In recent years

the rapid advances of nanofabrication technology have made it possible to work with quasi-zero-dimensional quantum dot in laboratories. The electron energy spectrum of such quantum dot is fully quantized. Such systems are of great interest in fundamental studies

as well as in practical applications for microelectronic devices. There exist numerous works that have demonstrated the significant influence of electron-phonon interactions on the electronic

optical

and transport properties of microstructures such as quantum wells

quantum wires

and quantum dots. Other people studied the effects of electron-phonon on the quantum dots by the second-order Rayleigh-Schrodinger perturbation theory or the Landau-Pekar variational treatment. In this paper

we investigate weak-coupling polarons properties in a parabolic quantum dot by the linear combination operator in the first time. It is shown that the ground state energy and binding energy decrease with increasing the effective confined length of the quantum dot. The result presents the polaron ground state energy of parabolic quantum dot is as a function of the effective confinement length. They show the variation of the electron-phonon coupling strength has very small effect on the polaron binding energy of parabolic quantum dot in the profoundly weak effective confinement length field. But

the variation of electron-phonon coupling strength has obvious effect in the strong effective confinement length field. They will turn to bulk structure crystal when the effective confinement length intend to infinite. Also

indicate the polaron binding energy of parabolic quantum dot is as a function of effective confinement length. It is evident that the variation of the electron-phonon coupling constant has no effect on the polaron binding energy of parabolic quantum dot. In one word

the confine of parabolic quantum dot strengthens the ground state energy.