InGaN-based visible light-emitting diodes (LEDs) find commercial applications for solid-state lighting and displays, but lattice mismatch limits the thickness of InGaN quantum wells that can be grown on GaN with high crystalline quality. Since narrower wells operate at a higher carrier density for a given current density, they increase the fraction of carriers lost to Auger recombination and lower the efficiency. The incorporation of boron, a smaller group-III element, into InGaN alloys is a promising method to eliminate the lattice mismatch and realize high-power, high-efficiency visible LEDs with thick active regions. In this work, we apply predictive calculations based on hybrid density functional theory to investigate the thermodynamic, structural, and electronic properties of BInGaN alloys. Our results show that BInGaN alloys with a B:In ratio of 2:3 are better lattice matched to GaN compared to InGaN and, for indium fractions less than 0.2, nearly lattice matched. Deviations from Vegard's law appear as bowing of the in-plane lattice constant with respect to composition. Our thermodynamics calculations demonstrate that the solubility of boron is higher in InGaN than in pure GaN. Varying the Ga mole fraction while keeping the B:In ratio constant enables the adjustment of the (direct) gap in the 1.75–3.39 eV range, which covers the entire visible spectrum. Holes are strongly localized in non-bonded N 2p states caused by local bond planarization near boron atoms. Our results indicate that BInGaN alloys are promising for fabricating nitride heterostructures with thick active regions for high-power, high-efficiency LEDs.
InGaN light-emitting diodes (LEDs) with an electricity-to-light conversion efficiency of ∼39% provide large efficiency gains and cost reductions compared to incandescent and fluorescent light sources.1 However, InGaN LEDs suffer from decreasing internal quantum efficiency at high currents (efficiency droop), particularly at longer wavelengths (green gap). The cause of the droop has been extensively studied, with Auger recombination identified as a major loss mechanism.2 For equal electron and hole densities, the Auger recombination rate is equal to the carrier density cubed times a material-dependent Auger coefficient . Since the Auger coefficient is an intrinsic property of InGaN that does not depend strongly on composition, temperature, or strain,3 the carrier density at a given current density must be lowered to reduce the Auger losses.
Increasing the active-region volume is a straightforward approach to reduce the carrier density and hence the Auger loss, yet growth challenges limit its practicality. Devices using a single thick InGaN layer exhibit higher high-power efficiency than thinner quantum wells both for polar4 and for semipolar5 growth orientations. The thickness of InGaN active layers is however limited by the lattice mismatch with the underlying GaN layers and the subsequent appearance of performance-degrading dislocations. On the other hand, the efficiency of multiple-quantum-well (MQW) structures is lower than a single InGaN layer of the same total active thickness5 since hole injection is poor and only the first few QWs near the p-layer in a MQW structure emit light.6,7
The co-alloying of InGaN with wurtzite boron nitride (w-BN) can produce BInGaN alloys lattice-matched to GaN with gaps spanning the visible range (Fig. 1). Co-alloying has been demonstrated in, e.g., GaAsPBi8 and GaAsNBi,9 in which the co-incorporation of P (N) and Bi atoms yields alloys lattice-matched to GaAs with a reduced bandgap. Under ambient conditions, BN adopts the hexagonal layered structure that can be exfoliated to form 2D materials. Wurtzite BN is a high-pressure polytype with the same crystal structure as InGaN. For the low BN content, alloys of BN with InGaN are expected to also adopt the wurtzite structure to minimize dangling-bond formation. Moreover, although w-BN is an indirect-gap semiconductor, the gap of BInGaN is determined by the lower gap of the InGaN component, which is direct. The in-plane lattice constant of BInGaN is approximated, to first order, by Vegard's law. Based on the experimental in-plane lattice constants of GaN (3.181 Å), InN (3.538 Å), and w-BN (2.536 Å),10–12 the optimal B mole fraction to lattice-match ByIn1−yN to GaN is y = 0.356 ≈ 0.4. Therefore, an approximate ratio of B:In ≈ 2:3 is expected to yield BInGaN alloys nearly lattice-matched to GaN. Hence, varying the Ga mole fraction while keeping the B:In ratio constant is a promising method to tune the gap of BInGaN while keeping the in-plane lattice constant nearly matched to GaN.
