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) 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

=

4.73 x 10

-4 eV/K, and = 636 K. Plot

E

g versus

T

over the range

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. ( 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 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.