School of Physics and Nanotechnology Yachay Tech
Professor
: Ernesto Medina Homework 1 1)
Effective mass
: a) A simplified
E
versus
k
curve for an electron in the conduction band is given in Fig. I. The value of
a
is 10 Å. Determine the relative effective mass
m
*/
m
0 b) Compute the hole mass ratio
m
*/
m
0 using Fig. II I II 2)
Density of states in d dimensions
: a) Derive the density of states as a function of the energy for materials with a free dispersion in an arbitrary dimension d. Include the full calculation of the prefactors. b) For a free electron in 3d, calculate the density of quantum states (#/cm 3 ) over the energy range of (i) 0<
E <
2.0 eV and (ii) 1<
E<
2 eV. c) Determine the number (#/cm 3 ) of quantum states in Si between
E c
and
E c
+
kT
at
T
300 K. 3) 4)
Fermi distribution
: Assume the Fermi energy level is 0.30 eV below the conduction band energy
E c
. Assume
T =
300 K. (a) Determine the probability of a state being occupied by an electron at
E = E c
+
kT/
4. (b) Repeat part (a) for an energy state at
E
=
E c
+
kT
.
Band gap I
modeled by : The bandgap energy in a semiconductor is usually a slight function of temperature. In some cases, the bandgap energy versus temperature can be
E g
=
E g
(0) −
β α T
2 +
T
parameter values are where
E g
(0) is the value of the bandgap energy at
T =
0 K. For silicon, the
E g
(0) =
eV,
α
=
4 eV/K, and = 636 K. Plot
E
g versus
T
over the range
T <
K. In particular, note the value at
T
300 K.
5)
Band gap II
: (
a
) The forbidden bandgap energy in GaAs is 1.42 eV. ( electron and elevate the electron to the conduction band. (
ii i
) Determine the minimum frequency of an incident photon that can interact with a valence ) What is the corresponding wavelength? (
b
) Repeat part (
a
) for silicon with a bandgap energy of 1.12 eV. 6)
Effective masses
: The
E
versus
k
diagrams for a free electron (curve A) and for an electron in a semiconductor (curve B) are shown in the Figure. Sketch (
a
)
dE
/
dk
versus
k
and (
b
)
d
2
E/ dk
2 versus
k
for each curve. (
c
) What conclusion can you make concerning a comparison in effective masses for the two cases? 4)
Effective mass II
: The energyband diagram for silicon is shown in Figure 3.25b. The minimum energy in the conduction band is in the [100] direction. The energy in this onedimensional direction near the minimum value can be approximated by
E
=
E
0 +
E
1 cos (
k

k
0 ) where
k
0 is the value of
k
at the minimum energy. Determine the effective mass of the particle at
k k
0 in terms of the equation parameters. 5)
KronigPenney Model
: In AshcroftMermin follow through the solution to the general KronigPenney model. Derive all the intermediate steps of the calculation, and draw figure 8.6 with your own plotting tools. NOTE: The exercises requested in class will give you additional points. Please add them to your HW. Please send me a note if you find typos in the homework, so I can correct the exercises for all to benefit.