Previous experimental and theoretical studies have explored the structural and electronic properties of B-containing GaN, InN, and InGaN. Ougazzaden et al. grew BGaN thin films with up to 3.6% boron.13 They reported lower gaps than GaN and a large bowing parameter.14 Kadys et al. used metal-organic chemical vapor deposition (MOCVD) to grow up to 2.9, 4.3, and 5.5% boron BGaN on GaN, AlN, and SiC substrates, respectively.15 Cramer et al. reported high-crystal-quality BGaN with up to 3% boron grown with plasma-assisted molecular beam epitaxy and observed statistically random atomic distributions.16 Gunning et al. grew up to 7.4% boron BGaN on AlN using metalorganic vapor phase epitaxy (MOVPE) at low temperatures and pressure (750–900 °C and 20 Torr) but reported severe structural degradation and a shift to a twinned cubic structure at higher boron concentrations.17 Approximately equimolar nanocolumnar BInN has also been synthesized with a reported gap of 3.75 eV.18 Quaternary BInGaN has been grown on GaN/sapphire and ZnO-buffered Si substrates with MOVPE by Gautier et al.19,20 They reported smaller lattice constants and gaps than GaN for up to 2% B and up to 14% In content.19,20 The extracted bowing parameters were applied to predict the gap for a broader composition range. However, their InGaN bowing parameter is larger than subsequent predictive calculations,21 while the BN gap was set to the indirect Γ-K value instead of the larger direct Γ-Γ one.13,14,22 McLaurin also reported the growth of BInGaN.23 Theoretically, Park and Ahn examined BInGaN/GaN quantum wells with effective mass theory. They reported a lower mismatch to GaN than InGaN and a reduction of the polarization fields.24 Assali et al. examined ordered BInGaN in the metastable zinc blende phase with density functional theory (DFT). They found a near-lattice-match to GaN for a composition with the same 2:3 B:In ratio as our estimate and a small increase of the gap (0.1–0.3 eV) compared to InGaN of the same In content.25 However, they did not examine the thermodynamically stable wurtzite phase, which has a larger gap than zinc blende,26 and did not account for disorder. Overall, the properties of GaN-lattice-matched disordered wurtzite BInGaN alloys over their full composition range and their potential for reducing the LED droop problems remain unexplored.
In this work, we explore the thermodynamic, structural, and electronic properties of statistically random quaternary wurtzite BInGaN alloys with hybrid-functional DFT. BInGaN alloys with a B:In ratio of 2:3 are better lattice mismatch to GaN than InGaN. Co-alloying with In lowers the enthalpy of mixing and facilitates higher B incorporation. Our results show that BInGaN alloys can be designed nearly lattice-matched to GaN with a direct bandgap adjustable over the entire visible range and are therefore promising active-layer materials to overcome the efficiency-droop and green-gap problems of nitride LEDs.
We performed DFT calculations based on the projector augmented wave (PAW) method27,28 using the Vienna Ab initio Simulation Package (VASP).29–32 The GW-compatible pseudopotentials including 3, 13, 13, and 5 valence electrons were employed for B, In, Ga, and N, respectively, with a 600 eV plane-wave cutoff. Structural relaxations were performed using the optB86b-vdW functional33 and a Γ-centered Wisesa-McGill-Mueller Brillouin-zone grid with a minimum period distance of 21.48 Å.34 Forces on atoms were relaxed to 1 meV/Å. Band-gap calculations were performed with the Heyd–Scuseria–Ernzerhof (HSE06) functional.35,36 Random alloys were modeled using Special Quasi-random Structures (SQSs) generated with the Alloy Theoretic Automated Toolkit37 and a 3 × 3 × 2 wurtzite supercell. Cations were arranged to approximate the pair-correlation functions of random alloys up to 5.125 Å. Five SQSs were generated at each composition and relaxed to obtain structural parameters. The SQSs that most closely match random pair-correlation functions at each composition were used for electronic and thermodynamic calculations (Fig. S1, supplementary material). The projected density of states (pDOS) was calculated using the optB86b-vdW functional and a Γ-centered 8 × 8 × 8 Brillouin-zone grid, and the gap was rigidly shifted to the HSE06 value.
Our thermodynamic analysis reveals that the solubility of B into InGaN is higher than into GaN. To calculate the transition temperature between the solid-solution and the miscibility-gap regimes as a function of composition, we evaluated the enthalpy of mixing as a function of alloy composition, , by subtracting the total energy of ternary and quaternary alloys from the linear combination of the binaries. The entropy was evaluated using the regular solution model, ( is the mole fraction for each of the alloy ingredients, and is Boltzmann's constant). The transition temperatures (Fig. 2) for BxIn1.5xGa1−2.5xN, BxGa1−xN, and In1.5xGa1−1.5xN are well above typical growth temperatures for x > 0.05, which is expected since nitrides are typically grown with epitaxial techniques (e.g., MOCVD or molecular beam epitaxy) that take place far from thermodynamic equilibrium. However, the relative equilibrium phase boundaries can be combined with experimental literature data to predict the limits of boron incorporation into InGaN. For boron mole fractions less than 0.2, the temperature needed to dissolve B into InGaN is approximately 2/3 of the temperature needed to dissolve it into GaN. The increased solubility of B into InGaN is due partly to the increased configurational entropy in the quaternary alloy and partly to the partial cancellation of local stress by the opposite size mismatch between the smaller B and the larger In atoms. We therefore estimate that 1.5× as much boron can be incorporated into InGaN compared to GaN. Since high-quality, single-phase BxGa1−xN alloys with boron concentration up to x = 0.03 have already been demonstrated,16 we anticipate that up to ∼4.5% boron incorporation should be possible in BInGaN with existing growth approaches.
