We present a comprehensive study of the crystal structure of the thin-film, ferromagnetic topological insulator (Bi, Sb)$_{2-x}$V$_x$Te$_3$. The dissipationless quantum anomalous Hall edge states it manifests are of particular interest for spintronic
s, as a natural spin filter or pure spin source, and as qubits for topological quantum computing. For ranges typically used in experiments, we investigate the effect of doping, substrate choice and film thickness on the (Bi, Sb)$_2$Te$_3$ unit cell using high-resolution X-ray diffractometry. Scanning transmission electron microscopy and energy-dispersive X-ray spectroscopy measurements provide local structural and interfacial information. We find that the unit cell is unaffected in-plane by vanadium doping changes, and remains unchanged over a thickness range of 4--10 quintuple layers (1 QL $approx$ 1 nm). The in-plane lattice parameter ($a$) also remains the same in films grown on different substrate materials. However, out-of-plane the $c$-axis is reduced in films grown on less closely lattice-matched substrates, and increases with the doping level.
We investigate mesoscopic Josephson junction arrays created by patterning superconducting disks on monolayer graphene, concentrating on the high-$T/T_c$ regime of these devices and the phenomena which contribute to the superconducting glass state in
diffusive arrays. We observe features in the magnetoconductance at rational fractions of flux quanta per array unit cell, which we attribute to the formation of flux-quantized vortices. The applied fields at which the features occur are well described by Ginzburg-Landau simulations that take into account the number of unit cells in the array. We find that the mean conductance and universal conductance fluctuations are both enhanced below the critical temperature and field of the superconductor, with greater enhancement away from the graphene Dirac point.
