Confirming the photocatalytic inactivity of Cu2O nanocubes through the formation of Au‐decorated–Cu2O heterostructures, spiky ZnO nanostructures are grown on Cu2O cubes, octahedra, and rhombic dodecahedra to demonstrate that charge transfer across semiconductor heterojunctions is also strongly facet dependent. Unintended CuO formation in the growth of ZnO on perfect Cu2O cubes makes them slightly active toward methyl orange photodegradation. Under optimal ZnO growth conditions without CuO presence, Cu2O cubes remain inactive, while rhombic dodecahedra show an enhanced photocatalytic activity due to better charge transfer according to normal Cu2O–ZnO band alignment. Surprisingly, photocatalytically active Cu2O octahedra become inactive after ZnO deposition. An extensive interfacial microscopic examination reveals preferential formation of the ZnO (101) planes on the {111} surfaces of Cu2O octahedra, while different ZnO lattice planes are observed to deposit on Cu2O cubes and rhombic dodecahedra. The photocatalytic inactivity of ZnO‐decorated Cu2O octahedra is explained in terms of an unfavorable band alignment arising from an unusual degree of band bending for the ZnO {101} face relative to the band energy of the Cu2O {111} surface. The efficiency of charge transfer across semiconductor heterojunctions strongly depends on the band edge energies of the contacting planes.

Large PbS protruded cubes, edge and cornertruncated cubes and octahedra, and perfect octahedra with sizes over 200 nm have been synthesized in aqueous solution. By using two surface oxidefree tungsten probes to contact a clean particle, these PbS nanocrystals displayed facet dependent electrical conductivity behaviors. Both {110} and {100} faces are highly conductive at applied voltages beyond 4 V, but the {111} faces can remain nonconductive even at 5 V. An asymmetric I-V curve was recorded when electrical contacts were made simultaneously on the {110} and {111} facets of a truncated cube. A modified band diagram of PbS is constructed to account for the observed facetdependent effect. Density of states plots for varying numbers of PbS surface planes show larger areas of conduction band electron occupancy for the (110) and (100) planes than that for the (111) planes at a layer thickness of 3.0-3.4 nm. The work represents that, for the first time, the facetdependent electrical properties of an ntype semiconductor nanocrystal are directly probed. Facetdependent electrical conductivity should be a general semiconductor property and can be exploited to fabricate singlenanocrystal operating electronic components. 

Abstract. In this paper, we give some characterizations of matrices which have defect index one. Recall that an n -by-n matrix A is said to be of class S n (resp., S −1 n ) if its eigenvalues are all in the open unit disc (resp., in the complement of closed unit disc) and rank (I n − A * A) = 1 . We show that an n -by-n matrix A is of defect index one if and only if A is unitarily equivalent to U ⊕ C , where U is a k -by-k unitary matrix, 0 k < n , and C is either of class S n−k or of class S −1 n−k . We also give a complete characterization of polar decompositions, norms and defect indices of powers of S −1 n -matrices. Finally, we consider the numerical ranges of S −1 n -matrices and S n -matrices, and give a generalization of a result of Chien and Nakazato on tridiagonal matrices (cf. [3, Theorem 7]). Mathematics subject classification (2010): 47A12, 15A60.