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The authors report the experimental observation of a critical diameter (CD) of III-V compound semiconductor epitaxial nanowires (NWs) grown on lattice-mismatched substrates using Au-catalyzed vapor-liquid-solid growth. The CD is found to be inversely proportional to the lattice mismatch. NWs with well-aligned orientation are synthesized with catalysts smaller than the CD. Well-aligned InP NWs grown on a Si substrate exhibit a record low photoluminescence linewidth (5.1 meV) and a large blueshift (173 meV) from the InP band gap energy due to quantization. Well-aligned InAs NWs grown on a Si substrate are also demonstrated.