Fermi level of intrinsic semiconductor

Since Nc is nearly equal to N, the Fermi level of intrinsic semiconductors is located midway in the band gap as shown in Fig. 2-16. All the equations given in the foregoing are valid under the condition that rii Nc or Ny. this condition is frilfilled with usual intrinsic semiconductors. 

Fig. 2-16. Electron state density distribution and electron-hole pair formation in the conduction and valence bands of intrinsic semiconductors Cf > Fermi level of intrinsic semiconductors.

Fig. 8-16. Electron state density in a semiconductor electrode and in hjrdrated redox partides, rate constant of electron tunneling, and exchange redox current in equilibrium with a redox electron transfer reaction for which the Fermi level is close to the conduction band edge eF(sc) = Fermi level of intrinsic semiconductor at the flat band potential 1. 0 (tp.o) = exchange reaction current of electrons (holes) (hvp)) - tunneling rate constant of electrons (holes).

These expressions agree with the formula of Kingston and Neustadter (14), where exp (Ep - EjAT) = n0Ai exp (Ej - EpAT) - p0Ai Ep is the Fermi level of the semiconductor and Ej is die Fermi level of an intrinsic semiconductor. 

As = surface area of a semiconductor contact [A ] = concentration of the reduced form of a redox couple in solution [A] = concentration of the oxidized form of a redox couple in solution A" = effective Richardson constant (A/A ) = electrochemical potential of a solution cb = energy of the conduction band edge Ep = Fermi level EF,m = Fermi level of a metal f,sc = Fermi level of a semiconductor SjA/A") = redox potential of a solution ° (A/A ) = formal redox potential of a solution = electric field max = maximum electric field at a semiconductor interface e = number of electrons transferred per molecule oxidized or reduced F = Faraday constant / = current /o = exchange current k = Boltzmann constant = intrinsic rate constant for electron transfer at a semiconductor/liquid interface k = forward electron transfer rate constant = reverse electron transfer rate constant = concentration of donor atoms in an n-type semiconductor NHE = normal hydrogen electrode n = electron concentration b = electron concentration in the bulk of a semiconductor ... 

Let us imagine a situation where an -type semiconductor is brought in contact with a redox electrolyte (Fig. 4a). Let us also assume that the electrochemical potential, that is, its redox potential, is almost equal to the intrinsic Fermi level of the semiconductor. This situation forces the electrons (majority carriers) to flow from the semiconductor to the electrolyte. This migration continues until the two Fermi levels achieve an equilibrium position (Fig. 4b). At this condition, no further migration of electron occurs and a dynamic equflibrium is established. In this situation, instead of electrochemically reacting with the redox electrolyte, these majority carriers accumulate at the interface of semiconductor-electrolyte to maintain neutrahty of the material. 

What would be the magnitude of potential developed between the charges accumulated at the interface (x = 0) and at plane X = —Wn (or for p-type semiconductor at X = —Wp) The driving force for these carriers depends on the magnitude of the potential difference between the Fermi level of the semiconductor and the redox potential of the electrolyte (i.e. Ey — F(redox))-This potential is known as the contact potential [6). Can we fabricate a PEC cell, which gives a contact potential equal to the band gap value of the semiconductor In other words, can we form a PEC cell with a semiconductor (whose Ec Ef) and redox electrolyte (whose / redox v). such that the contact potential (9) = g The approximate Fermi level of the semiconductor (i.e. the intrinsic semiconductor) is approximately equal to half the band gap of the semiconductor (i.e. jfg). Therefore, redox electrolyte cannot lower the Fermi level of -type semiconductor beyond jEg. This condition puts a restriction to the maximum achievable contact potential 6), and is equal to jEg value. This also suggests that for a given semiconductor, the most suitable electrolyte would be the one that has a redox potential that is almost equal to the intrinsic Fermi level of the semiconductor. 

The Fermi level, Cp, of intrinsic semiconductors is obtained from Eqn. 2-13 as shown in Eqn. 2-15 ... 

Figure 2-41 compares the electron level diagram of intrinsic semiconductors with that of hydrated redox particles at the standard concentration. The two diagrams resemble each other in that the Fermi level is located midway between the occupied level and the vacant level. It is, however, obvious that the occupied and vacant bands for semiconductors are the bands of delocalized electron states, whereas they are the fluctuation bands of localized electron states for hydrated redox particles. 

