Origin of Energy Bands | Solid State Physics, Devices & Electronics PDF Download

Origin of Energy Bands

The failure of the free electron model is due to the over simplified assumption that a conduction electron in a metal experiences a constant or zero potential due to the ion cores and hence is free to move about crystal.
Now the periodic potential described below forms the basis of the band Theory of solids. The behavior of an electron in this potential is describe by constructing the electron wave functions using one-electron approximates.

As we shall discuss later, the motion of an electron in a periodic potential yields the following results.
(a) There exist allowed energy bands separated by forbidden region or band gap.
(b) The electronic energy function E(K) is periodic in the wave vector K
In the free electron Theory E varies with K  

The Block Theorem

The 1-D Schrödinger equation for an electron moving in a constant potential V₀ is  
 The solution ψ(x) = e^±ikx
For periodic potential with period equal to the lattice constant a we have
ψ(x) = e^±ikx U_k(x)    where U_k = U_k (x + a)  
Note: Let g(x) and f(x) be two real and independent solution to the second order differential The general equation can be written as   ψ(x) = A f(x) + B g(x)
while   = constant
In three dimensions, the block Theorem is expressed as  

Thus the wave function becomes the one of a free electron. 

The Kronig-Penney Model 

This model illustrates the behaviour of electrons in a periodic potential by assuming a relatively simple one-dimensional model of periodic potential as shown figure.

For this potential write down the Schrödinger wave equation and its general solution with taking potential constant finally. We get

which is a measure of the area V₀b of the potential barrier. Thus increasing P has the physical meaning of bonding an electron more strongly to a particular potential well

We know that

It may be noted that since α² is proportional to the energy E the abscissa is a measure of the energy. The following conclusion may be drawn from figure.
(i) The energy spectrum of the electrons consists of alternate regions of allowed energy bands (solid lines on abscissa) and forbidden energy band (broken lines)
(ii) The width of the allowed energy bands increases with αa or the energy
(iii) The width of  particular allowed energy band decreases with increase in  value of P.

Figure- Allowed (shaded) and forbidden (open) energy ranges as a function of P

 Energy Verses Wave-Vector relationship
The energy E is also an even periodic function of k with period of 2π/a i.e k = ± nπ/a
dn = (1/2π)dk
Velocity is

Concept Effective Mass and Holes  

In one dimension, an electron with wave-vector k has group velocity  
 ...(i)
If an electric field ε acts on the electron, then in time δ t . It will do work  
δ E = force times distance= −eε vδ t  (ii)  
But    (iii)
So, comparing equation (ii) with (iii), we have  
 (iv)
In terms of force F,
Generalising to three dimensions:

where
From equation (i) v
Differentiating with respect to time

But from equation (iv),
 So
But from Newton’s equation we expect
which leads us to define an effective mass
That is  
The dynamics of electrons is modified by the crystal potential;  
The effective mass depends on the curvature of the bands;  
Flat bands have large effective ma      
Near the bottom of a band, m* is positive, near the top of a band, m* is negative.  

Hole Concept 

➤ Hole wavevector: The total k of a full band is zero: if we remove an electron with wavevector k_e the total k of the band is  k_h +k_e = 0⇒ k_h = −k_e
➤ Hole energy: Take the energy zero to be the top of the valence band. The lower the electron energy, the more energy it takes to remove it; thus
E_h(k_h) = -E_e(k_e)
But bands are usually symmetric, E(k) =E (−k)
So E_h(k_h) = E_h(-k_h) = -E_e(k_e)
➤ Hole velocity: In three dimensions
But k_h =−k_e So  
and so  
The group velocity of the hole is the same as that of the electron.  

➤ Hole effective mass: The curvature of E is just the negative of the curvature of − E ,  
So m_h* = −m_e*
Note that this has the pleasant effect that if the electron effective mass is negative, as it is at the top of the band, the equivalent hole has a positive effective mass.  

➤ Hole dynamics
We know that
Substituting k_h =−k_e and v_h = v_e gives  

Exactly the equation of motion for a particle of positive charge.  
Under an electric field, electrons and holes acquire drift velocities in opposite directions, but both give electric current in the direction of the field.