A simply and reproducible way is proposed to significantly suppress the nucleation density of graphene on the copper foil during the chemical vapor deposition process. By inserting a copper foil into a tube with one close end, the nucleation density on the copper foils can be reduced by more than five orders of magnitude and an ultra-low nucleation density of ~10 nucleus/cm2 has been achieved. The structural analyses demonstrate that single crystal monolayer graphene with a lateral size of 1.9 mm can be grown on the copper foils under the optimized growth condition. The electrical transport studies show that the mobility of such single crystal graphene is around 2400 cm2/Vs.

The third-order linear piezoelectricity tensor seems to be simpler than the fourth-order linear elasticity one, yet its total number of symmetry types is larger than the latter and the exact number is still inconclusive. In this paper, by means of the irreducible decomposition of the linear piezoelectricity tensor and the multipole representation of the corresponding four deviators, we conclude that there are 15 irreducible piezoelectric symmetry types, and thus further establish their characteristic web tree. By virtue of the notion of mirror symmetry and antisymmetry, we define three indicators with respect to two Euler angles and plot them on a unit disk in order to identify the symmetry type of a linear piezoelectricity tensor measured in an arbitrarily oriented coordinate system. Furthermore, an analytic procedure based on the solved axisdirection sets is also proposed to precisely determine the symmetry type of a linear piezoelectricity tensor and to trace the rotation transformation back to its natural coordinate system.

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