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Applet Worksheet

Fermi distribution function, its meaning on the electron occupancy of energy states, and the effect of temperature.

• Calculate f(EF + 0.06eV) and 1 - f(EF - 0.06eV) using the Fermi function f(E) = 1 / [ 1 + e(E-EF)/kT ] and fill the following table.  Use kT = 0.0259 * ( T / 300) eV.
•  Temperature f(EF + 0.06eV) 1 - f(EF - 0.06eV) 300K ___________ ___________ 500K ___________ ___________ 900K ___________ ___________
• (1)  Set the temperature in the applet at one of the above value and make EF coincide with one of the energy levels. Count the number of occupied states at the level 60meV above EF and divide it by 16 (total number of states at that level) to estimate f(EF + 0.06 eV). Compare with the calculated value.  Count the number of vacant states for the level at EF - 60 meV and divide it by 16 to estimate 1 - f(EF - 60 meV). Compare with the calculation.
(2) Is it true that f( EF + E0) = 1 - f( EF - E0) for a given energy value E0 ?  Say what this equation means in English, i.e., in terms of the probability of occupancy and the probability of vacancy of the states or the symmetry of the electron distribution f(E) with respect to EF.
(3) At a higher temperature the Fermi distribution around EF is more smooth.  Explain this in terms of how the increased thermal energy is affecting the electron energy distribution.
• How many kT's away from Ef are the states completely full or completely empty ??
• Does this applet help you understand better why they use 3kT or 4kT as an energy limit in making approximation to f(E) ??
• Please comment about the following aspects and send me an email:
• Compared to your (previous) understanding of the Fermi function based on the mathematical formula (eg, Eq. 1-10 of Yang) and a static illustration (eg, Fig.1-17, p.18 of Yang), did this applet improve the clarity of the physical meaning of the distribution function ?  How did this applet affect your understanding or perception ?
• Before using this applet (ie, from formula and some static figures alone), did you have an immediate 'feeling' that the energy states below Ef must be occupied by electron ? Did this applet help you augment this notion ?
• Please make any comments as to how to improve this applet for a more effective learning of the concepts/principles ?

Foot Notes

1 Completely full or completely empty:  Because in this applet there are only 16 energy states at each level, if the probability of occupancy is less than 1/16 or so, none of the states may appear to be occupied at that energy level (completely empty).  Likewise, if the probability of occupancy is greater than 1.0 - 1/16, then all the states may appear to be occupied (completely full).   This discussion is valid only within an accuracy of 6x10-2 for the probability.  It must NOT lead you to conclude that every states will be empty, no matter how many states there are at that energy level, if E - Ef > 4kT, for example.

A very good example of a very small f(E) value, but a large number of state-occupying electrons comes in the semiconductor energy band.  At 300K (kT = 0.0259 eV), let us consider the conduction band edge, Ec, which is at Ec - Ef = 0.5 eV. Note that Ec - Ef = 19.3kT >> 4kT. Does it mean that ALL states at E = Ec are vacant and no conduction electons ? No !  There are in fact 1.16x1011 electrons per cm3 at energy states at Ec because n = Nc * f(E) or 1.16E11 = 4.13E-9 * 2.8E19. That is, one hundred and sixteen billion electrons per cm3.   This is due to an arithmatic that a zero (or a very small number) times infinity (or a very big number) is not zero.  Here, f(E) = f(Ec - Ef) = f(0.5 eV) = 4.13 x 10-9, meaning that just about 4 states will be occupied for every one billion states !  But in the conduction band of Si, you have about 2.8x1019 states per cm3 (Nc = 2.8x1019 cm-3), a number so large that I can not even spell it out in English.  Out of 2.8x1019 states, a mere one hundred and sixteen billion states are occupied by electrons.