Funkar Fermi Dirac

Dawwamai funksiya suna da wani abubuwa da ake amfani da ita don bayyana tsarin yawan zama da zan iya samun shi a kan yanayin kasa. Idan an maimaito da Fermi-Dirac distribution function, muna son in a nemi da kyau cewa muna tabbatar da adadin shi da za a iya samun fermion a kan yanayin kasa na atom (zaka so kuɗi a cikin rubutun “Atomic Energy States”). A nan, da fermions, muna nufin electrons na atom wadanda suka haɗa da Pauli exclusion principle.

Yakin Fermi Dirac Distribution Function

A cikin fanni masu elektronika, wani batu mai muhimmanci shine conductivity na tashar. Wannan siffofin tashar ya gudana daga adadin electrons da suka fi siffar a kan tashar don tattara.

Idan a yi tasiri game da energy band theory (refer to the article “Energy Bands in Crystals” for more information), waɗannan su ne adadin electrons da suka samun conduction band na tashar. Saboda haka, don in iya samun yadda wannan tattara ta faru, yana buƙata a bayyana adadin carriers a kan conduction band.

Fermi Dirac Distribution Expression

Daga baya, adadin da ake iya samun electron a kan yanayin kasa E a tafukar T ya nuna a matsayin

Na biyu, adadin da ake iya samun electron a kan yanayin kasa E a tafukar T ya nuna a matsayin

k suna Boltzmann constant
T suna tafukar absolute
Ef suna Fermi level ko Fermi energy

Iman, in ba ni ma'anar Fermi level. Don in iya samun wannan, zaka so kuɗi a cikin rubutun “
a cikin equation (1). Idan a yi haka, muna samun

Wannan na nufin cewa Fermi level shine level da ake iya samun electron a kan shi exactly 50% of the time.

Fermi Level in Semiconductors

Intrinsic semiconductors su ne tashar masu sahihi wadanda babu impurities a kan su. Saboda haka, suke canzawa da adadin hole da ke sama da adadin electron. Wannan na nufin cewa suke da Fermi-level exactly in between the conduction and the valence bands as shown by Figure 1a.

Iman, in ba ni ma'anar Fermi level. Don in iya samun wannan, zaka so kuɗi a cikin rubutun “n-type semiconductor”. A nan, muna tabbatar da adadin electrons da za a iya samun a kan tashar da ke da yawa da holes. Wannan na nufin cewa Fermi-level ya gudana a kan conduction band as shown by Figure 1b.
Following on the same grounds, one can expect the Fermi-level in the case of p-type semiconductors to be present near the valence band (Figure 1c). This is because, these materials lack electrons i.e. they have more number of holes which makes the probability of finding a hole in the valence band more in comparison to that of finding an electron in the conduction band.

Effect of temperature on Fermi-Dirac Distribution Function

At T = 0 K, the electrons will have low energy and thus occupy lower energy states. The highest energy state among these occupied states is referred to as Fermi-level. This inturn means that no energy states which lie above the Fermi-level are occupied by electrons. Thus we have a step function defining the Fermi-Dirac distribution function as shown by the black curve in Figure 2.
However as the temperature increases, the electrons gain more and more energy due to which they can even rise to the conduction band. Thus at higher temperatures, one cannot clearly distinguish between the occupied and the unoccupied states as indicated by the blue and the red curves shown in Figure 2.

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