Semiconductor in Equilibrium

Semiconductor in Equilibrium

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Carriers in Semiconductors Dopant Atoms and Energy Levels Extrinsic Semiconduct Statistics of Donors and Acceptors Charge Neutrality Position of Fermi Level

Equilibrium Distribution of Electrons and Holes :
* The distribution of electrons in the conduction band is given by the     density of allowed quantum states times the probability that a     state will be occupied. The thermal equilibrium conc. of electrons no is given by   * Similarly, the distribution of holes in the valence band is given by the    density of allowed quantum states times the probability that a state will    not be occupied by an electron. * And the thermal equilibrium conc. Of holes po is given by

Equilibrium Distribution of Electrons and Holes
Density of states functions Fermi-Derac probability function   and areas representing electrons and holes concentrations for the case when Ef is near the midgab energy

The no and po eqs.
* Recall the thermal equilibrium conc. of electrons *Assume that the Fermi energy is within the bandgap. For electrons in   the conduction band, if Ec-EF >>kT, then E-EF>>kT, so the Fermi   probability function reduces to the Boltzmann approximation, * We may define                            ,(at T=300K, Nc ~1019 cm-3), which is called the effective density of states  function in the conduction band

Geometry
* The thermal equilibrium conc. of holes in the valence band is given by * For energy states in the valence band, E<Ev. If (EF-Ev)>>kT, * Then   * we may define                                  ,(at T=300K, Nc ~1019 cm-3), which is called the effective density of states  function in the valence  band

nopo product
* The product of the general expressions for no and po are given by for a semiconductor in thermal equilibrium, the product of no and po is always a constant for a given material and at a given temp. * Effective Density of States Function

Intrinsic Carrier Concentration
* For an intrinsic semiconductor, the conc. of electrons in the conduction    band, ni , is equal to the conc. of holes in the valence band, pi. * The Fermi energy level for the intrinsic semiconductor is called the     intrinsic Fermi energy, EFi. * For an intrinsic semiconductor ,   * For an given semiconductor at a constant temperature, the value of ni is    constant, and independent of the Fermi energy.

Intrinsic Carrier Conc.
Commonly accepted values of ni at T = 300 K Silicon ni = 1.5x1010 cm-3 GaAs ni   = 1.8x106 cm-3 Germanium ni = 1.4x1013 cm-3	the intrinsic carrier concentration

Intrinsic Fermi-Level Position
* For an intrinsic semiconductor, ni = pi,   *  Emidgap =(Ec+Ev)/2: is called the midgap energy. *  IF mp*  = mn*  , then EFi = Emidgap (exactly in the center of the bandgap) *  IF mp*  > mn*  , then EFi > Emidgap (above the center of the bandgap) *  IF mp*  < mn*  , then EFi < Emidgap ( below the center of the bandgap)

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Dopant and Energy Levels
	Two dimensional representation of silicon lattice doped with phosphorus		The energy band diagram showing (a) discrete donor energy state  (b) The effect of donor state being ionized

Acceptors and Energy Levels

Ionization Energy
● Ionization energy is the energy required to elevate the donor electron into the conduction band.
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Extrinsic Semiconductor
● Adding donor or acceptor impurity atoms to a semiconductor will change the distribution of electrons and holes in the material, and therefore, the Fermi energy position will change correspondingly. Recall

Extrinsic Semiconductor
● When the donor impurity atoms are added, the density of electrons is greater than the density of holes,  (no > po)  --> n-type; EF > EFi When the acceptor impurity atoms are added,  the density of electrons is less than the density of holes,  ( no < po) --> p-type; EF < EFi)

Degenerate and Nondegenerate
● If the conc. of dopant atoms added is small compared to the density of   the host atoms, then the impurity are far apart so that there is no   interaction between donor electrons, for example, in an n-material.      -->  nondegenerate semiconductor ● If the conc. of dopant atoms added increases such that the distance    between the impurity atoms decreases and the donor electrons begin to    interact with each other, then the single discrete donor energy will split    into a band of energies.  --> EF move toward Ec ● The widen of the band of donor states may overlap the bottom of the    conduction band. This occurs when the donor conc. becomes    comparable with the effective density of states, EF ³  Ec     --> degenerate semiconductor

Degenerate and Nondegenerate
Simplified energy band diagram for degeneratey   doped  (a) n- type semi-conductor   (b) p- type semi-conductor
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Statistics of Donors and Acceptors
●  The probability of electrons occupying the donor  energy state was given by ●  where Nd is the conc. of donor atoms, nd is the density of electrons    occupying the donor level and Ed is the energy of the donor level.    g = 2 since each donor level has two spin orientation, thus each donor level    has two quantum states. ●  Therefore the conc. of ionized donors Nd+ = Nd – nd      Similarly, the conc. of ionized acceptors Na- =  Na – pa, where                                     g = 4 for the acceptor level in Si and GaAs

Complete Ionization
●  If we assume Ed-EF >>  kT or EF-Ea >> kT ( e.g. T= 300 K), then
	that is, the donor/acceptor states are almost completely ionized and all the donor/acceptor impurity atoms have donated an electron/hole to the conduction/valence band.

Freeze-out
At T = 0K, no electrons from the donor state are thermally elevated into the conduction band; this effect is called freeze-out. At T = 0K, all electrons are in their lowest possible energy state; that is for an n-type semiconductor, each donor state must contain an electron, therefore, nd =Nd or Nd+=0, which means that the Fermi level must be above the donor level.

Charge Neutrality
* In thermal equilibrium, the semiconductor is electrically neutral.    The electrons distributing among the various energy states creating    negative and positive charges, but the net charge density is zero. * Compensated Semiconductors: is one that contains both donor and    acceptor impurity atoms in the same region.    A n-type compensated semiconductor occurs when Nd > Na and a p-type semiconductor    occurs when Na > Nd. * The charge neutrality condition is expressed by                                                               * where no and po are the thermal equilibrium conc.   of e- and h+ in the conduction band and valence band, respectively.   Nd+ is the conc. Of positively charged   donor states and Na - is the conc. of negatively   charged acceptor states.

Compensated Semiconductor
Energy band diagram of compansated semiconductor showing ionized and un-ionized donor and acceptor

Compensated Semiconductor
* If we assume complete ionization, Nd+ = Nd and Na- = Na, then * If Na = Nd = 0, ( for the intrinsic case),  no =po * If Nd >> Na, no = Nd  is used to  calculate the conc. of holes in valence band

Compensated Semiconductor
Electron concentration versus temperature showing the 3 regions Partial Ionization Extrinsic intrinsic
Energy band diagram showing the redistribution of electrons when donor are added
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Position of Fermi Level
* The position of Fermi level is a function of the doping concentration     and a function of temperature, EF(n, p, T). * Assume Boltzmann approximation is valid, we have

EF(n, p, T)
position of Fermi level as a function of: donor concentration  ( n- type ) acceptor concentration (p - type )	position of Fermi level for an : (a) n - type Nd > Na (b) p - type Na > Nd

EF(n, p, T)
The Fermi- leveles of : (a)  Material A in thermal Equilibrium (b)  Material B in thermal Equilibrium (c)  Materials A and B at the instance that they are place in contact (d)  Materials A and B in contact at thermal Equilibrium

Anderson Jose Mariño Ortega

C.I. 17.456.750

E.E.S.

http://osp.mans.edu.eg/rehan/solid_2004/ch5.htm