  This work studies high-temperature superconducting spiral resonators as a
viable candidate for realization of RF/microwave metamaterial atoms. The theory
of superconducting spiral resonators will be discussed in detail, including the
mechanism of resonance, the origin of higher order modes, the analytical
framework for their determination, the effects of coupling scheme, and the
dependence of the resonance quality factor and insertion loss on the parity of
the mode. All the aforementioned models are compared with the experimental data
from a micro-fabricated YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) spiral resonator.
Moreover, the evolution of the resonance characteristics for the fundamental
mode with variation of the operating temperature and applied RF power is
experimentally examined, and its implications for metamaterial applications are
addressed.
