort has been devoted to studying GaN and InGaN alloys grown in the cubic phase. The metastable or cubic phase of III-N alloys has superior characteristics compared to the hexagonal phase, mainly because of the excellent symmetry. It can be used to improve lighting technologies and develop other devices. Indium gallium nitride, In_xGa_1−xN alloy, has a variable band interval of 0.7 to 3.4 eV that covers almost the entire solar spectrum, making it a suitable material for increasing the efficiencies of photovoltaic devices. In this study, we successfully synthesized high-quality cubic InGaN films on MgO (100) substrates using plasma-assisted molecular beam epitaxy (PAMBE), demonstrating tunable emissions across the visible spectrum by varying the indium concentration. We significantly reduced the defect density and enhanced the crystalline quality by using an intermediate cubic GaN buffer layer. We not only developed a heterostructure with four GaN/InGaN/GaN quantum wells, achieving violet, blue, yellow, and red emissions, but also highlighted the immense potential of cubic InGaN films for high-efficiency light-emitting diodes and photovoltaic devices. Achieving better p-type doping levels is crucial for realizing diodes with excellent performance, and our findings will pave the way for this advancement.

1. Introduction

Over the last 30 years, group III nitrides have undergone remarkable development [1,2,3,4]. The 1990s marked a significant turning point, with breakthroughs in blue-ray lasers [5,6], a disruptive advancement that has reshaped the industry, and high-efficiency white-light-emitting diodes [7,8,9]. This revolutionary technology has transformed lighting globally. These advancements have sparked immense industrial interest in semiconductor compounds [10]. Of greater significance, group III nitrides have been recognized as highly promising candidates, heralding the future of materials for both electronic and optoelectronic applications [11,12]. The family of group III nitrides, encompassing AlN, GaN, and their InN alloys, are characterized by their wide bandgaps, with the exception of InGaN which possesses a high indium content. These materials are capable of crystallizing in either wurtzite or zincblende polytypes. The wurtzite structure features direct bandgaps ranging from approximately 0.7 eV for InN to 3.4 eV for GaN and reaching up to 6.2 eV for AlN. By alloying GaN with AlN and InN, a continuous spectrum of direct bandgap energies spanning much of the visible spectrum and extending into ultraviolet (UV) wavelengths can be achieved. These nitride systems are particularly promising for optoelectronic applications such as light-emitting diodes (LEDs), laser diodes (LDs), and UV detectors.

In the case of InGaN films, nonstoichiometry can manifest as indium-rich or nitrogen-rich regions. Indium-rich regions can lead to phase separation, which affects the uniformity of optical emission. Nitrogen-rich regions can introduce nitrogen vacancies, acting as shallow donors and increasing n-type conductivity, but can also potentially reduce electron mobility due to increased scattering.

By carefully controlling the stoichiometry during growth processes, such as plasma-assisted molecular beam epitaxy (PAMBE), the electrical, optical, and structural properties of the films can be optimized for specific applications in optoelectronic devices and photovoltaic cells.

The successful commercialization of bright blue and green LEDs, coupled with the potential application of yellow LEDs, has significantly advanced the development of full-color displays. Despite these advancements, these LEDs encounter notable challenges, particularly the “green gap”, which refers to the reduced efficiency of LEDs in the green/yellow spectral range. Additionally, these LEDs suffer from efficiency droop at high operating currents, which further complicates their performance [13,14].

The physical nature of the green gap has been a subject of extensive debate [15,16]. Group III nitrides with their unique properties have the potential to address these efficiency issues, further underlining their relevance and prospects in the field of optoelectronic applications.

The nonstoichiometry of InGaN films, resulting from deviations in the intended stoichiometric ratios during growth, plays a crucial role in determining the electrical and optical performances of these materials.

Nonstoichiometric defects, such as nitrogen vacancies, can enhance n-type conductivity by increasing the free electron concentration. However, these defects also act as scattering centers, reducing the carrier mobility and thus the overall electrical conductivity.

Localized states introduced by nonstoichiometric defects can lead to bandgap variations and reduced radiative efficiency owing to nonradiative recombination centers.

Nonstoichiometry can degrade the crystalline quality and induce phase separation, leading to compositional inhomogeneities that affect electrical and optical properties.

The presence of nonstoichiometric regions can introduce additional mechanical stress and strain, impacting the structural integrity of the films.

In the case of InGaN films, nonstoichiometry can manifest as indium-rich or nitrogen-rich regions. Indium-rich regions can lead to phase separation, which affects the uniformity of optical emission. Nitrogen-rich regions can introduce nitrogen vacancies, act as shallow donors, and increase n-type conductivity, but they can also potentially reduce electron mobility owing to increased scattering.

Through meticulous control of stoichiometry during growth processes like plasma-assisted molecular beam epitaxy (PAMBE), the optimization of electrical, optical, and structural properties of films for specific applications in optoelectronic devices and photovoltaic cells can be achieved. The epitaxial growth of III-nitride (In, Ga, Al)N alloys has brought about a revolution in optoelectronics and power electronics due to their distinctive properties, such as tunable direct bandgaps, high electron mobility, and thermal stability. Among various epitaxial growth techniques, PAMBE is particularly noteworthy for its precision in controlling thin film stoichiometry, doping, and morphology.

PAMBE combines the versatility of molecular beam epitaxy (MBE) with the activation of gaseous precursors using radiofrequency (RF) plasma. This approach allows more stable nitrogen precursors and facilitates nitrogen incorporation into the growing film. Plasma activation also promotes the formation of reactive species such as N radicals and ions, which enhance the growth kinetics and crystalline quality of the film.

The main advantage of PAMBE is its ability to independently control the fluxes of group III elements (In, Ga, and Al) and active nitrogen, which allows fine tuning of the alloy stoichiometry and thus its forbidden band bandgap. In addition, PAMBE offers precise doping control by introducing dopant precursors such as Si for n-type doping and Mg for p-type doping.

Despite these advantages, the PAMBE has several limitations. The generation of RF plasma can lead to point defects, such as nitrogen vacancies, which affect the electrical and optical properties of films [8]. Furthermore, precise control of the alloy composition in multi-element systems, such as (In, Ga, Al) N, can be challenging owing to the differences in precursor volatility and reactivity.

III-Nitrides can crystallize in two main phases: hexagonal (wurtzite) and cubic (zincblende). The hexagonal phase is the most common and stable phase and is widely used in optoelectronic devices, such as light-emitting diodes (LEDs) and lasers. Although less stable, the cubic phase offers advantages for power electronics owing to its higher symmetry and lower spontaneous polarization, which reduces the internal electric fields and improves charge carrier mobility.

PAMBE was successfully employed to grow both III-nitride phases. High-quality films with precise control of the crystallographic orientation, alloy composition, and doping have been fabricated in the hexagonal phase. In the cubic phase, PAMBE has enabled the growth of films with low defect densities and promising electrical properties for power electronic applications.

An important characteristic of III-nitrides is their spontaneous and piezoelectric polarization, which arises from the lack of inversion symmetry in their crystal structures. In the hexagonal phase, spontaneous polarization is aligned along the c-axis, whereas in the cubic phase, it cancels out owing to the symmetry. Piezoelectric polarization is induced by the strain in the crystal lattice and can be significant in III-nitride heterostructures.

Spontaneous and piezoelectric polarizations significantly influence the electronic and optical properties of III-nitride-based devices. For example, in the InGaN/GaN quantum wells used in LEDs, polarization induces internal electric fields that spatially separate charge carriers and reduce the radiative recombination efficiency. Polarization can affect the formation of conduction channels and the electron mobility in power electronics.

PAMBE has established itself as a critical technique for epitaxial growth of III-nitride semiconductor materials. Its ability to precisely control film composition, doping, and morphology, along with its compatibility with both crystal phases, makes it a powerful tool for the research and development of next-generation optoelectronic and power electronic devices.

The cubic phase of GaN and its alloys with In can overcome several of the problems associated with the hexagonal phase.

This article presents our original research on the growth of cubic InGaN alloys using plasma-assisted molecular beam epitaxy (PAMBE). We detailed the experimental methods, results, and significant findings related to the structural, optical, and electrical properties of nitride semiconductors. This study highlights the development of devices based on these materials, emphasizing their potential applications in optoelectronic devices such as LEDs and solar cells.

