  We present a theory pertaining to the asymptotic behavior of Rashba energy
splitting in a quantum-well state (QWS). First, unlike previous studies, we
derive $\textbf{k}$-linear Rashba term from a first-principles Hamiltonian in a
physically convincing manner. The $\textbf{k}$-dependent in-plane intrinsic
magnetic-field term originates from the spin--orbit interaction and hybridized
$s$-$p_z$ orbital, whereas a steep nucleus potential realizes the linearity for
the $\textbf{k}$ of the effective magnetic field. Next, we analyze the Rashba
effect of a QWS using a one-dimensional tight-binding model developed based on
the bottom-up approach that is aforementioned. The Rashba-splitting behavior of
this system is captured from the density at the interface. The density can be
expressed analytically as a function of the monolayer number and well depth.
Finally, we apply our formula to the QWS of a few-monolayers Ag on an Au(111)
surface to validate the theory based on a realistic system. Our tight-binding
analysis qualitatively fits the first-principles result using only two fitting
parameters and predicts the optimal condition for achieving a large Rashba
splitting.
