Homework 4

ECE 6163 Fall 2009
Homework Assignment #4
Due 9/23/09
1. A carbon nanotube is created by taking a graphene sheet and rolling into a cigar-like
shape (chemically of course!). Assuming it is oriented along the x-axis, the 1D E-k
dispersion of the nanotube can be written as
E(kx) = 0 ± √[E2 + (takx)2].
where 0, E t, a are system constants, and the ± means there are two bands, one with +
sign in the formula and one with the – sign.
(a) What are the effective masses for E = 0?
(b) What are the effective masses for E ≠ 0?
(c) Find the 1D density of states (per unit length) of the nanotube, using the steps worked
out in class. Leave it in analytical form for an arbitrary E.
(d) The expression for E = (2at/d)(-2m/3), where d is the diameter of the tube, a =
0.231nanometers, t=2.5 eV, and ,m are integers. The density of states of this tube is
obtained by taking the result from (c) and summing over integer values of . For a
nanotube with diameter d=15.4 nanometers, m=200. Plot the density of states (units of
/eV/nm) vs energy (units of eV).
2. (a) Establish a general expression (involving integrals) for the average kinetic energy,
<KE>, of conduction band electrons in a semiconductor.
(b) Taking the semiconductor to be non-degenerate, simplify the integrals in part (a) and
obtain a closed form expression for <KE>. Hint:  3  x
 x 2e dx 
3. In Si, where mhh*/m0=0.537 and mlh*/m0=0.153, what fraction of the holes are heavy
holes? (for simplicity, assume the effective masses are not temperature dependent).
4. Because mn* and thus NC are relatively small in GaAs, the donor doping at which the
material becomes degenerate is also relatively small. Determine the ND doping at the
non-degenerate/degenerate transition point in GaAs at 300K.