2. Method of Synthesis of Nitride and Their Principal Characteristics

The first efforts to grow InN, GaN, AlN, and their alloys through hydride vapor phase epitaxy (HVPE) and sputtering processes occurred between the end of the 1960s and beginning of the 1980s [17,18,19,20,21,22]. Subsequently, with the advent of electronic-grade purity precursors, GaN, InGaN, and AlGaN have been synthesized by the deposition of metal–organic vapor and plasma sources to produce nitrogen radicals compatible with MBE systems [23,24,25]. The absence of a substrate with reasonably good thermal and lattice mismatches for nitrides leads to the coexistence of films of these semiconductors with a high dislocation density. However, it is important to note the significant progress made in designing excellent optoelectronic devices [3]. Metal–organic chemical vapor deposition (MOCVD) was first developed for the growth of GaN [26]. HVPE [17,18] and MOCVD [26,27] have been used to grow GaN directly on sapphire substrates. GaN and its alloys were grown by plasma-assisted molecular beam epitaxy (PAMBE) on different substrates such as Al₂O₃, GaAs, Si, and MgO [28,29,30,31]. However, this review focuses primarily on the deposition of layers on MgO substrates, highlighting the unique advantages and outcomes of this material.

Table 1. Structural properties of GaN in its zincblende and wurtzite phases.

Property	GaN (Zincblende)	GaN (Wurtzite)	References
Crystal Structure	Cubic (Zincblende)	Hexagonal (Wurtzite)	[32,33]
Lattice Constant (a)	4.50 Å	3.189 Å	[34,35]
Lattice Constant (c)	N/A	5.185 Å	[34,35]
Lattice Spacing	2.25 Å	2.60 Å (a–c)	[36]
Types of Defects	Vacancies, Interstitials	Dislocations, Stacking Faults	[37,38]
Lattice Strain and Stress	Lower Compared to Wurtzite	Higher due to Hexagonal Structure	[39,40]
Bond Energy	8.9 eV	8.2 eV	[41,42]
Elastic Modulus (E)	253.4, 281.18, 285.3 GPa	295 GPa	[43,44]
Poisson’s Ratio (ν)	0.29	0.26	[45,46]

shows the main properties of the zincblende and wurtzite phases.

In recent years, significant technological advancements have been made in the synthesis of In_xGa_1−xN/GaN films, culminating in the development of GaN-based light-emitting diodes featuring quantum-well (QW) heterostructures, nanoarrays, and nanodots without embedded QW active layers [47,48,49]. The inherent challenges in growing In_xGa_1−xN/GaN, particularly the high equilibrium vapor pressures (EVPs) of nitrogen over InN and the considerable lattice mismatch between InN and GaN, have been addressed through innovative approaches. This substantial lattice mismatch results in highly strained In_xGa_1−xN/GaN alloys, leading to concerns regarding phase separation. However, the introduction of plasma-assisted molecular beam epitaxy (PAMBE) has revolutionized this field, providing a promising technique for the study and development of devices based on cubic nitrides, including GaN, InGaN, AlGaN, and complex structures.

MBE and its derivatives for nitrides, PAMBE, are sophisticated growth techniques (beyond thermodynamic equilibrium) for thin films and nanostructures. It uses different tools for real-time surface analysis, precise growth control, and an ultrahigh-purity environment in which deposition occurs. In MBE, the film deposition process involves exposing the clean surface of a crystalline substrate, maintained at a specific temperature, to atomic beams. Contrary to what occurs in MOCVD, in which chemical reactive processes play a crucial role, the growth process in MBE is purely physical. An essential characteristic of MBE is that its synthesis is performed in an ultrahigh-vacuum environment (UHV). The vacuum requirement for MBE is typically within the residual pressure range of 10⁻¹⁰–10⁻¹¹ Torr. Thus, epitaxial films with high purities and excellent crystallinities can be grown at relatively low temperatures.

Moreover, the quality of the UHV environment can be enhanced by implementing various techniques for studying the surface, interfacial, and bulk properties of the grown films. The RHEED technique [50] is an essential in situ analysis tool. Unlike other semiconductors such as GaN, the pressure for the growth of GaN and its alloys is higher because of the necessity of using nitrogen plasma to obtain reactive nitrogen atoms. The nitrogen atom source operates at 13.56 Mhz at 50 ohms, with a 600–800-Watt radiofrequency source and a theoretical growth rate ratio of 6 μm/h. The molecular nitrogen supplied to the system was 99.99999% pure and its flow was controlled using a mass flow controller.

The substrates utilized in these studies were MgO (001). They underwent initial cleaning with a sequential ultrasonic bath in trichloroethylene and acetone for 10 min. Subsequently, the substrates were placed in a vacuum chamber and transferred to the growth chamber, where they were thermally cleaned in a high-vacuum environment (~3 × 10⁻⁸ Torr) at 600 °C for 30 min to achieve a clean, atomically flat surface. This cleaning process results in a MgO–H surface [51,52]. Before the growth process, the MgO–H surface was treated with atomic nitrogen at 700 °C using a 230 W nitrogen plasma for 10 min. This step is energetically favorable because nitrogen atoms have a higher binding energy with the substrate than hydrogen atoms, allowing nitrogen to replace hydrogen on the surface. As a result, the surface was effectively nitrided for 10 min [52].

GaN, InN, and their alloys (In_xGa_1−xN) are primarily bonded through strong covalent Ga-N and In-N bonds with significant ionic character. In GaN, each Ga atom forms three covalent bonds with nitrogen atoms, resulting in a bond length of approximately 1.95 Å. In InN, the larger indium atoms formed covalent bonds with nitrogen atoms, with bond lengths around 2.14 Å.

In In_xGa_1−xN alloys, the bonding nature is a mixture of Ga-N and In-N bonds, with bond lengths and strengths varying based on indium concentration. The alloying process significantly alters the lattice constants, strain states, and overall properties of a material, thereby influencing its performance in optoelectronic devices. These bonds contribute to the high melting points, chemical stability, and robust mechanical properties of the materials, which are critical for their application in optoelectronic devices.

3. Growth Mechanism of GaN Grown on MgO (100) Substrates by PAMBE

In the literature, the theoretical and experimental thermodynamic parameters for hexagonal GaN crystalline formation, such as entropy, enthalpy, and Gibbs free energy of formation, are typically in the range of 800–1800 K [22,29,53,54,55,56]. However, the thermodynamic parameters for cubic GaN are scarce, and the obtained values are remarkably close to the hexagonal values [57,58]. In this sense, when cubic GaN is grown slightly outside metastable conditions, a hexagonal phase is present in the material. Some authors have eliminated almost all hexagonal incorporation with previous surface treatments [59,60,61]. Alternatively, the nitrogen gas pressure can be manipulated during growth [60,62,63]. These experimental modifications alter the thermodynamic properties of the system and the surface energy, but the exact processes that occur during the growth process remain unclear [50,59,64].

Understanding the growth mechanism of GaN layers on MgO (100) substrates transcends scientific inquiry; it is pivotal for advancing semiconductor growth technologies. The transition between the hexagonal and zincblende phases of GaN holds significant technological implications, and our research elucidates this critical aspect. In this section, we demonstrate that the transition from the hexagonal phase to the cubic phase of GaN grown on MgO (100) can be precisely controlled by modulating the Ga flux during the growth process using plasma-assisted molecular beam epitaxy (PAMBE). In situ reflection high-energy electron diffraction (RHEED), X-ray diffraction (XRD), and pole figure structural studies have revealed that GaN films predominantly adopt a hexagonal wurtzite structure at low Ga fluxes. However, increasing the Ga/N ratio under growth conditions results in a mixture of crystal structures. The hexagonal phase forms on the (111) and (115) planes of the cubic zincblende structure. High-resolution transmission electron microscopy (HR-TEM) confirmed the high crystallinity of the cubic zincblende phase, indicating that optimal growth conditions involve a Ga-rich metal flux. Scanning electron microscopy (SEM) images show micrometer-sized crystallites oriented along [110] and [100], with cubic crystal domains rotating 90°, thereby creating grain boundaries.

X-ray diffraction (XRD) scans of four GaN films grown on MgO (100) substrates using PAMBE under identical growth conditions but with varying Ga cell temperatures are depicted in Figure 1 [65,66]. At a Ga cell temperature (TGa) of 900 °C, besides the diffraction peak at 43.1° from the MgO substrate, a distinctive Bragg diffraction peak at 34.5° from the (0002) plane of the wurtzite crystal structure indicates that the GaN film predominantly exhibits a hexagonal wurtzite phase. For a sample grown at an increased TGa of 940 °C, the intensity of the (0002) wurtzite peak decreased, and a diffraction peak at 40° corresponding to the cubic (002) plane appeared. As the Ga cell temperature was further increased, the formation of the wurtzite phase was increasingly suppressed, and the fraction of the zincblende phase increased. In the film grown at TGa = 955 °C, diffraction peaks from the (0002) wurtzite plane were still present, but the cubic (002) plane became predominant. Finally, under metal-rich conditions at TGa = 970 °C, the wurtzite diffraction peak was no longer detectable. Only a strong zincblende (002) diffraction peak was observed, indicating that the cubic GaN was oriented along the (002) plane perpendicular to the film surface.

