In Eq. 41-6 let, . (a) At what temperature does the result of using this equation differ by 1% from the result of using the classical Boltzmann equation (which is Eq. 41-1 with two changes in notation)? (b) At what temperature do the results from these two equations differ by 10%?
The given equation,
The probability that an electron will occupy a certain level, according to Fermi-Dirac statistics and Boltzmann's statistics, is known as occupancy probability. Fo the Fermi level the occupancy probability is 0.5.
The occupancy probability due to Fermi-Dirac statistics,
The Boltzmann occupation probability,
Let the fractional difference between the probability differences be f.
Thus, using equations (1) and (2), the fractional difference can be given as follows:
The above equation can also be written as-
For f = 0.01,
Taking logarithm on both sides
Hence, the value of the temperature is 2520 K.
If f = 0.1
Then, the temperature is given as-
Hence, the value of the temperature is .
At 1000K, the fraction of the conduction electrons in a metal that have energies greater than the Fermi energy is equal to the area under the curve of Fig. 41-8b beyond divided by the area under the entire curve. It is difficult to find these areas by direct integration. However, an approximation to this fraction at any temperature T is .
Note that frac = 0 for T = 0 K, just as we would expect. What is this fraction for copper at (a) 300 K and (b) 1000 K? For copper . (c) Check your answers by numerical integration using Eq. 41-7.
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.)
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