  The current paper is devoted to the study of existence, uniqueness and
Lifshitz tails of the integrated density of surface states (IDSS) for
Schr\"{o}dinger operators with alloy type random surface potentials. We prove
the existence and uniqueness of the IDSS for negative energies, which is
defined as the thermodynamic limit of the normalized eigenvalue counting
functions of localized operators on strips with sections being special cuboids.
Under the additional assumption that the single-site impurity potential decays
anisotropically, we also prove that the IDSS for negative energies exhibits
Lifshitz tails near the bottom of the almost sure spectrum in the following
three regimes: the quantum regime, the quantum-classical/classical-quantum
regime and the classical regime. We point out that the
quantum-classical/classical-quantum regime is new for random surface models.