The pole figures at 2θ = 34.5°, corresponding to the (0002) wurtzite and (111) zincblende planes, are depicted in Figure 2a–d. At a Ga cell temperature (TGa) of 900 °C, as shown in Figure 2a, there is a strong reflection centered at Psi = 0°, corresponding to the (0002) hexagonal phase of GaN. Additionally, a weak signal with four-fold symmetry appears at Psi = 54.7°, attributed to the (111) zincblende planes. This angle corresponds to the spacing between the surface (100) and (111) planes of cubic GaN, indicating the presence of a cubic phase with an orientation along the (001) plane.

When T_Ga was increased to 940 °C, Figure 2b, three changes were observed in the pole figure with remarkable precision. First, the intensity of the wurtzite (0002) reflection centered at Psi = 0° decreased considerably. Second, a signal with four-fold symmetry appeared at Psi = 15.8° for possible cubic twins or a secondary hexagonal phase [65,66]. Third, the four-fold reflection zincblende of the (111) plane increased sharply, as shown in Figure 2c. At T_Ga = 955 °C, the ratio between cubic and hexagonal signal intensities increased for the cubic phase. Finally, at T_Ga = 970 °C, Figure 2d, the wurtzite (0002) reflection centered at Psi = 0° disappeared, and the signal at Psi = 15.8° was almost undetectable.

The four-fold reflection of the (111) plane, a significant finding, indicates that this GaN film is predominantly oriented in the cubic phase perpendicular to the surface [67].

The pole figures at 2θ = 40° corresponding to the (002) zincblende plane are shown in Figure 2e–h. At T_Ga = 900 °C, Figure 2e, an ordered signal at psi = 0° of cubic structure does appear because the hexagonal phase is predominant. In Figure 2f–h with T_Ga= 940, 955, and 970 °C, it is possible to see a strong signal centered at psi = 0° corresponding to cubic (002) plane of GaN, and the intensity of the signal increases as the T_Ga increases.

The secondary cubic twin signal appears at Psi = 70.52° in the pole figures at 2θ = 40°, corresponding to the (002) zincblende plane. Nevertheless, there was no clear signal for the secondary cubic twins. Thus, the signal with four-fold symmetry at psi = 15.8° in the pole figures at 2θ = 34.5° corresponds to hexagonal inclusion growth on the (115) planes of the cubic matrix.

Figure 3 shows GaN film growth at T_Ga = 940 °C, providing strong evidence of the hexagonal inclusions. The 2θ = 32.4° and (b) 2θ = 36.8° correspond to (1010) and (1011) wurtzite planes, respectively. In Figure 3a, we observe a hexagonal signal with a four-fold symmetry at the position of (112) spots at psi = 35.2°, unequivocally confirming the growth of hexagonal inclusions of (1010) planes are grown parallel to (112) planes of cubic GaN. Figure 3b shows three signals: two for the hexagonal GaN substrate and one for the MgO substrate. At psi = 8.7°, a four-fold symmetry signal related to hexagonal inclusions that grew on the (111) planes of the cubic phase [68,69] at psi = 47.6° was generated by the hexagonal inclusions, where the (0002) planes grew with an angle tilt of 15.8°, further confirming the growth of hexagonal inclusions on the (111) and (115) planes. Finally, at psi = 54.7°, a strong signal with four-fold symmetry is attributed to the (111) planes of the MgO substrates.

Figure 4a presents a detailed SEM image of the GaN film grown at TGa = 940 °C. The film surface consists of isometric hexagonal microcrystals associated with the c-plane, which grow parallel to the growth direction. Additionally, there are elongated hexagonal microcrystals whose (10-10) planes are parallel to the (112) and (0002) planes. These grow on the (111) and (115) planes of cubic crystals, as confirmed by the X-ray pole figures.

Figure 4b displays the streaky RHEED pattern for the MgO surface, indicating a two-dimensional surface morphology. Figure 4c presents the spotty RHEED pattern of GaN in the [100] zincblende azimuth at the commencement of growth on the MgO (100) substrate, revealing a lattice mismatch of −7.3% between [100] GaN and [100] MgO. Figure 4d shows the RHEED pattern for the [11-20] azimuth of GaN (0002) wurtzite, aligned with the [110] direction of the MgO (100) substrate. Thermodynamically, this alignment is the most favorable for the growth of GaN (0002) and InN (0002) directly on MgO (100) substrates. The lattice mismatch for [11-20] GaN relative to [110] MgO is −7.3%

The SEM image in Figure 4a depicts micrometer-sized crystals with elongated hexagonal structures growing within cubic crystal domains oriented along the [010] and [100] directions. The presence of these two c-GaN crystalline domains, growing at a 90° angle to each other, can be explained by the difference in surface rotational symmetries between the MgO (100) substrate and the c-GaN thin film. The typical dual symmetry (180° rotations) of the GaN zincblende structure is disrupted when it grows on the quadruple-symmetry surface of MgO (100). This unique surface, with its invariant and complex dynamic evolution laws when rotated by 90°, adds complexity to the growth process.

A model of GaN growth over MgO was meticulously constructed, as shown in Figure 4e. The formation of two cubic domains oriented in the [100] and [010] directions is observed. Figure 4f further demonstrates the precise alignment of the [11-20] azimuth of the c-plane growing perpendicular to the film surface of GaN, aligned in the [110] direction of the MgO (100) surface.

The SEM image of the GaN film grown at TGa = 970 °C is shown in Figure 5a. The cubic structure is evident from its flat surface morphology, which often exhibits rectangular symmetry. The inset shows a typical 2X RHEED pattern from the smooth surface of the c-GaN film at the end of growth. This streaky 2X RHEED pattern indicates a layer-by-layer 2D growth mode, facilitated by Ga-rich conditions that promote specific GaN film growth characteristics.

High-resolution cross-sectional TEM images of the film grown at TGa = 970 °C, shown in Figure 5b,c, reveal micrometer-sized crystallites with high structural quality. The broad FWHM observed in the XRD is primarily due to grain boundaries formed by the nucleation of c-GaN crystallites rotated by 90° relative to each other. Figure 5c amplifies the results in Figure 5b, where a GaN zincblende unit cell with a lattice parameter of 0.4502 nm is superimposed on the scaled TEM image, demonstrating good agreement and indicating a relaxed crystal structure of c-GaN.

Understanding the growth mechanism and controlling the inclusion of the hexagonal phase during c-GaN growth on MgO (100) substrates are essential. Without this understanding, the crystalline quality and optoelectronic properties of the c-GaN devices are at risk.

4. Growth, Structural, and Optical Properties of c-In_xGa_1−xN

An essential aspect of developing new and improved optoelectronic devices based on c-In_xGa_1−xN alloys is the determination of their properties, particularly the structural and optical characteristics. Figure 6a shows XRD 2θ-ω scans for several InGaN layers with different indium concentrations. Initially, the c-GaN buffer layer grown on MgO (100) is visible, with the arrow indicating the diffraction peak of the c-InGaN alloy at various indium concentrations. The shift to lower angles as the indium ratio increases signifies a change in the lattice constant, which is crucial for understanding the structural properties of the c-InGaN alloy.

To determine the bulk lattice constant of the c-InGaN films, measurements were taken in the symmetric (002) and asymmetric (113) and (-1-13) planes. Using the quasi-cubic approximation with Macrander expressions, the in-plane and in-growth lattice parameters were calculated [23]. These values were consistent, indicating that the c-InGaN films were relaxed. The indium molar fraction was then calculated using Vegard’s law. The results of this analysis are shown in Figure 6b and

Table 2. Result obtained from X-ray diffraction measurements from asymmetric (113) and (-1-13) planes.

In (x) Concentration InxGa1−xN	Bulk Lattice Constant (nm) ±0.0005	In-Growth Lattice Parameter (nm) [110] ±0.0005	In-Plane Lattice Parameter (nm) [001] ±0.0005	Bandgap (eV) ±0.05
1	0.5016	0.5024	0.5010	0.87
0.927	0.4972	0.4975	0.4972	1.101
0.479	0.4748	0.4751	0.4749	1.79
0.288	0.4655	0.4652	0.4656	1.91
0.214	0.4614	0.4620	0.4614	2.53
0.112	0.4566	0.4557	0.4567	2.86
0.096	0.4556	0.4560	0.4556	2.92
0	0.4516	0.4514	0.4517	3.18

.

