We provide an analysis of the electromagnetic modes of three-dimensional metamaterial resonators in the THz frequency range. The fundamental resonance of the structures is fully described by an analytical circuit model, which not only reproduces the resonant frequencies but also the coupling of the metamaterial with an incident THz radiation. We also evidence the contribution of the propagation effects, and show how they can be reduced by design. In the optimized design the electric field energy is lumped into ultra-subwavelength ($lambda$/100) capacitors, where we insert semiconductor absorber based on the collective electronic excitation in a two dimensional electron gas. The optimized electric field confinement is evidenced by the observation of the ultra-strong light-matter coupling regime, and opens many possible applications for these structures for detectors, modulators and sources of THz radiation.
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