# HomeWork1Semiconductors # 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)

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  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.