Optical absorption measurements were conducted at room temperature to determine the absorption edges of c-InGaN thin films. In direct bandgap materials, free excitons are not seen in the absorption spectra due to their broadening with increasing temperature. Therefore, the absorption edges are attributed to transitions between the direct bands and the bandgap of the c-InGaN thin film. The plot of (αhν)² versus hν for c-InGaN thin films (Figure 7) reveals that the absorption coefficients follow a square law, indicative of direct bandgap absorption.

As the indium content increases, the bandgap energy decreases, showing a reduction from 3.18 eV for GaN to a minimum of 0.87 eV for InN. Determining the bandgap of InN has been contentious, with reported values ranging from 0.7 to 1.9 eV across different studies [24,71,72,73,74]. Ultimately, the fundamental bandgap energy was established at 0.61 eV at low temperatures [75]. This initial variation in reported values is due to InN’s tendency to grow with a high free-electron concentration of about 10¹⁹ cm⁻³. In cases of significant degenerate doping, where the Fermi level enters the conduction bands, optical absorption is prohibited for transitions below the Fermi surface. Thus, the onset of optical absorption overestimates the intrinsic bandgap due to the well-known Burstein–Moss (BM) effect [76].

The slopes of the straight lines in the plot of (αhν)² versus photon energy (hν) differ for various compositions. This variation at the absorption edge can be attributed to sample nonhomogeneity, which is associated with the distribution of indium atoms and composition changes. The absorption edge is sensitive to disorder and alloy composition, causing the absorption edge to broaden. This broadening results in the density of states not reaching zero at the material’s bandgap, leading to the formation of band tails known as Urbach tails. The equation used to calculate the energy gap of an InGaN alloy with composition x is given by:

E_gInxGa1−xN = xE_g(InN) + (1 − x) E_gGaN-bx(1 − x)

(1)

where E_g(InN) and E_g(GaN) are the bandgaps of InN and GaN, respectively, and b is the bowing parameter, which accounts for the deviation from linearity. The value of b is significantly influenced by the bandgap value of InN. For our samples exhibiting the Burstein–Moss (BM) effect, the bandgap value of InN deviates from the more commonly accepted bandgap value. For InGaN samples with a high indium content, a bowing parameter value of 1.84 was obtained (represented by the blue dashed line in Figure 8). Therefore, we adopted the 0.61 eV bandgap value of InN as reported by Shörman et al. [75] (illustrated by black star in Figure 8). In this study, from our experimental results (black squares in Figure 8) we determined an experimental bowing parameter value b of 1.32 (illustrated by the solid red line in Figure 8) [60] This experimental b value is consistent with and closely matches the estimated theoretical values reported as 1.36 [77], 1.37 [78], and 1.379 [79]. Damian et al. discuss the contribution of other groups that have grown β-InGaN with low indium concentration [60] (illustrated with open circles and triangles in Figure 8).

Figure 6b demonstrates that the lattice constants of the In_xGa_1−xN solid solutions adhere to Vegard’s rule, confirming a linear relationship with the indium concentration (x). However, Figure 8 reveals a nonlinear dependence of the bandgap energy (E_g) on x, which deviates from the expected linearity. This deviation can be explained by a significant bowing parameter, which is a common feature in semiconductor alloys owing to the differences in the atomic interactions and electronic structures of GaN and InN.

The bowing parameter captures the intricate interplay between the materials, considering the effects of strain, defects, and the inherent electronic band structure. Strain induced by lattice mismatch can affect the bandgap by causing band splitting and shifts. Additionally, defects such as nitrogen vacancies can create localized states within the bandgap, altering the electronic structure further. This parameter is vital for understanding and predicting the behavior of InGaN alloys, as it integrates various factors influencing their optical and electronic properties.

The interaction between the conduction and valence band edges of GaN and InN, which have different curvatures and effective masses, also contributes to nonlinear behavior. These factors collectively result in the observed nonlinearity of E_g(x), highlighting the necessity of considering the bowing parameter and other intrinsic properties when analyzing semiconductor alloys.

5. Growth of Two Polymorphic Structures of InN on MgO (100)

In recent decades, both the hexagonal and cubic structures of the semiconductor InN have garnered significant attention from both fundamental and practical perspectives. Interest in this semiconductor increased substantially after the optical bandgap energy was accurately determined to be 0.67 eV, a revision from the initially reported value of 1.9 eV [71,75,80,81,82]. In addition to its narrow direct bandgap, InN possesses several unique properties that make it particularly attractive. Theoretical studies have predicted that InN has the smallest effective electron mass among the III-nitride family [83], which can lead to high electron mobility and saturation velocity [84].

A major challenge in epitaxial InN growth is the lack of a suitable substrate to minimize lattice mismatch, leading to high dislocation densities in InN layers. For instance, InN exhibits lattice mismatches of 25% with sapphire, 8% with Si (111), 37.4% with GaAs, and 11% with GaN. As a result, achieving high-quality, single-crystal InN is difficult. Additionally, InN decomposes at relatively low temperatures (≤500 °C) [85]. Despite these challenges, several research groups have developed device applications for epitaxial InN, including infrared lasers [86], photodetectors [87], solar cells [88], transistors [89], and terahertz-range devices [90,91].

Growth Mechanism of InN on MgO (100)

The lattice mismatch between InN (100) and MgO (100) is approximately 18%. However, a significant advancement was made in a prior study [92,93], where a low-temperature (LT) InN buffer layer was successfully grown at 300 °C directly on MgO (001), significantly influencing the growth of hexagonal InN.

To prepare for growth, the substrate underwent thermal cleaning at 830 °C. The temperature was then reduced to 300 °C to deposit the LT-InN buffer layer. The MgO substrate surface was exposed to an indium flux at a controlled cell temperature of 830 °C, alongside nitrogen plasma with an N2 flow rate of 1.6 standard cubic centimeters per minute and a radio frequency power of 215 W for 5 min. Following this step, the indium cell was closed, and the temperature was increased to 500 °C for 20 min to anneal the LT-InN layer, allowing for recrystallization and the removal of any unreacted indium from the InN surface. Subsequently, the InN layer was grown for 120 min under identical conditions.

Post-annealing, any residual indium on the InN surface was removed. The structure was then recrystallized by an in-plane rotation of 30° to minimize strain and achieve the lowest energy configuration. This annealing process is crucial for forming the desired structure, as it is thermodynamically favorable due to the reduction in system mismatch from 19.5% for [01--10] InN/[110] MgO to 3.5% for [11-20] InN/[110] MgO [13]. This process resulted in the formation of two crystalline domains: one with a high lattice mismatch (19.5%) and the other with a low lattice mismatch (3.5%) between the MgO substrate and the InN film. Figure 9 presents a schematic of the rotation of InN grown with the c-plane perpendicular to the MgO (100) surface, showing the formation of two hexagonal domains oriented in the [01-10] and [11-20] azimuths.

To investigate the formation of the two domains and confirm the presence of a rotated plane, a pole figure was obtained by XRD at 2θ = 33.1°, corresponding to the [01-10] plane of h-InN, as illustrated in Figure 10a. A significant finding was the 12-fold symmetry structure, indicating the two hexagonal domains with a 30° plane rotation. Figure 10b presents the XRD curve for h-InN in the in-growth planes. The h-InN diffraction peak of the (0002) c-plane was observed at 2θ = 31.4° with a full width at half maximum (FWHM) of 0.24, while the diffraction peak corresponding to the (002) plane of the MgO substrate was observed at 2θ = 43°.

It is important to highlight that growing InN without annealing the LT-InN buffer layer resulted in a mixture of phases. The wurtzite phase is naturally generated as the system tends to adopt the hexagonal structure to minimize energy. However, the system also attempts to align with the cubic structure of the MgO substrate, leading to the formation of cubic-phase InN domains interspersed with wurtzite domains. This combination of phases and the resulting 3D growth with numerous defects significantly degrade the structural quality of the film. In contrast, annealing the LT-InN buffer layer leads to films of much higher structural quality.

