Is there a relationship between the crystallinity nature "single crystal, polycrystalline or amorphous" and semiconductor material property at the nanoscale?
It is a very broad question; and there is no generalized answer for all materials. Anyway, crystallinity of most samples depend on the synthetic technique. For the same semiconductor, one can prepare single crystals, polycrystalline and amorphous samples by different methods.
As stated above, your question is very broad and as a result is likely to get a broad answer. Here is my take to try and help you,
If you are asking how would a semiconductor device behave if made on all three: a single Si crystal substrate, a polycrystalline Si substrate, or an amorphous Si substrate???? Then there is an impact on fabricating a nanoscale device on each of these substrates. Semiconductors are fabricated on single crystal substrates. If a Si substrate is polycrystalline and full of poly-grain boundaries then there would be a consistency issue with the device's transistors electrical properties as transistors would be fabricated in and around grain boundaries. Again, there is a reason a high quality single crystal (Si ingot) is used in producting SI wafers for semiconductor processing. A single crystal is by definition a single crystal and if fabricated correctly provides a consistent substrate with a a clean lattice structure to work with. A polycrystalline Si substrate would have a range of electrical properties with a range of varying functionality transistors.
On the side, material properties at the nanoscale very much are impacted by their nano-structure. For example, pure carbon in various nano-structured forms will have different properties. So generally speaking nano-structure material properties vary with the crystalline nature of materials. Look at polymers. Based on fabrication and treatment you can change the amount of a polymer's crystalline phase my how it is heated; seem the multitude of research in DSC studies with polymers.
Joo, Ji Bong, Qiao Zhang, Michael Dahl, Ilkeun Lee, James Goebl, Francisco Zaera, and Yadong Yin. "Control of the nanoscale crystallinity in mesoporous TiO 2 shells for enhanced photocatalytic activity." Energy & Environmental Science 5, no. 4 (2012): 6321-6327.
As you know that the electronic properties in semiconductor depend on electrons, holes and lattice vibrations. In crystalline nanomaterials, the atoms are arranged in a periodic array or layers of the lattice. Thus each atom and the as created hole has a definite value of the potential gradient. The regular arrangement of the atoms in the lattice helps it to have a single value of the work function and thus maximum number of electron-hole pairs are produced even at a small applied voltage which otherwise very difficult or impossible to acheive with amorphous nature.