Figure 41-21 shows three leveled levels in a band and also the Fermi level for the material. The temperature is 0K. Rank the three levels according to the probability of occupation, greatest first if the temperature is (a) 0K and (b) 1000K. (c) At the latter temperature, rank the levels according to the density of states N(E) there, greatest first.
Using the occupancy probability and the energy at the three levels, we can get the required probabilities of occupying the band at the two different temperatures. Again, the density of states equation indicates that the density value is directly proportional to the square root of energy. Hence, it determines the value of density of all three leveled energies.
The occupancy probability of the energy level (E) , according to Fermi-Dirac Statistics, having Fermi energy and absolute temperature (T) , is given as-
The density of states associated with the conduction electrons of a metal, is given as-
and is the mass of the electron.
At absolute zero temperature ( T = 0 K ), the probability using equation (i) is 1 for all energies less than the Fermi energy and zero for energies greater than the Fermi energy.
Thus, energy at level 3 is 1, while at the other two levels is zero as their energies are greater than Fermi energy.
Hence, according to the diagram, according to occupancy probability the levels are ranked as
level 3 > level 2 = level 1.
At temperature, T = 1000 K a few electrons whose energies were less than that of the Fermi energy at T = 0 K move up to states with energies slightly greater than the Fermi energy. That means probability at Fermi energy is .
Thus, the occupancy probability of the levels below the Fermi energy level is higher as compared to the ones above the Fermi level. Also, for the levels present above the fermi level, the higher the level, the less is the occupancy probability.
Hence, at 1000 K , the three levels are ranked as-
level 3>level 2>level 1.
From equation (ii), we get that the density of states is proportional to square root of energy
According the diagram, the ranking of the energies of the three levels is .
For an ideal p-n junction rectifier with a sharp boundary between its two semiconducting sides, the current I is related to the potential difference V across the rectifier by , where , which depends on the materials but not on I or V, is called the reverse saturation current. The potential difference V is positive if the rectifier is forward-biased and negative if it is back-biased. (a) Verify that this expression predicts the behavior of a junction rectifier by graphing I versus V from to . Take and . (b) For the same temperature, calculate the ratio of the current for a 0.50 V forward bias to the current for a 0.50 V back bias.
Figure 41-1a shows 14 atoms that represent the unit cell of copper. However, because each of these atoms is shared with one or more adjoining unit cells, only a fraction of each atom belongs to the unit cell shown. What is the number of atoms per unit cell for copper? (To answer, count up the fractional atoms belonging to a single unit cell.)
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