According to quantum free electron theory of metals, a electron in a metal experiences constant(or zero) potential and free to move inside the crystal but will not come out of the metal because an infinite potential exists at the surface. Bloch Theorem: According to this theorem, the periodic potentials due to the positive ions in metal have been considered. (i.e. the electrons moves in a periodic potential provided by lattice). If the electron moves through these ions, it experiences varying potentials. The potential of an electron at the positive ion site zero and is maximum in between two ions. i.e. the potential experienced by an electron varies periodically with the same period as the lattice. The potential is negative because of attractive force between electrons and positive ions.
Along X- direction in the crystal, the potential function V(x) has the periodicity of lattice given by V(x) = V(x + a ) where, ‘x’ is distance of the electron from the core and ‘ a’ is the periodicity of the lattice. The energies of electron can be known by solving Schreodinger’s time independent wave equation for motion of an electron along x-direction is given by
Bloch showed a type of solution for the above equation, given by ψ(x) = uk (x) eikx where uk(x) represents periodic function given by uk (x) = uk(x+a) Here, K = 2π/λ = propagation vector and λ = wavelength of debroglie’s wave associated with the moving electron and eikx represents a plane wave. However, it is extremely difficult to solve the Schrödinger’s equation with periodic potential described above. Hence the Kronig – Penney Model is adopted for simplification. Kronig – Penney Model
In 1930, Kronig- penny proposed a one dimensional model for the shape of rectangular potential wells and barriers having lattice periodicity. According to this theory, the electrons move in a periodic potential produced by the positive ion cores. The potential of electron varies periodically with periodicity of ion core ‘a’ which is nothing but inter-atomic spacing. It is assumed that the potential energy of the electrons is zero near nucleus of the positive ion core and maximum when it is lying between the adjacent nuclei as shown fig.
Fig. One dimensional periodic potential
The width of the potential well and barrier are ‘a’ and ‘b’ respectively. The potential energy of an electron in the well is zero and in the barrier is V0. The periodicity of the potential is a+b. This model is an highly artificial, but it illustrates many of the characteristics features of the behavior of electrons in periodic lattice. The energies and wave functions of electrons associated with model can be calculated by solving the Schrödinger’s wave equation for two regions I and II.
The Schrödinger’s equations are
for region-I 0 < x < a and for region-II -b < x < a
Since E less than Vo, define two +ve quantities, α2 = 2mE/ ħ2 and β2 = 2m(V0– E )/ħ2 for region 0 < x < a for region -b < x < a
According to Bloch Theorem, the solutions of above equations can be written as Ψ(x) = uK (x) eiKx
The consists of a plane wave eiKx modulated by the periodic function uK(x), where this UK(x) is periodic with the periodicity of the lattice. uK(x) = uK(x+a)
where K is propagating constant along x-direction and is given by K =2π/λ is a Propagation wave vector. In order to simplify the computations, an assumption made regarding the potential barrier. As Vo increases the width of the barrier ‘b’ decreases so that the product Vob remains constant. After tedious calculations, the possible solutions for energies are obtain from the relation P sin〖∝a〗/αa+cos〖∝a〗=cos〖Ka 〗------------ (1) where P = mVo ab /ħ2 is scattering power of the potential barrier. It is a...