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We find examples of cohomogeneity one metrics on$$S^4$$$S^4$and$$\mathbb {C}P^2$$$C{P}^2$with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of Grove–Ziller metrics with flat planes that become instantly negatively curved under Ricci flow.

 

Abstract Gallium nitride (GaN) has emerged as one of the most attractive base materials for next-generation high-power and high-frequency electronic devices. Recent efforts have focused on realizing vertical power device structures such as in situ oxide, GaN interlayer based vertical trench metal–oxide–semiconductor field-effect transistors (OG-FETs). Unfortunately, the higher-power density of GaN electronics inevitably leads to considerable device self-heating which impacts device performance and reliability. Halide vapor-phase epitaxy (HVPE) is currently the most common approach for manufacturing commercial GaN substrates used to build vertical GaN transistors. Vertical device structures consist of GaN layers of diverse doping levels. Hence, it is of crucial importance to measure and understand how the dopant type (Si, Fe, and Mg), doping level, and crystal quality alter the thermal conductivity of HVPE-grown bulk GaN. In this work, a steady-state thermoreflectance (SSTR) technique was used to measure the thermal conductivity of HVPE-grown GaN substrates employing different doping schemes and levels. Structural and electrical characterization methods including X-ray diffraction (XRD), secondary-ion mass spectrometry (SIMS), Raman spectroscopy, and Hall-effect measurements were used to determine and compare the GaN crystal quality, dislocation density, doping level, and carrier concentration. Using this comprehensive suite of characterization methods, the interrelation among structural/electrical parameters and the thermal conductivity of bulk GaN substrates was investigated. While doping is evidenced to reduce the GaN thermal conductivity, the highest thermal conductivity (201 W/mK) is observed in a heavily Si-doped (1–5.00 × 1018 cm−3) substrate with the highest crystalline quality. This suggests that phonon-dislocation scattering dominates over phonon-impurity scattering in the tested HVPE-grown bulk GaN substrates. The results provide useful information for designing thermal management solutions for vertical GaN power electronic devices.