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Polymer matrix composites have high strengths in tension. However, their compressive strengths are much lower than their tensile strengths due to their weak fiber/matrix interfacial shear strengths. We recently developed a new approach to fabricate composites by overwrapping individual carbon fibers or fiber tows with a carbon nanotube sheet and subsequently impregnate them into a matrix to enhance the interfacial shear strengths without degrading the tensile strengths of the carbon fibers. In this study, a theoretical analysis is conducted to identify the appropriate thickness of the nanocomposite interphase region formed by carbon nanotubes embedded in a matrix. Fibers are modeled as an anisotropic elastic material, and the nanocomposite interphase region and the matrix are considered as isotropic. A microbuckling problem is solved for the unidirectional composite under compression. The analytical solution is compared with finite element simulations for verification. It is determined that the critical load at the onset of buckling is lower in an anisotropic carbon fiber composite than in an isotropic fibfer composite due to lower transverse properties in the fibers. An optimal thickness for nanocomposite interphase region is determined, and this finding provides a guidance for the manufacture of composites using aligned carbon nanotubes as fillers in the nanocomposite interphase region.