Fermi Level, Fermi Function and
Electron Occupancy of Localized Energy States

• To move the Fermi level : mouse drag the magenta-colored ^_____ Ef
• To change temeprature: use the scollbar near the bottom.

All magenta-colored horizontal bars (i.e., Ef and scroller) may be mouse-dragged.

1. Set the temperature to the lowest possible value.  (1) Sketch the shape of f(E).  (2) Move the Fermi level by mouse-drag so that it coincides with one of the energy states.  Verify that the probability of occupancy is exactly 1/2. (there are 16 states per energy. how many are occupied ?)
2. The energy states, 16 of them per energy, are separated by 60 meV.   Increase the temperature, observe that the electron population is more smeared out in the states around Ef.   
• Slect a Temperature, and find the electron occupancy (for states above Ef) and the electron vacancy (for states below Ef) from the Figure on the right hand side.  Are these probabilities consistent with the Fermi function f(E) ?  (That is, count the number of states occupied, divide it by the number of states at that energy.  Calculate the numerical value of Fermi function for that energy. Compare.)   
• How many kT's away from Ef are the states completely full or completely empty¹ ??  
• Does this applet help you understand better why they use 3kT or 4kT as an energy limit in making approximation to f(E) ??
3. After answering the above two questions, make a comment about the following aspects (This part will be also graded as per your Technical Communication Skills, which should convey your technical opinions/ideas clearly in both quantitative and qualitative aspects.)
• Compared to your (previous) understanding of the Fermi function based on the mathematical formula (eg, Eq. 1-10 of Yang) and the 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 and perception. Make specific comments.
• Before using this applet (ie, from formula and static figure alone), did you have an immediate 'feeling' that the energy states below Ef must be occupied by electron ? Did this applet reinforce this notion ?  Provide some specific comments.
• Make comments on how helpful this applet is in learning the concepts of Fermi level and Fermi funtion.  Make any other comments here.
• Make comments on how to improve this applet for a more effective learning of the concepts/principles ?   Are you willing to participate in developing these applets ?
  
  

Foot Notes

¹ Completely full or completely empty:  Because there are 16 energy states at each level, if the probability of occupancy is less than 1/16 or so, none of the states may be occupied at that level (completely empty).  Likewise, if the probability of occupancy is greater than 1.0 - 1/16, then all the states may be occupied (completely full).   This discussion is only valid within an accuracy of 6x10^-2 of the probability.  It must not lead you to conclude that EVERY states will be empty, no matter how many states there are, whenever the E - Ef > 4kT.    A very clear example of a very small f(E) value, but a large number of state-occupying electrons comes in the semiconductor band.  For example, at 300K (kT = 0.0259 eV), let us consider Ec - Ef = 0.5 eV. You note that Ec - Ef = 19.3kT >> 4kT. Does it mean that ALL states at or above E = Ec (the conduction band edge) are empty and thus there are no conduction electons ? No !  There are in fact 1.16x10¹¹ electrons per cm³ at energy states at or above Ec !   That is, one hundred and sixteen billion electrons per cm³ !   This is due to the simple arithmatic that (a very small number) times (a very big number) is not necessarily very small !  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 ( n = Nc * f(E) or 1.16E11 = 4.13E-9 * 2.8E19) !

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