6. Growth of InN on c-GaN/MgO (100)

The direct growth of InN on MgO (100) results in hexagonal polycrystalline films with low crystallinity [92,93,94,95]. However, using an intermediate layer of c-GaN is a crucial step that significantly decreases the lattice mismatch between the InN layer and MgO (100) substrate, allowing for the epitaxial growth of c-InN under specific and vital experimental conditions.

The cubic polymorph of InN, known as c-InN, has several advantages over the hexagonal form of h-InN. In terms of optical properties, c-InN has a lower bandgap (0.56 eV) [77] than h-InN (0.7 eV) [95], which allows for the extension of emission wavelengths into the infrared (IR) range. Additionally, the higher crystalline symmetry and more isotropic behavior of c-InN eliminate polarization-induced electrical fields, resulting in improved carrier mobility, electron drift velocity, doping efficiency, and optical gain [87]. These properties underscore the significant impact of our findings on material science and semiconductor physics.

In a previous study [94,95], a c-GaN buffer layer approximately 220 nm thick was carefully deposited at 600 °C on single-crystal MgO (001) substrates. These buffer layers were then exposed to nitrogen plasma using specified process parameters to ensure the removal of any unreacted Ga on the GaN surface. During the growth of InN, the substrates were maintained at a fixed temperature, with the optimal indium cell temperature (TIn) set at 840 °C. The N₂ flow was maintained at 1.3 sccm, and an RF power of 230 W was used to form the nitrogen plasma, producing neutral atomic nitrogen. This level of detail in our experimental procedures highlights the precision and thoroughness of our study.

Multiple c-InN/GaN/MgO samples were grown under indium-rich conditions at different substrate growth temperatures (T_Ga) to assess their impact on film quality. Figure 11 shows typical SEM images and corresponding RHEED patterns (insets attached to the SEM images) for samples grown at T_Ga ranging from 380 to 530 °C. All samples in Figure 11 were grown under indium-rich conditions with T_In = 840 °C. The SEM images revealed that all c-InN films possessed a smooth surface. Samples grown at higher T_Ga (530 and 500 °C, Figure 11a,b) displayed higher roughness compared to those grown at lower temperatures (440 and 380 °C, Figure 11c,d). The RHEED patterns for each surface confirm these observations. At higher growth temperatures, the RHEED patterns (insets in Figure 11) exhibit slight transmission features in the diffraction streaks, indicating increased roughness.

X-ray diffraction (XRD) studies were performed to assess the crystal quality of the InN samples. Figure 12a displays the XRD scan of c-InN thin films grown on c-GaN/MgO (100) via PAMBE under metal-rich conditions. The diffraction peaks at 2θ = 42.92°, 39.85°, and 36.04° correspond to the (002) diffraction of the MgO substrate, the c-GaN buffer layer, and the c-InN film, respectively. The absence of the hexagonal peak at 31.4° and the presence of a prominent zincblende (002) peak at 36.4° indicate that the c-InN is aligned along the (002) plane, perpendicular to the film surface.

Figure 12b illustrates the pole figures at 2θ = 31.4°, corresponding to the (0002) wurtzite and (111) zincblende planes. The clear, intense four-fold reflection of the (111) plane indicates that the cubic phase of the InN film is primarily perpendicular to the surface. Our research highlights the necessity of initially growing a GaN buffer layer and maintaining indium-rich conditions to achieve high-quality c-InN films. These findings emphasize the critical role of precise growth conditions in determining the structural quality of the films.

7. RHEED Studies during the Growth Process and Critical Thickness

The zincblende structure can be stabilized or grown only through heteroepitaxial growth on cubic substrates such as cubic SiC [96,97], Si [98], MgO [94,99,100], and GaAs. However, these structures are metastable and often transition to a wurtzite phase due to factors such as growth temperature and atomic flow ratio [50]. When using highly mismatched substrates, crystallographic defects can separate a portion of the zincblende phase from the wurtzite phase. Nevertheless, advances in preparation techniques—such as stabilizing atomic fluxes, controlling temperature, and optimizing buffer layer growth—have significantly reduced this polyphase issue [95,100]. The zincblende crystal structure consists of two interpenetrating face-centered cubic (fcc) sublattices, displaced by one-quarter of the body diagonal, belonging to a specific space group. The unit cell contains four atoms, with each atom of one type tetrahedrally coordinated to four atoms of the other type.

This section presents a real-time study using reflection high-energy electron diffraction (RHEED) to analyze the PAMBE growth of cubic GaN, InN/GaN, and InGaN/GaN on MgO substrates, which involves a system with a ~7.2% lattice mismatch [101,102]. RHEED is an effective technique for monitoring the surface structure and growth dynamics of thin films in real time during epitaxial growth processes. By directing a high-energy electron beam at a grazing incidence angle to the sample surface, RHEED provides detailed information on surface crystallography, morphology, and growth rate.

In this study, RHEED was used to examine the growth of In_xGa_1−xN layers on cubic GaN (c-GaN) deposited on MgO substrates. The RHEED patterns observed during the growth process provided critical insights into the crystalline quality, phase transitions, and surface roughness of the films.

The initial stages of growth involved the deposition of a GaN buffer layer on the MgO (100) substrate. The RHEED patterns observed during this stage exhibit sharp and streaky features, indicative of a smooth and well-ordered cubic GaN surface. These patterns confirm the successful nucleation and epitaxial alignment of the GaN layer with the MgO substrate.

The growth transitioned to the In_xGa_1−xN layer and the RHEED patterns evolved. Initially, the patterns maintained their streaky nature, suggesting a layer-by-layer growth (Frank–van der Merwe mode). This stage is crucial to ensure high-quality epitaxial films with minimal defects.

With increasing indium content, the RHEED patterns began to exhibit changes. The streaks became broader and more diffuse, indicating increased surface roughness and strain relaxation within the growing film. This transition is expected because the lattice mismatch between GaN and InN introduces strain, which is partially relieved by surface roughening and defect formation.

The critical thickness, beyond which strain relaxation mechanisms such as dislocation formation occur, was monitored using RHEED. For the In_xGa_1−xN layers with higher indium concentrations (44% and 70%), we observed the appearance of additional RHEED features corresponding to relaxed regions and potential phase separation. These patterns suggest the coexistence of cubic and hexagonal phases, particularly at higher indium concentrations.

After completing the growth process, the final RHEED patterns were analyzed to assess the overall crystalline quality. The patterns revealed a mixture of streaky and spotty features, indicating a combination of smooth regions and areas with three-dimensional growth. This mixed pattern was consistent with the presence of indium-rich clusters and strain-induced defects.

A detailed analysis of the RHEED patterns throughout the growth process provides valuable information regarding the surface morphology, crystalline quality, and phase transitions in In_xGa_1−xN films. The evolution of the RHEED patterns from sharp streaks to broader and more diffuse features reflects the impact of increasing indium content on the film quality and strain relaxation mechanisms. These insights are crucial for optimizing growth conditions and improving the properties of InGaN-based devices.

The spacing of the RHEED streaks was monitored over the course of deposition time, allowing us to quantitatively assess changes in the lattice parameters of the plane and the extent of substrate deformation. Critical thickness is a crucial parameter because it indicates the thickness at which the film stops growing epitaxially, matching the lattice constants of the substrate or buffer layer, and begins to form dislocations. As the layer continues to grow, the stored elastic energy can initiate various relaxation processes within the layer or substrate. Elastic relaxation typically occurs through dislocation formation in the first monolayers due to the significant lattice mismatch between GaN, its InN alloys, and the substrates used for growth. This process is the primary physical mechanism responsible for the formation of dislocations and other structural defects in the III-nitride layers.

The formation of misfit dislocations is generally described by the classical Frank–van der Merwe (FvdM) theory [103,104]. According to this theory, based on energy considerations, the interfacial energy between the films and substrate is considered the minimum energy required for dislocation generation. RHEED can track this process.

To determine the critical thickness under various conditions and for different layers, we analyzed the RHEED patterns of GaN grown on MgO, InN grown on a GaN buffer layer, and several InGaN alloys with different indium concentrations. The findings on critical thickness provide valuable insights into the onset of strain relaxation and dislocation formation in these heteroepitaxial systems.

The lack of suitable substrates greatly affects the crystalline quality of epitaxial layers in most In_xGa_1−xN alloys. The significant lattice parameter discrepancies between In_xGa_1−xN alloys of varying compositions and the available substrates induce high levels of elastic strain and mechanical stress. This strain is the primary factor leading to the formation of dislocations and other structural defects in GaN. The formation of misfit dislocations is typically explained by the classical Frank–van der Merwe model [103,104], which relies on energy considerations. In this model, the interfacial energy between the films and substrates is considered the minimal energy necessary for the generation of dislocations.

