A certain metal has conduction electrons per cubic meter. A sample of that metal has a volume of and a temperature of 200K. How many occupied states are in the energy range of that is centered on the energy ? (Caution: Avoid round-off in the exponential.)
There are occupied states in the given energy range.
At first, using the number of conduction electrons per unit volume, we can calculate the Fermi energy of the material. Then, we calculate the occupancy probability. Then, using Fermi-Dirac statistics we can calculate the value of the density of occupied states. Now, the value of the density of occupied states is calculated using the density of states at the energy level and the occupancy probability value is used in the formula of the number of occupied density states to get the required value.
The equation of Fermi energy (i)
The occupancy probability is , (ii)
The density of states associated with the conduction electrons of a material,
The density of occupied states, (iv)
The number of occupied states in the given energy range ,
Using the value of conduction electrons per unit volume in equation (i), we can calculate the Fermi energy of the material as follows:
Thus, using the above value in equation (ii), we can get the occupancy probability of the material as follows:
Now, the density of the states associated with the conduction electrons for the given energy level can be calculated using equation (iii) as follows:
The density of the occupied states is given using equation (iv) as follows:
With the given energy range of and the given volume of material, we can get the number of occupied states using equation (v) as follows:
Hence, the value of the occupied states is .
Doping changes the Fermi energy of a semiconductor. Consider silicon, with a gap of 1.11eV between the top of the valence band and the bottom of the conduction band. At 300K the Fermi level of the pure material is nearly at the mid-point of the gap. Suppose that silicon is doped with donor atoms, each of which has a state 0.15eV below the bottom of the silicon conduction band, and suppose further that doping raises the Fermi level to 0.11eV below the bottom of that band (Fig. 41-22). For (a) pure and (b) doped silicon, calculate the probability that a state at the bottom of the silicon conduction band is occupied. (c) Calculate the probability that a state in the doped material (at the donor level) is occupied.
A certain computer chip that is about the size of a postage stamp contains about 3.5 million transistors. If the transistors are square, what must be their maximum dimension? (Note: Devices other than transistors are also on the chip, and there must be room for the interconnections among the circuit elements. Transistors smaller than are now commonly and inexpensively fabricated.)
In the biased p-n junctions shown in Fig. 41-15, there is an electric field in each of the two depletion zones, associated with the potential difference that exists across that zone. (a) Is the electric field vector directed from left to right in the figure or from right to left? (b) Is the magnitude of the field greater for forward bias or for back bias?
(a) Using the result of Problem 23 and 7.00eV for copper’s Fermi energy, determine how much energy would be released by the conduction electrons in a copper coin with mass3.10g if we could suddenly turn off the Pauli exclusion principle. (b) For how long would this amount of energy light a 100 W lamp? (Note: There is no way to turn off the Pauli principle!)
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