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


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 (#/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



T 2


+ T where E g

(0) is the value of the bandgap energy at T = 0 K. For silicon, the parameter values are

E g


= 1.170 eV,



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


+ E

1 cos ( k k


) 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.


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.