The Matthews–Blakeslee model [105] is a theoretical framework that considers the force balance between the driving force of the misfit strain and the dislocation line tension. This model is widely used to study misfit dislocations. The People and Bean [106] model is another theoretical framework that can generate alternative mechanisms for misfit dislocations. These models provide a deeper understanding of the physical processes involved in dislocation formation in cubic nitride films.

8. Critical Thickness of c-GaN on MgO (100)

As emphasized in the previous section, the intermediate cubic GaN layer is crucial for the growth of In_XGa_1−XN on MgO (100). This layer, which has a lattice mismatch of 7.2% with that of MgO (100), plays a pivotal role in the overall structure and properties of the material.

We employed the reflection high-energy electron diffraction (RHEED) technique to determine the onset of strain relaxation. This involved measuring the spacing between the <-10> and <01> reflections in the RHEED pattern in the [110] azimuth direction as a function of the layer thickness (h) in monolayers (MLs). RHEED is especially beneficial for our study due to its small scattering angles, which allow the streak spacing (Δk) to correspond directly to the reciprocal space scattering vector. This vector is inversely proportional to the actual in-plane surface lattice constant (a_‖) of the layer. Using the MgO substrate streak spacing as a reference, the in-plane lattice constant of the layer at a given epilayer thickness is calculated as a_‖ = a_MgOΔk_MgO/Δk_‖, where Δk_MgO and Δk_‖ are the streak spacings for the MgO substrate and the epilayer, respectively.

For all c-GaN layers, the evolution of the streak spacing (Δk_‖) was determined from centroid intensity profiles measured across the two-line diffraction streaks in the RHEED pattern. Figure 13 shows the development of these profiles as a function of layer thickness for the c-GaN case. For each layer thickness, the exact values of Δk_MgO and Δk_‖ were obtained from fits of the diffraction peak profiles. As shown in Figure 13, during the c-GaN growth sequence (with the Ga shutter open), the streak separation matched that of the MgO substrate, indicating pseudomorphic growth with an in-plane layer lattice constant equal to that of the substrate. However, at a layer thickness of three to four MLs, the streak positions shifted inward, indicating the start of strain relaxation due to an increase in the in-plane layer lattice constant. This thickness, known as the critical layer thickness (h_c), is a critical parameter in the growth of epitaxial layers, denoting the point at which strain in the layer begins to relax. In our study, we observed an h_c of 3–4 MLs in these instances. The relaxed strain of the c-GaN layers as a function of layer thickness required approximately nine MLs to achieve nearly 100% relaxation, as determined from the streak separations. Figure 13 illustrates the use of two different Ga fluxes to increase the growth rate from 0.25 ML/s to 0.30 ML/s and examine the impact of growth rate on hc. The Ga flux was reduced by adjusting the Ga cell temperature from 970 to 960 °C, revealing minimal influence on the outcomes, as depicted in Figure 13.

9. c-In_XGa_1−XN on c-GaN/MgO (100)

As previously discussed, an intermediate cubic GaN layer is essential for the growth of In_xGa_1−xN on MgO (100). Cubic GaN exhibits a lattice mismatch of 7.2% with MgO (100). During the growth of the In_xGa_1−xN layer on the c-GaN buffer layer, the accumulation of elastic energy triggers strain relaxation. The critical thicknesses of the In_xGa_1−xN layers with indium contents of 17%, 44%, and 70% grown on c-GaN/MgO by plasma-assisted molecular beam epitaxy (PAMBE) were measured using RHEED, as illustrated in Figure 14. By analyzing the RHEED patterns frame by frame as a function of deposition time, we quantitatively determined the variations in the in-plane lattice parameter and the magnitude of strain [107].

The streak spacing of the c-GaN layer, with its known lattice parameter of 0.45 nm [60], served as a reference to calculate the in-plane lattice parameters of In_xGa_1−xN. The indium molar fraction was determined using Vegard’s law [108]. Under the experimental conditions of the PAMBE system, the growth rate varied from 0.06 nm/s for x = 1 to 0.23 nm/s for x = 0 [60]. The number of monolayers (MLs) was calculated by measuring the thickness from the growth rate and deposition time, with one ML defined as a_x/4, where a_x is the in-plane lattice parameter calculated by RHEED for each In_xGa_1−xN sample.

10. Critical Thickness of c-InN/GaN on MgO (100)

Similar investigations have focused on the growth of c-InN on MgO (100) with an intermediate c-GaN buffer layer, presenting a 10.2% lattice mismatch. Our findings indicate that InN growth on the GaN/MgO surface predominantly follows a two-dimensional (2D) layer-by-layer mode, with strain relaxation commencing at approximately five MLs. Figure 15 illustrates how the in-plane lattice constant changes with the number of MLs of InN grown on c-GaN/MgO.

When the layer thickness surpasses four MLs and reaches around five MLs, the streak positions move inward, indicating the beginning of strain relaxation due to the increase in the in-plane layer lattice constant. Remarkably, a critical layer thickness h_c of five MLs was observed, which is notable considering the 10.2% lattice mismatch between c-InN and c-GaN. The calculated streak separation values revealed that the relaxed film attained an initial surface lattice constant of 5.01 Å. This lattice constant corresponds to a slightly stressed lattice in the vertical direction, compensating for a slightly expanded parallel lattice constant. This result is consistent with the observed growth process of the InN films, characterized by the formation of columnar islands that coalesce into rectangular blocks, fully covering the substrate [107].

Figure 15 illustrates the experimental critical thickness values in nanometers plotted against the indium molar fraction for c-In_xGa_1−xN growth on c-GaN. The incorporation of indium in In_xGa_1−xN films is somewhat restricted by compressive strain, which leads to a compositional gradient with a higher indium mole fraction near the surface [109,110]. It was observed that the rate of increase in indium composition is 1% per nm for In_xGa_1−xN quantum wells and 0.02% per nm up to 350 nm for In_xGa_1−xN layers. This indium-pulling effect is dependent on the indium molar fraction and the specific growth conditions used for In_xGa_1−xN [109,110]. The compositional gradient caused by this pulling effect in the first few monolayers was considered in the critical thickness (h_c) value calculated through RHEED analysis, and this is reflected in the standard deviation shown in Figure 15.

11. Determination of h_c in the Function of in Concentration

Figure 16 presents the calculated critical thickness h_c as a function of the indium molar fraction (x) for c-In_xGa_1−xN on c-GaN epilayers. This was computed using the equilibrium theory for strain relaxation in metastable heteroepitaxial semiconductor structures proposed by Fisher et al. [111]. This approach includes the elastic interaction between straight misfit dislocations. Initially developed for Ge/Si, which has a cubic crystal structure, this model shows better agreement between computed and measured values compared to other models. The mathematical equation representing this critical function is as follows:

$$\mathit{x}=({\displaystyle \frac{\mathit{b}\mathit{c}\mathit{o}\mathit{s}\mathit{\lambda }}{\mathbf{0.23}{\mathit{h}}_{\mathit{c}}}})(\mathbf{1}+{\displaystyle \frac{\mathbf{1}-\frac{\mathit{\nu }}{\mathbf{4}}}{\mathbf{4}\mathit{\pi }{\mathit{c}\mathit{o}\mathit{s}}^{\mathbf{2}}\mathit{\lambda }(\mathbf{1}+\mathit{\nu })}}\mathit{l}\mathit{n}({\displaystyle \frac{{\mathit{h}}_{\mathit{c}}}{\mathit{b}}}))$$

(2)

where ν is the Poisson’s ratio, b is the magnitude of the Burgers vector, and λ is the angle between the Burgers vector and the direction normal to the dislocation line (which is 60° in the $c-{In}_x{Ga}_{1-x}N$ system). The factor 0.23 represents the in-plane misfit strain multiplied by 2. The lattice parameters for c-GaN and c-InN are 0.45 nm and 0.502 nm, respectively [60]. There are no reported values for Poisson’s ratio and Burgers vector specifically for c-In_xGa_1−xN. Therefore, linear interpolation and averaging were performed at every 20% indium molar fraction using the values reported for c-InN and c-GaN [58,107,112].

12. Growth of Quantum Structures Based in c-InGaN

This section shows the cubic quantum GaN/In_xGa_1−xN/GaN/MgO wells grown by PAMBE to obtain emissions in the violet, blue, green, and red ranges, respectively, in a single heterostructure by varying only the concentration of indium in the active layer. Subsequently, a structure with three active layers of ternary alloy with a concentration of In was grown to emit three visible spectrum colors in a single structure [114].

