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*/︎m0 b) Compute the hole mass ratio m*/︎m0 using Fig. 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 (#/cm3) over the energy range of (i) 0<︎ E < 2.0 eV and
(ii) 1<︎ E<︎ 2 eV. c) Determine the number (#/cm3) of quantum states in Si
between Ec and Ec ︎+ kT at T ︎ 300 K.
3) Fermi distribution: Assume the Fermi energy level is 0.30 eV below the
conduction band energy Ec. Assume T = 300 K. (a) Determine the probability of a
state being occupied by an electron at E =︎ Ec ︎+kT/︎4. (b) Repeat part (a) for an
energy state at E ︎= Ec ︎+ kT.
4) Band gap I: The bandgap energy in a semiconductor is usually a slight function
of temperature. In some cases, the bandgap energy versus temperature can be
modeled by
αT 2
Eg = Eg(0) −
β +T
where Eg(0) is the value of the bandgap energy at T = 0 K. For silicon, the
parameter values are Eg(0) = ︎1.170 eV, α = ︎ ︎4.73 x︎ 10︎-4 eV/K, and β = ︎636 K.
Plot Eg versus T over the range 0 <︎ T <︎ 600 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. (i) Determine
the minimum frequency of an incident photon that can interact with a valence
electron and elevate the electron to the conduction band. (ii) 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) dE︎/
dk versus k and (b) d2E/︎dk2 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 energy-band 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 one-dimensional direction near the minimum value can be
approximated by
E︎=E0 ︎+E1 cos︎(k︎-k0)
where k0 is the value of k at the minimum energy. Determine the effective mass of
the particle at k ︎ k0 in terms of the equation parameters.
5) Kronig-Penney Model: In Ashcroft-Mermin follow through the solution to the
general Kronig-Penney model. Derive all the intermediate steps of the calculation,
and draw figure 8.6 with your own plotting tools.
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.