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Question
A semiconducting material has a band gap of 1 eV. Acceptor impurities are doped into it which create acceptor levels 1 meV above the valence band. Assume that the transition from one energy level to the other is almost forbidden if kT is less than 1/50 of the energy gap. Also if kT is more than twice the gap, the upper levels have maximum population. The temperature of the semiconductor is increased from 0 K. The concentration of the holes increases with temperature and after a certain temperature it becomes approximately constant. As the temperature is further increased, the hole concentration again starts increasing at a certain temperature. Find the order of the temperature range in which the hole concentration remains approximately constant.
(Use Planck constant h = 4.14 × 10-15 eV-s, Boltzmann constant k = 8·62 × 10-5 eV/K.)
Solution
Given:
Band gap = 1 eV
After doping,
Position of acceptor levels = 1 meV above the valence band
Net band gap after doping = (1 − 10−3) eV = 0.999 eV
According to the question,
Any transition from one energy level to the other is almost forbidden if kT is less than 1/50 of the energy gap.
\[\Rightarrow k T_1 = \frac{0 . 999}{50}\]
\[ \Rightarrow T_1 = \frac{0 . 999}{50 \times 8 . 62 \times {10}^{- 5}}\]
\[ \Rightarrow T_1 = 231 . 78 \approx 232 . 8 \] K
\[T_1\] is the temperature below which no transition is possible.
If kT is more than twice the gap, the upper levels have maximum population; that is, no more transitions are possible.
For the maximum limit,
\[\text{ K T}_2 = 2 \times {10}^{- 3} \]
\[ \Rightarrow T_2 = \frac{2 \times {10}^{- 3}}{8 . 62 \times {10}^{- 5}}\]
\[ \Rightarrow T_2 = \frac{2}{8 . 62} \times {10}^2 = 23 . 2 \] K
\[T_2\] is the temperature above which no transition is possible.
∴ Temperature range = 23.2−231.8
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