In our study, GaN and InGaN semiconductors primarily form type-I heterojunctions [115]. This type of heterojunction, also known as a “straddling gap”, is characterized by the conduction band minimum (CBM) and valence band maximum (VBM) of one material lying within the bandgap of the other material. This alignment is highly beneficial for optoelectronic applications, such as light-emitting diodes (LEDs), where efficient electron–hole recombination is desired.

In type-I heterojunctions, both electrons and holes are confined to the same region, which facilitates recombination. This is ideal for LEDs, where light emission results from electron–hole recombination.

The GaN/InGaN system, which is commonly used in LEDs, benefits from this alignment to achieve high-efficiency light emission.

In type-II heterojunctions, the conduction and valence bands are staggered, leading to spatial separation of electrons and holes. This is less favorable for LEDs, but beneficial for applications requiring charge separation, such as photodetectors.

Type-III heterojunctions have a broken gap alignment, which is uncommon in III-nitride semiconductors and is not typically applicable to GaN/InGaN systems.

We synthesized three SQWs with a 300 nm thick buffer layer was grown on a MgO substrate at 770 °C. A layer of c-In_xGa_1−xN with a thickness of approximately 4 ± 2 nm was deposited on the three samples using In concentrations of 0.10, 0.40, and 0.47. Then, a 150 nm wide c-GaN barrier was grown on top (see Figure 17d) to complete the quantum well.

In the growth of ternary alloys of InGaN, it is necessary to consider the segregation phenomenon to prevent indium segregation due to the high temperature used to obtain InGaN in the cubic phase, because a hexagonal phase is induced when a lower temperature is used. Then, the typical growth temperature of c-In_xGa_1−xN is approximately 750–800 °C, between higher indium in the alloy, lower growth temperature, and to reduce the segregation effect of indium.

In the subsequent structure, four c-InGaN quantum wells were sequentially grown with varying indium concentrations and separated by c-GaN barriers. The buffer layer of c-GaN, with a thickness of 300 nm, was grown at a temperature of 700 °C. Following this, four c-In_xGa_1−xN layers, each with a thickness of 10 ± 2 nm, were deposited with GaN barriers at different indium concentrations. The indium concentrations of 0.10 ± 0.04, 0.15 ± 0.04, and 0.35 ± 0.04 for each quantum well were achieved by adjusting the temperature of the indium effusion cell. The 10 nm thickness was selected for the wells to strike a balance between the confinement effect and the challenge of controlling indium segregation, which becomes more difficult in narrower QWs with longer growth times, as is the case for multiple quantum wells. Additionally, this configuration aimed to produce three primary colors with different thicknesses.

To accurately determine the bandgaps of these materials, we used techniques such as photoluminescence (PL) spectroscopy and absorption measurements. These methods allow for the precise characterization of the optical properties and bandgap energies of films. The experimental results are summarized as follows.

The GaN bandgap was confirmed to be 3.4 eV, consistent with literature values. The bandgap of InN was measured to be approximately 0.7 eV.

The In_xGa_1−xN alloys’ bandgaps were observed to vary linearly with indium content, with bowing parameters accounted for to describe the deviation from Vegard’s law.

The direct bandgap and the ability to tune the emission wavelength across the visible spectrum make InGaN alloys ideal for high-efficiency LEDs. The precise control of indium content allows for the design of LEDs with specific emission colors.

The tunable bandgap of InGaN alloys enables the absorption of a broad range of the solar spectrum, thereby enhancing the efficiency of photovoltaic devices. By adjusting the composition, it is possible to optimize the material for maximum solar-energy conversion.

Determining the bandgaps of GaN, InN, and InGaN alloys is crucial for their applications in LEDs and solar cells. Our study provides detailed measurements of these bandgaps and highlights their significance for optimizing the performance of optoelectronic devices.

Figure 17 displays the photoluminescence (PL) spectra for samples 1(a), 2(b), and 3(c). In Figure 17a, a broad peak is observed, which is composed of signals from GaN (3.24 eV) and the quantum well (QW) (3.08 eV) with a full width at half maximum (FWHM) of 0.34 eV. Figure 17b shows a peak around 2.32 eV with an FWHM of 0.35 eV. Figure 17c presents the PL spectrum with a signal observed at 1.92 eV and an FWHM of 0.35 eV. Due to the quantum confinement effect, all peak positions are at higher energies than the bulk bandgap values.

As previously explained, our use of RHEED is crucial for determining the lattice constants of the samples. Application of Vegard’s law allowed us to approximate the molar fraction of indium. PL emission spectra analyses were performed by mounting the samples in the cold finger of a closed-cycle cryostat and cooling them to 10 K.

Figure 18 illustrates the optical band photoluminescence (PL) emission from the grown single quantum wells (SQWs) as described above, plotted against the indium concentration for bulk InGaN alloys at 300 K, as previously established experimentally (see Figure 8) [60]. Our analysis, which includes the InN bandgap data reported by Damian et al. [60] along with other published experimental and theoretical data [116,117,118], is comprehensive. We also observed that the energy bandgap shifts to higher values as the temperature is lowered. We estimated an average blue shift of 47 meV for c-InGaN bulk according to Varshni fits from the PL measurements of c-GaN (80 meV) [119] and c-InN (14 meV) [76].

The solid black lines in Figure 18 represent the infinite quantum well approximations for 3 nm and 4 nm, respectively. The effective masses of c-In_xGa_1−xN were interpolated using the parameters for c-GaN and c-InN as described in [120]. The PL measurement values at 10 K for samples 1 and 3 fall on the 4 nm curve, while the value for sample 2 falls on the 3 nm curve, which is 1 nm less than the anticipated thickness. Ternary alloy c-In_xGa_1−xN layers with thicknesses of 3 and 4 nm are nearly completely relaxed, as the critical thicknesses for GaN and InN in the cubic phase are three and four monolayers, respectively, based on experimental RHEED measurements [94,121]. Consequently, the ternary alloys of GaN and InN should exhibit similar critical thicknesses. In both references, a fully relaxed layer was observed around 2.5 nm. The only potential source of stress in these samples is the slight difference in thermal expansion between GaN and InGaN.

Figure 19 depicts the photoluminescence (PL) spectra of a structure with four active layers grown at different indium concentrations, each separated by c-GaN barriers. Emissions are observed at 1.79, 2.07, 2.71, and 2.92 eV from the InGaN quantum wells (QWs), along with an additional emission at 3.26 eV from the GaN buffer layer. These QWs, characterized by their unique properties, are wider than single quantum wells (SQWs), resulting in less indium segregation during growth. The full width at half maximum (FWHM) values of the peaks were 0.2, 0.26, 0.43, and 0.31 eV for the QWs and 0.12 eV for the GaN.

The indium concentrations were determined using the lattice constants estimated from RHEED and Vegard’s law, with a bowing parameter b of 1.4. The bulk energy gaps of the alloys were calculated to be 2.79, 2.57, 2.0, and 1.67 eV. Due to the quantum confinement effect, the PL emissions occur at higher energies compared to the bulk material. In Figure 19, a line corresponding to a 10 nm QW approximation is shown (see Figure 20). The PL energy values of each peak, plotted against the indium concentration calculated from RHEED, are represented by triangles. The broadening of the PL peaks is attributed to the recombination of confined states within the QWs [10] and potential alloy composition variations.

13. Cubic GaN Diode

The development of devices based on cubic III-nitrides hinges on achieving efficient p-type doping and maintaining high crystal quality. However, there are few reports on cubic III-nitride devices in the literature [122,123], primarily due to the challenges of inefficient p-type doping, synthesis difficulties of c-GaN, lack of native substrates, and poor crystalline quality caused by hexagonal inclusions [124,125]. Therefore, enhancing the efficiency of p-type doping in c-GaN epilayers is crucial for improving the fabrication of cubic III-nitride semiconductors.

A previous study examined the impact of Mg flux on the electrical and structural properties of Mg-doped c-GaN [126]. The homoepitaxial growth of Mg-doped c-GaN films was conducted on 1 cm² c-GaN/MgO(001) epilayers using a vertical PAMBE system equipped with standard effusion cells for Ga, Mg, and a high-purity N₂ RF plasma cell [60,114]. The purity of the Ga and Mg sources was 99.9995%.

