Effect of External Force

External force F acting on the wave packet would be any force other than the crystalline force associated with periodic potential.

External force may arise from dopants or external electric field.

Force F acting over a short distance dx work on the wave packet wave packet energy increases by

we know Rate of change of momentum =

= Mass x Acceleration

therefore, Effective mass

At k = 0, curvature of (b) band is more than the curvature of (a), i.e.

Fig.4.1

Heavy mass slower movement, larger transit time.

Mobility of a carrier    curvature of    seen in  Fig.4.2

Now consider the band segment of the Kronig-Penny model as shown in Fig. 4.3

  near the bottom or minimum of bands.

   near the top part of each band.

For 

In response to applied force the particle/electron will accelerate in a direction opposite to than expected from purely classical calculation.

In most cases we do not know the exact equation of the E-k diagram (difficult even for the Kronig-Penny model) but it been found that top or bottom of the band edge/Brillouin zone edge of the E-k relation is approximately parabolic as shown in Fig. 4.3         

where A is constant.

Therefore, constant at the edge of the Brillouin zone/top or bottom of an energy band.

Therefore, constant near the edge of the Brillouin zone/top or bottom of an energy band.

Current Flow

If N atoms are there in 1 dimensional crystal then the distinct k values in each band = N spaced apart by

For the sake of discussion we assume each atom gives 2 electrons to the crystal a total of 2N free electron in the crystal. 2N electrons will be distributed among the available energy states.

At temperature 2N electrons will fit the 2N states in the lowest 2 bands (valence bands).

Fig. 4.4

At room temperature sufficient thermal energy is there and a few electrons from the top of the 2 nd band will move up to the bottom of the 3 rd band as shown in Fig. 4.5.

If a voltage applied to the crystal a current will flow through the crystal.

4 th band - no electron at room temperature - no current as totally empty bands do not contribute to the charge transport process.