2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/220812A class of thick branes in the background of sine-Gordon kinks with a scalar potential $V(ϕ)=p(1+\cos\frac{2ϕ}{q})$ was constructed by R. Koley and S. Kar [Classical Quantum Gravity \textbf{22}, 753 (2005)]. In this paper, in the background of the warped geometry, we investigate the issue of localization of spin half fermions on these branes in the presence of two types of scalar-fermion couplings: $η\barΨϕΨ$ and $η\barΨ\sinϕΨ$. By presenting the mass-independent potentials in the corresponding Schrödinger equations, we obtain the lowest Kaluza--Klein (KK) modes and a continuous gapless spectrum of KK states with $m^2>0$ for both types of couplings. For the Yukawa coupling $η\barΨϕΨ$, the effective potential of the right chiral fermions for positive $q$ and $η$ is always positive, hence only the effective potential of the left chiral fermions could trap the corresponding zero mode. This is a well-known conclusion which had been discussed extensively in the literature. However, for the coupling $η\barΨ\sinϕΨ$, the effective potential of the right chiral fermions for positive $q$ and $η$ is no longer always positive. Although the value of the potential at the location of the brane is still positive, it has a series of wells and barriers on each side, which ensures that the right chiral fermion zero mode could be trapped. Thus we may draw the remarkable conclusion: for positive $η$ and $q$, the potentials of both the left and right chiral fermions could trap the corresponding zero modes under certain restrictions.22 pages, 21 figures, published version to appear in Phys. Rev. DHigh Energy Physics - Theorytext