A transparent conductor structure includes a transparent, insulating substrate and a discontinuous, transparent conducting layer established on the substrate. The discontinuous conducting layer is partitioned into a plurality of segments, each of which has a thickness ranging from about 1 nm to about 10 μm. Two or more of the segments are connected together by i) a metal trace disposed on or between them, or ii) a mesh, formed from a plurality of metal traces, disposed across a surface of the segments.