With the development of the solid theory and experimental technology

the properties of the surface polaron in a crystal have been of considerable interest.Many investigators studied the properties of the surface polaron in many aspects by a variety of theoretical and experimental methods.Sak and Evans

et al

.studied theoretically the surface polaron in polar crystals for the first time.Gu

et al

.discussed ideal surface polaron and the weak

the intermediate and the strong coupling polaron in semi-infinite polar crystal.The properties of the surface polaron in polar crystal and the effective mass of a slow-moving surface polaron in a semi-infinite crystal are investigated using perturbation and linear combination operator methods by the present authors and co-workers

respectively.However

the research of these methods only was restricted to the case of crystals having only one mode of the longitudinal optical(LO) phonon.A large number of polar crystals

with several atoms per unit cell

have more than one LO phonon branch.For example

in cuprite such as CuO

2

there are two LO phonon modes and for SiO

2

GaAs

1-x

P

x

and a large number materials with perovskites structure(SrTiO

3

LiNbO

3

BaTiO

3

…) there are more than two modes.The polaron problem with many LO phonon branches has been studied by Matsura and Lepine.The properties of the polaron and the magnetopolaron in a polyatomic polar crystals are investigated by the present authors.However

so far

research of the properties of surface polaron in polyatomic crystals considering both the electron-bulk-longitudinal-optical(LO)-phonon and electron-surface-optical(SO)-phonon interaction

has been very scarce.The properties of the excited state of polaron in polyatomic semi-infinite crystal

which are the strong-coupling with bulk longitudinal optical phonon and surface optical phonon

are studied by using a linear combination operator and unitary transformation.Results show that when the election is infinitely close to the surface of the crystal

we can neglect the contribution of the LO phonons

otherwise

the distance between the election and the surface of the crystals is far more than the semi-diameter of the polaron;we must take into account the SO phonons effect

the LO phonons can be omitted.More commonly

when the distance between the election and surface is at the same magnitude grade of polaron's radii LO and so phonon's contribution must be studied at the same time.