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
i) Calculate the eigenvalues of a 100 hp, 1150 rpm dc machine at rated flux with no extra load inertia. Use the "hot" value of the armature resistance. Calculate estimates of t

Low Investments in Transmission and Distribution System The investment on T&D system in our country has commonly been lower than that on generation. It is hugely believed in

what is the limitations of maximum power transfer thorem ?

Q. A 230-V, single-phase, 60-Hz source supplies two loads in parallel. One draws 10 kVA at a lagging power factor of 0.80 and the other draws 6 kWat a lagging power factor of 0.90.

Q. With a direct current of I A, the power expended as heat in a resistor of Ris constant, independent of time, and equal to I 2 R. Consider Problem and find in each case the effe

I need I circuit that calculates f(n) = 2^2^n, where n is a 4bit number. I tried with a decoder but unsuccessfully.

Q. Write a short note on the frequency response of a R-C coupled amplifier. At low frequencies ( , the reactance of coupling capacitor Cc is quite high and hence very small pa

Q. A magnetic circuit is found to have an ac hysteresis loss of 10 W when the peak current is Im = 2 A. Assuming the exponent of Bm to be 1.5, estimate Ph for Im = 0.5A and 8A.

ELECTRICAL STARTER MOTOR This usually consists of a heavy duty, compound wound, DC motor, which draws its electrical supply from an external source. The motor works in conjunc

Discuss about signal distortion in optical waveguides Ans: The major reason for signal distortion within optical waveguide is dispersion. Dispersion of the transmitted