Calculating the Motion of Electron (With Formula) | Electrical Engineering

According to de-Broglie, the motion of an electron with a velocity v is equivalent to the motion of a wave packet with group velocity v_g.

The velocity of free electrons is proportional to k.

Now on the basis of band theory, the variation of E with k is shown in Fig. 5.15 (a). Since the curve is symmetrical having points of inflexion at M and N, the value of slope dE/dk is different for different portions of the curve. As the curve is horizontal at A, O and B, therefore, for these regions dE/dk = 0.

The variation of v with k is shown in Fig. 5.15 (b). The absolute value of velocity becomes maximum at k = k₀, where k₀ corresponds to the point of inflexion in the E-k diagram. Beyond this point, the velocity decreases with the increasing energy which is different in behaviour from that of free electrons.

Effective Mass:

Let us now consider the motion of an electron in a crystal in the presence of an external applied field. According to quantum theory, the velocity of the electron v is equal to the group velocity of the wave packet representing the electron.

Let ԑ = external applied electric field, then the gain in the energy of the electron in time dt is;

dE = -eԑ v dt … (xxxxii)

where e is the charge on the electron.

Using equations (xxxx), equation (xxxxii) can be written as-

Equation (xxxxiii) shows that time rate of change of crystal momentum is equal to the impressed force -eԑ. The acceleration of the electron is-

Substituting the value of dk/dt from equation (xxxxiii), we get-

For free electron, we have-

a = -eԑ/m … (xxxxvi)

Equation (xxxxv) and (xxxxvi) suggest that on the basis of band theory, the electron behaves as if it had an effective mass m* equal to h²/d²E/dk².

Thus effective mass of an electron is-

Equation (xxxxvii) shows that effective mass is determined by d²E/dk². Since for the lower positions of E-k curve, d²E/dk² is +ve, so m* is +ve and for upper position d²E/dk² is -ve, so m* is -ve. At the points of inflexion, d²E/dk² = 0, so at these points m* becomes infinite. Physically speaking an electron behaves as a particle with +ve charge in the upper half of the band. The variation of m* with k is shown in Fig. 5.15(c).

Sometimes, it is convenient to introduce a factor which determines the extent upto which the electron in a k-state is a free electron. If f_k is small, i.e., m* is large, the particle behaves as a heavy particle or we can say that the wave functions centred on neighbouring atoms overlap very little, then the overlap integral will be small and as a result the width of the band will be narrow.

Here the overlap integral determines the rate of quantum tunneling of an electron from one ion to another and in case of large effective mass, the electron tunnels very slowly from one ion to an adjacent ion in the lattice, e.g. 4f electrons of the rare-earth metals.