Our calculations verify that BInGaN with a 2:3 B:In ratio is better lattice-matched to GaN substrates than InGaN and can therefore be grown to larger thicknesses. Figure 3 shows the calculated configurationally averaged a and c lattice constants of BxIn1.5xGa1−2.5xN relative to GaN. The trend for a is not described well by a straight line and therefore deviates from Vegard's law, agreeing with previous calculations for BGaN.38 Instead, it follows a bowing relationship, , where , , and . The a lattice-constant mismatch of BInGaN is smaller than ±0.25% for indium fractions under 0.2 [Fig. 3(a)], which facilitates growth on the c-plane of GaN. Due to the bowing, the mismatch of B0.045In0.0675Ga0.8875N to GaN is predicted to be only . The c lattice constant of BInGaN [Fig. 3(b)] remains near the GaN value.
Our band-gap calculations demonstrate that BxIn1.5xGa1−2.5xN has a tunable direct gap that spans the entire visible range. In contrast to BGaN, which transitions from direct to indirect gap for increasing B content,39 the gap of BInGaN remains direct throughout the entire explored composition range. Figure 4 shows the calculated gap of BxIn1.5xGa1−2.5xN, BxGa1−xN, and In1.5xGa1−1.5xN as a function of composition. For the 2:3 B:In ratio, the gap ranges from 3.14 to 1.50 eV for decreasing Ga content. Increasing our calculated values by 0.25 eV to match the room-temperature gap of GaN (3.39 eV)40 brings our BInGaN gap estimate to the 3.39–1.75 eV range, spanning the entire visible spectrum. The gap of BxIn1.5xGa1−2.5xN is also approximately equal to that of In1.5xGa1−1.5xN for the same In mole fraction. To explore this behavior, we determined the orbital character of the conduction and valence band edges. The pDOS of B0.278In0.417Ga0.306N, calculated with the optB86b-vdW functional, (Fig. 5; comparison to the HSE pDOS in Fig. S2, supplementary material) shows that the edge states consist primarily of N and In orbitals, while B states lie higher in the conduction band. Similar to InGaN, the valence-band edge consists primarily of localized N 2p states, except in BInGaN the states reside near planarized B atoms (Fig. 5 inset). The hole localization energy ranges from 0.1 to 0.4 eV as the boron content ranges from 15% to 40%, the maximum value occurring for B0.278In0.417Ga0.306N (Fig. 6). Localized states were not observed for boron mole fractions lower than 15%. The conduction-band edge is primarily formed by N 2p, N 2s, and In 5s states. Our calculations therefore show that partial substitution of B for Ga in InGaN has only minor effects on the gap and edge states.
The addition of boron into InGaN active layers is a promising method of overcoming the LED droop and green-gap problems. The better lattice match with GaN and the resulting reduced strain allows the growth of thicker active layers, thus decreasing the carrier concentration for a given current density and hence the fraction of carriers that recombine via Auger. While the spatial separation of carriers by the polarization fields in polar wells is amplified for thicker layers at low carrier concentrations, the polarization fields are screened by free carriers under high-power operation and flat-band conditions prevail.2 Our estimate for the Debye screening length is approximately 0.5 nm for typical carrier densities ( 5 × 1018 cm−3) of the internal quantum efficiency maximum.2 Therefore, for sufficiently thick wells (e.g., thicker than 10 nm) and high carrier densities, the polarization fields are completely screened by free carriers.
One challenge regarding the growth of BInGaN is the different temperatures needed for the ingredient materials. GaN is typically grown at high temperature to achieve higher crystalline quality, but InGaN and BGaN are typically grown at lower temperatures to facilitate In and B incorporation.15,17 However, the lower enthalpy of mixing B into quaternary BInGaN than in ternary BGaN may facilitate the growth of BInGaN in a wider temperature window. For example, Gautier et al. successfully grew BInGaN with up to 2% boron and 14% indium using MOVPE at 730 °C.19,20 Another potential challenge is the appearance of secondary phases during growth. Gunning et al. found that BGaN creates a twinned cubic structure at their selected growth conditions.17 Similar cubic secondary phase inclusions may also form during BInGaN growth and deteriorate the structural quality. Hence, the thermodynamics of both hexagonal and cubic phases of BInGaN need to be further investigated both experimentally and theoretically to facilitate the development of device-quality materials.
In conclusion, we examine the effects of co-alloying boron and indium into GaN with first-principles calculations. Alloying 2B:3In into GaN creates alloys with gaps similar to InGaN of the same indium concentration, while reducing lattice mismatch and nearly eliminating it for alloys with In mole fractions less than 0.2 (i.e., bandgaps larger than 2.75 eV). Our thermodynamics analysis reveals that B is more easily incorporated into InGaN than into pure GaN. Our results point to BInGaN alloys as promising materials to fabricate thicker active regions than InGaN for higher-efficiency high-power visible LEDs.
See supplementary material for technical details on the band-gap variability between SQSs and on the comparison of the optB86b-vdW and the HSE06 pDOS.
This work was supported by the Designing Materials to Revolutionize and Engineer our Future (DMREF) Program under Award No. 1534221, funded by the National Science Foundation. This research used resources of the National Energy Research Scientific Computing (NERSC) Center, a DOE Office of Science User Facility supported under Contract No. DE-AC02-05CH11231.