For a doped semiconductor, the Fermi level position will be shifted from mid-gap, because the doping process will vary the tendency of the solid to either gain or lose electrons. For example, if donors are added to an intrinsic semicondnctor, the material will be more likely to lose electrons. The Fermi level of an n-type semiconductor will thus move closer to the vacuum level (i.e. will become more negative on the electrochemical potential scale) (Figure 9(b)). Similarly, if acceptors are added to an intrinsic material, the Fermi level will become more positive, because this phase will now have an increased tendency to accept electrons from another phase (Figure 9(c)). 

FIGURE 22.2 Electron energy levels for a standard pair of hydrated redox particles and for an intrinsic semiconductor ered = the most probable electron level of oxidant, eox = the most probable electron level of reductant, 8p(redox) = standard Fermi level of redox electrons, 8p = Fermi level of an intrinsic semiconductor, v = valence band edge level, and c = conduction band edge level. 

For clarification of the type of junctions formed at the semiconductor-electrolyte, let us take an example of n-type semiconductor. In addition to possessing free electrons (referred to as the majority carrier), n-type semiconductor also possesses holes (referred to as the minority carrier). The concentration of holes is temperature-dependent and is equivalent to the intrinsic concentration of the carrier (which is related to the concentration of Frankel defects). It can be shown mathematically that the Fermi level of minority carrier hes at almost half the band gap position. On the other hand, the concentration of majority carriers as well as the Fermi level depends on doping concentration. Thus, the Fermi level of the majority carrier can he anywhere between the conduction hand edge and the intrinsic Fermi level that is situated at i g. 

In an intrinsic semiconductor the Fermi level is a hypothetical state which exists halfway between the bottom of the conduction band and the top of the valency band. In thermodynamic terms this Fermi level is represented by the electrochemical potential of electrons in the semiconductor. The fact that the Fermi level exists halfway inside the energy gap, and where ideally no electrons or holes can exist, is of small consequence. The Fermi level represents the energy state at which the probability of existing electron and hole are equal and half each. The Fermi level within the semiconductor represents an ideal situation which is calculable and is in fact equivalent to the electrochemical potential inside the semiconductor. 

Additional information about the semiconductor can be obtained from the interface capacitance C , which arises because each interface state stores a charge. A surface potential C can be defined as the potential at the semiconductor-insulator interface which causes the center of the band gap of the semiconductor [the Fermi level of the intrinsic material, ( /),] to shift to a new value (Figure 4.3.5). This surface potential arises whenever the applied potential causes charge to build up at the interface. For example, for an -type material when E = Ef- ( /)ref is very much less than zero, ( /), will cross Ef as shown in Figure 4.3.5, leading to an accumulation of holes at the interface, that is, inversion as described above for the MOS devices. Now yZj can be calculated from the capacitance data described above by means of (NicoUian and Goetzberger [1967])... 

Figure 4.3.5. A simplified energy band diagram of an n-type EIS device which has an applied voltage E such that the semiconductor oxide interface is in the inversion regime [shown by crossing of Ef and ( /) ] leading to a buildup of holes at the interface. Two energy scales are shown, one referenced to an electron at infinity (e ) and the other referenced to the saturated calomel electrode (SCE). The surface potential the voltage drop across the insulator, Vj, and the Fermi levels of the reference electrode, the semiconductor under an applied voltage E, and the intrinsic semiconductor [(Ef) t, Ef, and (Ef)i respectively] are all shown. (After Diot et al. [1986].)...

The Fermi level of semiconductors is located in the band gap. For intrinsic semiconductors, the electron concentration in the conduction band is equal to the hole concentration Cp in the valence band. The equilibrium constant can be written as ... 

In addition to metals, there are semimetals,such as graphite, whose valence band and conduction band can overlap. In general, their minimum energy gaps are very narrow. The third class of solids is the intrinsic semiconductor its minimum energy gap Eg is generally below 3 eV. Thus, thermal excitation alone can create an electron-hole pair to enhance conduction. The Fermi level of the intrinsic semiconductor lies between the valence band (HOMO) and the conduction band (LUMO). Hence,... 

The Fermi level of the intrinsic semiconductor is halfway between the conduction band and valence band. However, this level fluctuates on addition of impurities into the pure semiconductor. There are two types of doped semiconductors n-type semiconductor and p-type semiconductor. 

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