Semiconductors
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) 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 .
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
E g
= E g
(0)
−
α
T 2
β
+ T where E g
(0) is the value of the bandgap energy at T = 0 K. For silicon, the parameter values are
E g
(0)
= 1.170 eV,
α
=
4.73 x 10 -4 eV/K, and = 636 K.
Plot E g 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 ) 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 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
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) 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.
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.