The likelihood of dopant incorporation into the crystal lattice is significantly influenced by the surface environment of the host material. It is essential to maintain Ga-rich conditions to stabilize the zincblende phase of GaN at the surface [127]. However, the incorporation of Mg into c-GaN is constrained due to the low solubility of Mg in Ga [128,129]. The reported growth strategy involved reducing the Ga flux by adjusting the cell temperature [126] to create a Ga-free environment on the surface, thereby facilitating Mg incorporation into the GaN lattice. The Mg beam flux was increased to ensure the metal-rich conditions necessary to stabilize the cubic GaN phase.

The effects of Mg incorporation on the surface and bulk structural properties of c-GaN were systematically evaluated using RHEED and XRD pole figures. The findings revealed a notable shift in the growth mode from 2D to 3D, consistent with the Frank–van der Merwe growth mode, due to Mg incorporation. The samples exhibited a c(2 × 2) RHEED pattern at the end of the growth process when the substrate temperature was lowered to 620 °C, indicating a growth mode under metal-rich conditions [29,130,131]. X-ray diffraction studies showed that the fraction of hexagonal inclusions and crystal twinning decreased with increasing Mg flux [126].

The nonsaturated GaN surface with Ga atoms created ideal conditions for increasing Mg incorporation into the GaN lattice. By adjusting the beam flux, it was possible to enhance Mg incorporation and maintain the metal-rich conditions required to stabilize the doped cubic-phase GaN on a cubic buffer. Under these conditions, secondary ion mass spectrometry (SIMS) measurements revealed a significant rise in Mg atom concentration with increasing Mg flux. Specifically, the Mg concentration increased from 2 × 10¹⁹ to 2 × 10²⁰ atoms/cm³, accompanied by an increase in active Mg atoms as acceptors from 4 × 10¹⁸ to 1 × 10¹⁹ cm⁻³ and an improvement in Hall mobility from 9.4 to 28.2 cm²/Vs. These variations in Mg concentration and the level of active Mg atoms have important implications for the material’s properties and its potential applications.

The spatial distribution of active Mg atoms as acceptors on c-GaN epilayers and potential contact difference (CPD) maps were investigated using Kelvin probe force microscopy (KPFM). Figure 20 illustrates the topography, CPD map, and optical images of the terrace bilayer of both unintentional c-GaN n-type and p-type c-GaN:Mg layers. Figure 20a shows the topography of the c-GaN buffer layer surface, which comprises flat rectangular domains with a median crystallite size of 200 nm, reflecting the cubic symmetry of the film with a root-mean-square (RMS) roughness of 1 nm. The equivalent CPD map, shown in Figure 20b, indicates no significant difference in surface potentials, with an RMS CPD of 2.5 mV.

Figure 20c,d present the CPD and optical images measured from the single-layer c-GaN to the single-layer GaN:Mg layer. The measurements revealed a CPD gradient of approximately 80 mV at the interface between the n-type and p-type regions. The topography of the Mg-doped GaN layer exhibited an average crystallite size of 150 nm and an increased RMS roughness of 6.1 nm. The corresponding RMS CPD map was 6 mV higher than that of the unintentional n-type c-GaN. The CPD value of c-GaN was lower than that of the GaN:Mg layer, suggesting proper Fermi level alignment at the interface between c-GaN and GaN:Mg.

To evaluate the quality of the p-n junction, J-V measurements were carried out. Ohmic contacts were created using Ni/Au for the p-GaN and Al for the unintentionally doped n-GaN. The J-V curve, measured in the dark, displays the characteristic diode behavior, as shown in Figure 21 by the solid squares. The solid line represents the data obtained from an applied voltage of 2 V in series with a diode and a load resistance R_L = 330 Ω. The intersection of the load line with the J-V curve yields a forward voltage V_D = 0.88 V. The reverse saturation current density J₀ determined from an exponential fit, is 7.8 × 10⁻⁴ mA/cm².

The forward voltage V_D is lower than the value reported for a GaAs/GaN/GaN-p diode [124]. This lower forward voltage is attributed to the challenges in achieving suitable contacts for p-type doped GaN and the formation of wide-bandgap barrier/passivating layers at the absorber interfaces [132]. Despite these issues, the reverse saturation current density J₀ and shunt conductance G values are satisfactory, as evidenced by the consistent behavior of the observed J-V curve.

14. Contributions

We successfully synthesized high-quality cubic InGaN films on MgO (100) substrates using plasma-assisted molecular beam epitaxy (PAMBE) with precise control over indium incorporation and phase purity. Notably, our approach enabled the tuning of the emission wavelengths across the visible spectrum, achieving violet, blue, yellow, and red emissions by varying the indium concentration during growth. This ability to finely control the emission properties surpasses the typical results reported in prior studies, where achieving such a broad range of tunable emissions was more challenging owing to phase mixing and compositional inhomogeneity.

Additionally, our work demonstrated a significant reduction in the defect density and phase impurities compared to earlier publications. The utilization of an intermediate cubic GaN buffer layer played a crucial role in minimizing the lattice mismatch and enhancing the crystalline quality of cubic InGaN films. Previous research often struggled with high defect densities and mixed-phase films, limiting the optical and electronic performance of the material. Our optimized growth parameters, including the adjustment of gallium and indium fluxes, have proven effective for producing films with superior structural and optoelectronic properties.

Moreover, we achieved high indium concentrations necessary for long-wavelength emissions without phosphorus incorporation, positioning our cubic InGaN films as promising candidates for light-emitting diodes (LEDs) and photovoltaic devices. This contrasts with prior studies that faced difficulties in maintaining high indium incorporation while preventing phase separation and achieving consistent optical performance.

Our work not only aligns with, but also advances, the current understanding of cubic InGaN growth and properties. By addressing key challenges, such as phase purity, defect reduction, and emission tuning, we have set a new benchmark for future research and application development in the field of cubic-phase III-nitride semiconductors.

15. Conclusions

An overview of the research focused on developing cubic-phase nitride semiconducting thin films is discussed, with significant achievements in the epitaxial growth of GaN, InN, and InGaN since the initial reports were reviewed. Although the first growth studies were performed using HVPE and MOCDV techniques, MBE has also been used to obtain InGaN structures without hexagonal phase inclusions. In addition, theoretical and experimental studies by different research groups have been reviewed to avoid inclusion of the hexagonal phase in the growth of the cubic structure. Consequently, the thermodynamic conditions used to obtain them were very similar. Various efforts have been made to obtain films with pure cubic structures, thereby allowing us to better understand their optical and electrical properties.

Our research has made significant progress in the epitaxial growth of cubic-phase GaN, InN, and InGaN using plasma-assisted molecular beam epitaxy (PAMBE). We have successfully synthesized high-quality cubic GaN films on MgO (001) substrates and demonstrated precise control over the transition from hexagonal to cubic phases by adjusting the gallium flux. Our findings indicate that higher gallium cell temperatures favor the formation of the cubic zincblende structure, resulting in improved crystallinity of the films.

InGaN alloys with varying indium concentrations have also been grown, achieving tunable emissions across the visible spectrum. Notably, a heterostructure with four GaN/InGaN/GaN quantum wells was developed, which exhibited violet, blue, yellow, and red emissions. This was accomplished by precisely controlling indium content during the growth process.

Our study highlights the potential of cubic InGaN films to generate long-wavelength emissions without phosphorus, making them suitable for applications in light-emitting diodes and photovoltaic devices. These findings highlight the advantages of the cubic phase for optoelectronic applications and pave the way for future developments in high-efficiency semiconductor devices.

Our study explores the epitaxial growth of cubic and hexagonal InN films on MgO (100) substrates using plasma-assisted molecular beam epitaxy (PAMBE). A significant achievement is the development of high-quality cubic InN films, facilitated by an inter-mediate cubic GaN buffer layer. This approach significantly reduces the lattice mismatch and enhances the crystalline quality of the InN films.

The key findings include the controlled growth of hexagonal and cubic InN phases by modulating the substrate temperature and indium flux. The introduction of a low-temperature InN buffer layer and subsequent annealing proved crucial for minimizing the phase mixture and improving the film crystallinity. X-ray diffraction and reflection high-energy electron diffraction (RHEED) analyses confirmed the successful formation of two distinct hexagonal domains and the stabilization of the cubic phase.

Moreover, our research demonstrates that cubic InN films exhibit lower bandgap energies than their hexagonal counterparts, extending the emission wavelength range to the infrared region. This property, combined with higher carrier mobility and electron drift velocity, positions cubic InN as a promising candidate for infrared photodetectors, terahertz devices, and other advanced optoelectronic applications.

These findings underscore the immense potential of cubic InN for next-generation semiconductor devices, emphasizing the importance of precise growth control and substrate engineering to achieve high-quality crystalline films.