  We study the discrete Schr\"{o}dinger operators $H_{\lambda,\alpha,\theta}$
on $\ell^2(\mathbb{Z}^{d+1})$ with surface potential of the form
$V(n,x)=\lambda \delta(x)\tan\pi(\alpha\cdot n+\theta)$, and
$H_{\lambda,\alpha,\theta}^{+}$ on $\ell^2(\mathbb{Z}^{d}\times \mathbb{Z}_+)$
with the boundary condition $ \psi_{(n,-1)}=\lambda \tan\pi(\alpha\cdot
n+\theta)\psi_{(n,0)} $, where $\alpha\in \mathbb{R}^d$ is rationally
independent. We show that the spectra of $H_{\lambda,\alpha,\theta}$ and
$H_{\lambda,\alpha,\theta}^{+}$ are $(-\infty,\infty)$ for all parameters. We
can also determine the absolutely continuous spectra and Hausdorff dimension of
the spectral measures if $d=1$.
