Find dispersion relation for free electron, Electrical Engineering

Find Dispersion Relation for Free Electron

Question: Find the dispersion relation for a free electron, and, thus, observe the relation between its rest mass and effective mass.

Solution: For a free electron, the electron momentum is  p =m0v = hk. Thus, E=m0V2/2=p2/ (2m0) = h2k2/(2m0). Therefore, the dispersion relation, i.e., E-k relation is parabolic. Hence, mx = h2/∂2Ε/∂K2=m0. This is a very interesting relation that says states that for a free electron, the rest mass and effective mass are one and similar, which is because of the parabolic band structure. Several materials have non-parabolic E-k relation, and, so they have quite different rest mass and effective mass for electrons.

Note: the effective mass may become a function of energy, for severely non-parabolic band structures, however, near the minima of the conduction band and towards the maxima of the valence band, the band structure can be taken to be parabolic, and, thus, an effective mass, which is independent of energy, may be obtained.

  • Thus, the effective mass is an inverse function of the curvature of the E-k diagram: weak curvature gives large mass, and strong curvature gives small mass.
  • Note that in general, the effective mass is a tensor quantity, however, for parabolic bands, it is a constant.
  • Another interesting feature is that the curvature d2E/dk2 is positive at the conduction band minima, however, it is negative at the valence band maxima.
  • Thus, the electrons near the top of the valence band have negative effective mass.
  • Valence band electrons with negative charge and negative mass move in an electric field in the same direction as holes with positive charge and positive mass.
  • Thus, the charge transport in the valence band can be fully accounted for by considering hole motion alone.
  • The electron and hole effective masses are denoted by m*n and m*p  respectively.


Table: the effective mass for electrons and holes (Χ m0, where m0 is the rest mass for electrons)311_Find Dispersion Relation for Free Electron.png

Posted Date: 1/11/2013 4:39:47 AM | Location : United States

Related Discussions:- Find dispersion relation for free electron, Assignment Help, Ask Question on Find dispersion relation for free electron, Get Answer, Expert's Help, Find dispersion relation for free electron Discussions

Write discussion on Find dispersion relation for free electron
Your posts are moderated
Related Questions
Q. Explain Working of Liquid-crystal display? This display needs only microwatts of power (over a thousand times less than a LED) and is used in such devices as electronic wris

Q. Two identical junction diodes whose volt-ampere relation is given by Equation in which IS = 0.1 µA, VT = 25 mV, and η = 2, are connected as shown in Figure. Determine the curren

Q. What do you mean by Quartz crystals? Synthetic quartz is composed of Silicon and Oxygen (Silicon Dioxide) and is cultured in autoclaves under high pressure and temperature.

We can use Ampere's Law to calculate the magnetic field in the core.

A 3-phase, wye-connected, 11 kV, 137.5 MVA turbo-alternator of 0.8 power factor lagging has a synchronous reactance XS = 1.4 Ω. Assume the machine to be ideal for convenience of an

Q. What is the basic difference between the weighted resistor and the R-2 R ladder D/A converters?

distingish between maximum and minimum modes of operations of 8086 with timing diagrames

Q. Give an account on the operation of the R-C coupled amplifie r. When a.c. signal is applied to the base of the first transistor, it appears in the amplified form across its c

1. The circuit shown below is a DC charging and discharging circuit. a. At t = 0 sec, switch S1 is thrown to position 1 ("pos1"). Write the mathematical expressions for V

#What are the types of Negative Feedback Amplifier?