EE 203. HW1 Due Monday, April 21.
() ( )
∗
=mk 2
2
hk
1. Calculate the product of the 1D density of states times the 1D velocity of an electron in the
conduction band.
2. Assuming a parabolic dispersion relation, , derive the expression for the
single-spin density of states g(E) for (a) a 1-dimensional crystal, (b) a 2-dimensional crystal,
and (c) a three dimensional crystal by explicitly evaluating.
()
[]
∑−=
k
k
1
)( E
L
Eg D
where D is the dimensionality.
3. Show that for an arbitrary, smooth dispersion relation of the form ε(k) = εx(kx) + εy(ky) +
εz(kz), (a) )()()( 112 ENENEN DDD ⊗= and that (b) )()()()( 1113 ENENENEN DDDD ⊗⊗=
where N1D, N2D, and N3D are the single spin 1D, 2D, and 3D density of states, respectively,
and ⊗ means convolution.
4. Assuming a dispersion relation
(a) Calculate the velocity of the electron at k = π/a.
(b) If the electric field is applied in the -x direction, derive the time dependence of k for an
electron initially at state k = -π/a and position x = 0; and
(c) Derive the time dependence of the electron velocity, v(t); and
(d) Derive the time dependence of the electron position, x(t).
(e) For a = 5 nm, E = 104V/cm, and m = 0.2 m0, what are the maximum and minimum values of
x that the electron will reach?
(f) What is the period of the oscillation?
(g) For the parameters of part (e), derive an expression for the effective mass as a function of k.
Sketch the function.
()
[]
ka
ma
EE Ccos1
2
2−+= h