hw10sol - Homework 10 Solution Prob. 10.1 We will look at...

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Homework 10 Solution Prob. 10.1 We will look at the density of states in a MOSFET inversion channel. This can be viewed as electrons being confined to a 3D box, but the gate direction (assigned as z here) has much more confinement than the channel length and width direction. Effectively, we can look at the system as a quantum box of dimension ( a, b, c ), where c << a, b . (a) Write down the eigenfunctions and eigenenergy for this system. Is there any difference from Eqs. (10.8) and (10.9)? (5 pts) ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 2 2 2 , , 2 2 2 sin sin sin 8 , , mc n mb n ma n E c z n b y n a x n abc z y x z y x z y x nz ny nx z y x z y x π φ + + = = = This is different from the plane waves in Eqs. (10.6) and (10.7), though similar in the way of composition of 3D wavefunctions. The eigenenergies are dominated by the last term since c << a, b . The wave function is similar to Eq. (10.9) with a constant z . There cannot be much variation in z anyway, or there will be many cycles of the sine wave, which means very high energy. (b) Construct the 3D plot of ( k x , k y , k z ) and the allowable points. What is the influence from the relation of c << a, b ? (5 pts) In the k x and k y plane, it is a dense uniform grid. The spacing between the 2D dense grid in the k z direction is much larger. (c) The density of state will seem to have discontinuities at energies 2 2 2 = c n m E z nz , establish expressions for the DOS in the energy ranges 0 ≤ E ≤ E 1 , E 1
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This note was uploaded on 11/26/2010 for the course ECE 3060 at Cornell University (Engineering School).

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hw10sol - Homework 10 Solution Prob. 10.1 We will look at...

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