  The compound HgBa$_2$CuO$_{4+\Delta}$ (Hg1201) exhibits a simple tetragonal
crystal structure and the highest superconducting transition temperature
(T$_c$) among all single Cu-O layer cuprates, with T$_c$ = 97 K (onset) at
optimal doping. Due to a lack of sizable single crystals, experimental work on
this very attractive system has been significantly limited. Thanks to a recent
breakthrough in crystal growth, such crystals have now become available. Here,
we demonstrate that it is possible to identify suitable heat treatment
conditions to systematically and uniformly tune the hole concentration of
Hg1201 crystals over a wide range, from very underdoped (T$_c$ = 47 K, hole
concentration p ~ 0.08) to overdoped (T$_c$ = 64 K, p ~ 0.22). We then present
quantitative magnetic susceptibility and DC charge transport results that
reveal the very high-quality nature of the studied crystals. Using XPS on
cleaved samples, we furthermore demonstrate that it is possible to obtain large
surfaces of good quality. These characterization measurements demonstrate that
Hg1201 should be viewed as a model high-temperature superconductor, and they
provide the foundation for extensive future experimental work.
