There is a potential energy barrier at the interface between a semiconductor and a wide band gap dielectric which the free carrier incident at the semiconductor/dielectric interface will encounter. According to the rules of quantum mechanics, the electron will penetrate into the dielectric, however, the amplitude of its wave-function will exponentially attenuate; which means that the carrier density will sharply decrease with distance from the interface. The electron/hole density attenuation will be larger for larger barrier and larger effective mass. For example, the carrier density will, roughly, attenuate by a factor of 1/e for each 0.16 nm into the dielectric, if the band offset at the interface is 2 eV and the effective mass is equal to 0.16 m. 

Dear Prof. Kar,

Thank you very much for your answer. The problem then turns into a case of electron penetrating a barrier. If we solve the problem from a classical point of view, i. e., solving the Maxwell eqn. in the semiconductor and the polar dielectric where the boundary conditions for potential are defined at the interface, charge distributions defined via the Maxwell-Boltzmann limit, how would one interpret the existence of an electron density inside the dielectric (near the interface with a penetration of a few nm's) if the polarization is pointing away from the interface (meaning positive pole is near the interface). The electron density inside the dielectric rapidly decays to zero after a few nm's.

Dielectrics have lots of bulk defects/ traps (though it is unintended). Again dielectric-semiconductor interface also have lots of interface traps which can eventually reduce the barrier height for carriers (though dielectric generally have wide bandgap). These defects help carrier penetration and result in considerable gate leakage.