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Disusun oleh: Agustian Noor
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| bet | 4/57 | | Sana | 09.05.2021 | | Hajmi | 0.62 Mb. | | #14427 |
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Fig. 2.4.2 Six of the 24 possible configurations in which 20 electrons can be placed having an energy of 106 eV.
A complete list of the 24 configurations is shown in the table below:
fddist.xls - occtable.gif
Table 2.4.1 All 24 possible configurations in which 20 electrons can be placed having an energy of 106 eV.
The average occupancy of each energy level as taken over all (and equally probable) 24 configurations is compared in the figure below to the expected Fermi- Dirac distribution function. A best fit was obtained using a Fermi energy of 9.998 eV and kT = 1.447 eV or T = 16,800 K. The agreement is surprisingly good considering the small size of this system.
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Fig. 2.4.3 Probability versus energy averaged over the 24 possible configurations of the example (red squares) fitted with a Fermi-Dirac function (green curve) using kT = 1.447 eV and EF= 9.998 eV.
The Fermi-Dirac distribution function
The Fermi-Dirac probability density function provides the probability that an energy level is occupied by a Fermion which is in thermal equilibrium with a large reservoir. Fermions are by definition particles with half-integer spin (1/2, 3/2, 5/2 ...). A unique characteristic of Fermions is that they obey the Pauli exclusion principle which states that only one Fermion can occupy a state which is defined by its set of quantum numbers n,k,l and s. The definition of Fermions could therefore also be particles which obey the Pauli exclusion principle. All such particles also happen to have a half-integer spin.
Electrons as well as holes have a spin 1/2 and obey the Pauli exclusion principle. As these particles are added to an energy band, they will fill the available states in an energy band just like water fills a bucket. The states with the lowest energy are filled first, followed by the next higher ones. At absolute zero temperature (T = 0 K), the energy levels are all filled up to a maximum energy which we call the Fermi level. No states above the Fermi level are filled. At higher temperature one finds that the transition between completely filled states and completely empty states is gradual rather than abrupt. The Fermi function which describes this behavior, is given by:
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