## Description

Ex 2.1

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

�” = �”(0) − ��*

(� + �)

where ��(0) is the value of the bandgap energy at � = 0 �. For silicon, the

parameter values are ��(0) = 1.170 ��, � = 4.73 × 109: ��/�, and � =

636 �. Plot �� versus � over the range 0 ≤ � ≤ 600 �. In particular, note the

value at � = 300 �.

Ex 2.2

Fig.1 shows the parabolic � versus � relationship in the valence band for a hole

in two particular semiconductor materials. Determine the effective mass (in units

of the free electron mass) of the two holes.

Figure 1. Figure for Ex 2.2.

Ex 2.3

Show that, Eq. (a) can be derived from Eq. (b).

�(�) = 4�(2�)B/*

ℎB √� (�)

�F(�)�� = ��*��

�B ⋅ �B (�)

2

Ex 2.4

The probability that a state at �J + �� is occupied by an electron is equal to the

probability that a state at �K − �� is empty. Determine the position of the Fermi

energy level as a function of �J and �K.

Ex 2.5

Calculate the energy range (in ��) between �M = 0.95 and �M = 0.05 for �M =

5.0 �� at (a) � = 200 � and (b) � = 400 �.

Ex 2.6

Calculate �MP with respect to the center of the bandgap in silicon for � = 200,

400 and 600 �.

Ex 2.7

If the density of states function in the conduction band of a particular

semiconductor is a constant equal to �, derive the expression for the thermalequilibrium concentration of electrons in the conduction band, assuming Fermi–

Dirac statistics and assuming the Boltzmann approximation is valid.

Ex 2.8

The value of �R in silicon at � = 300 � is 2 × 10ST ��9B. (a) Determine �M −

�K. (b) Calculate the value of �J − �M. (c) What is the value of �R? (d) Determine

�MP − �M.

Ex 2.9

Silicon at � = 300 � is doped with boron atoms such that the concentration of

holes is �R = 5 × 10SW ��9B . (a) Find �M − �K . (b) Determine �J − �M . (c)

Determine �R. (d) Which carrier is the majority carrier? (e) Determine �MP − �M.

Ex 2.10

(a) Determine the values of �R and �R in silicon at � = 300 � if �M − �K =

0.25 ��. (b) Assuming the value of �R in part (a) remains constant, determine the

values of �M − �K and �R at � = 400 �.