Surface sub bands in electric quantum limit consider the contact plane between an insulator and a semiconductor, as in a metal-oxide-semiconductor transistor or MOSFET. With a strong electric field applied across the SiO₂-Si interface, the potential energy of a conduction electron may be approximated by V(x) = eEx for x positive and by V(x) = ∞ for x negative, where the origin of x is at the interface. The wave function is 0 for x negative and may be separated as ψ(x, y, z) = u(x) exp [i (k_yy + k_zz)], where u(x) satisfies the differential equation – (h²/2m) d²u/dx² + V (x) u = εu with the model potential for V(x) the exact Eigen functions are Airy functions, but we can find a fairly good ground state energy from the variational trial function x exp(-ax). 

(a) Show that (ε) = (h²/2m)a² + 3eE/2a. 

(b) Show that the energy is a minimum when a = (3eEm/2h²)^1/3. 

(c) Show that (ε) _min = 1.89(h²/2m)^1/3 (3eE/2)^2/3. In the exact solution for the ground state energy the factor 1.89 is replaced by 1.78. As E is increased the extent of the wave function in the x direction is decreased. The function u(x) defines a surface conduction channel on the semiconductor side of the interface. The various Eigen values of u(x) define what are called electric sub bands. Because the Eigen functions are real functions of x the states do not carry current in the x direction, but they do carry a surface channel current in the yz plane. The dependence of the channel on the electric field E in the x direction makes the device a field effect transistor.