+ \frac{\kappa^2}{24}\xi^2(y)\bigg)\bigg]\tilde{\zeta}^n_{R,L}(y)\nonumber\\ 
   &=&-\tilde{m}_n\tilde{\zeta}^n_{L,R}(y)
   \label{6S}
  \end{aligned}$$

where $\tilde{m}_{n}$ is the mass of nth KK mode in St theory. The 4D fermions obey the canonical equation of motion $i\tilde{\gamma}^{\mu}\partial_{\mu}\tilde{\chi}^n_{L,R} = \tilde{m}_{n}\tilde{\chi}^n_{L,R}$. Moreover eqn.(\[6S\]) is obtained provided the following normalization conditions hold : $$\int_{0}^{\pi} dy e^{-3A(y)}\tilde{\zeta}_{L,R}^{m}\tilde{\zeta}_{L,R}^{n} = \delta_{m,n}
   \label{norm1S}$$ $$\int_{0}^{\pi} dy e^{-3A(y)}\tilde{\zeta}_{L}^{m}\tilde{\zeta}_{R}^{n} = 0
   \label{norm2S}$$

In the next two subsections, we discuss the localization scenario for massless and massive KK modes (for ST theory) respectively.

Massless KK mode
----------------

For massless mode, equation of motion of $\tilde{\zeta}_{L,R}$ takes the following form $$\begin{aligned}
   e^{-A(y)}&\bigg[&\pm\bigg(\partial_{y} - 2A'(y) + \frac{\kappa}{\sqrt{3}}\xi'(y)\bigg)\nonumber\\ 
   &+&m_5\bigg(e^{-\frac{\kappa}{2\sqrt{3}}<\Phi>} - \frac{\kappa}{2\sqrt{3}}\xi(y) + \frac{\kappa^2}{24}\xi^2(y)\bigg)\bigg]\tilde{\zeta}^n_{R,L}(y) 
   = 0
   \label{7S}
  \end{aligned}$$

Using the explicit solution of $\xi(y)$ and $A(y)$ (see eqn.(\[sol\_scalar\_final\]) and eqn.(\[sol\_warp\_final\])), we obtain the solution of eqn.(\[7S\]) as follows : $$\begin{aligned}
   \tilde{\zeta}_{L,R}(y) = \sqrt{\frac{k}{\big[e^{2k\pi r_c}-1\big]}} e^{2ky}
   \exp{\bigg[\frac{p}{2k}e^{(2+\nu)ky} - \frac{q}{4k}e^{-2(2+\nu)ky}}\bigg]
   \label{8S}
  \end{aligned}$$ for $m_5 = 0$; and

$$\begin{aligned}
   \tilde{\zeta}_{L,R}(y)&=&\sqrt{\frac{k \pm \tilde{m}_5}{e^{(\frac{p}{k}+\frac{q}{2k})}\big[\big(1\mp \frac{\kappa v_h}{2\sqrt{3}}
   m_5 e^{(2+\nu)k\pi r_c}\big)e^{(2k\pm m_5)\pi r_c}-1\big]}}\nonumber\\ 
   &\exp&{\bigg[(2k \pm \tilde{m}_5)y \mp \frac{\kappa v_h}{2\sqrt{3}}m_5 e^{(2+\nu)ky} \pm \frac{\kappa^2 v_h^2}{24}m_5 e^{2(2+\nu)ky}\bigg]}*\nonumber\\ 
   &\exp&{\bigg[\frac{p}{2k}e^{(2+\nu)ky} - \frac{q}{4k}e^{-2(2+\nu)ky}}\bigg]
   \label{9S}
  \end{aligned}$$

for $m_5 \neq 0$.

with $p$ and $q$ are given in eqn.(\[p\]) and eqn.(\[q\]) respectively. Moreover, the overall normalization constants in eqn. (\[8S\]) and eqn. (\[9S\]) are determined by using the normalization condition 