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We investigate and present a very low-profile, high-efficiency, and high-gain 2-D leaky-wave antenna (2DLWA) implemented on a high permittivity substrate operating in the millimeter-wave range, paving the way for the seamless integration of a high-gain and high-efficiency antenna with a frontend. In contrast to the typical air-filled 2DLWA, where a perturbation of the first higher-order mode (TE1/TM1) results in highly directed broadside radiation, in the proposed antenna, the propagation and leakage of a quasi-TEM (Q-TEM) mode results in an extremely low-profile ($0.065\lambda _{0}$), high-gain (~15 dBi) antenna. In this scenario, the transverse resonance condition for a Q-TEM mode is established by employing a capacitive partially reflective surface (PRS) in a thin ($0.065\lambda _{0}$) substrate with a relative permittivity of 10.2. The proposed periodic 2DLWA, unlike the conventional uniform/quasi-uniform air-filled counterparts, radiates based on the $\mathbf {n}=-\mathbf {1}$ space harmonic operation. Due to a strong mutual interaction between the PRS and ground plane, conventional design and analysis approaches are no longer relevant in this low-profile architecture. Instead, we employ the Floquet modal expansion theory to create an equivalent circuit model (ECM) for the PRS that takes into account the contributions and mutual interactions of the Floquet harmonics as well as the ground plane effect. By employing reflecting boundaries realized with edge truncation (air trenches) in a high permittivity substrate, we show that the aperture efficiency is enhanced by 15% compared to conventional counterpart. [ABSTRACT FROM AUTHOR]