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*Applet Worksheet*

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

Answer the following simple questions.

- Calculate f(E
_{F}+ 0.06eV) and 1 - f(E_{F}- 0.06eV) using the Fermi function f(E) = 1 / [ 1 + e(E-E_{F})/kT ] and fill the following table. Use kT = 0.0259 * ( T / 300) eV.

- (1) Set the temperature in the applet at one of the above value
and make E
_{F}coincide with one of the energy levels. Count the number of occupied states at the level 60meV above E_{F}and divide it by 16 (total number of states at that level) to estimate f(E_{F}+ 0.06 eV). Compare with the calculated value. Count the number of vacant states for the level at E_{F}- 60 meV and divide it by 16 to estimate 1 - f(E_{F}- 60 meV). Compare with the calculation.

(2) Is it true that f( E_{F}+ E_{0}) = 1 - f( E_{F}- E_{0}) for a given energy value E_{0}? 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 E_{F}.

(3) At a higher temperature the Fermi distribution around E_{F}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 ?

Temperature |
f(E _{F} + 0.06eV) |
1 - f(E _{F} - 0.06eV) |

300K |
___________ |
___________ |

500K |
___________ |
___________ |

900K |
___________ |
___________ |

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 1.16x10^{11}
electrons per cm^{3}
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 cm^{3}. 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 2.8x10^{19} states per cm^{3 }(Nc
= 2.8x10^{19} cm^{-3}), a number so large that I can not
even spell it out in English. Out of 2.8x10^{19} states,
a mere one hundred and sixteen billion states are occupied by electrons.