Fermi space charge

If the charge osc is fully accommodated by electronic states near the Fermi energy, no space charge is formed in the electrode phase, and any voltage applied to the electrode appears exclusively across the Helmholtz layer, i.e. the system behaves like a metal. 

Semiconductor - Electrolyte Interlace The electric field in the space charge region that may develop at the semiconductor electrolyte interface can help to separate photogenerated e /h 1 couples, effectively suppressing recombination. When a semiconductor is brought into contact with an electrolyte, the electrochemical potential of the semiconductor (corresponding to the Fermi level, Ey of the solid [50]) and of the redox couple (A/A ) in solution equilibrate. When an n-type semiconductor is considered, before contact the Ey of the solid is in the band gap, near the conduction band edge. After contact and equilibration the Ey will... 

The surface Fermi level, Cp, which depends on the surface state, is not the same as the interior Fermi level, ep, which is determined by the bulk impurity and its concentration. As electron transfer equilibrium is established, the two Fermi levels are equilibrated each other (ep = ep) and the band level bends downward or upward near the surface forming a space charge layer as shown in Fig. 2-31. 

The potential i sc of the space charge layer can also be derived as a fixnction of the surface state charge Ou (the surface state density multiplied by the Fermi function). The relationship between of a. and M>sc thus derived can be compared with the relationship between and R (Eqn. 5-67) to obtain, to a first approximation, Eqn. 5-68 for the distribution of the electrode potential in the space charge layer and in the compact layer [Myamlin-Pleskov, 1967 Sato, 1993] ... 

In the same way as described in Sec. 5.2 for a diifiise layer in aqueous solution, the differential electric capacity, Csc, of a space charge layer of semiconductors can be derived from the Poisson s equation and the Fermi distribution function (or approximated by the Boltzmann distribution) to obtain Eqn. 5-69 for intrinsic semiconductor electrodes [(Serischer, 1961 Myamlin-Pleskov, 1967 Memming, 1983] ... 

As the potential Ai )sc of an inversion layer increases and as the Fermi level at the electrode interface coincides with the band edge level, the electrode interface is in the state of degeneracy (Fermi level pinning) and both the capacity Csc and the potential A4>sc are maintained constant. Figure 5-48 shows schematically the capacity of a space charge layer as a function of electrode potential. As the electrode potential shifts in the anodic (positive) direction from a cathodic (negative) potential, an accumulation, a depletion, and an inversion layer are successively formed here, the capacity of the space charge layer first decreases to a minimum and then increases to a steady value. 

We consider, now, an electron-depleted space charge layer that is gradually polarized in the anodic direction. As long as the Fermi level is located away from the surface state, the interfacial capacity is determined by the capacity of the depletion layer that obeys a Mott-Schottlsy relation as shown in Fig. 5-61. 

In contrast to metal electrodes in which the electrostatic potential is constant, in semiconductor electrodes a space charge layer exists that creates an electrostatic potential gradient. The band edge levels and in the interior of semiconductor electrodes, thereby, differ from the analogous and at the electrode interface hence, the difference between the band edge level and the Fermi level in the interior of semiconductor electrodes is not the same as that at the electrode interface as shown in Fig. 8-14 and expressed in Eqn. 8—47 ... 

When the total overvoltage ti is distributed not only in the space charge layer t)8c but also in the compact layer tih, the Tafel constants of a and a each becomes greater than zero and the Tafel constants of a and each becomes less than one. In such cases, Kiv) and ip(T ) do not remain constant but increase with increasing overvoltage. Further, if Fermi level pinning is established at the interface of semiconductor electrodes, the Tafel constant becomes dose to 0.5 for... 

In the equilibriiun of interfacial redox reactions of the adsorbed protons and hydrogens, the Fermi level of semiconductor electrons at the electrode interface equals the Fermi level e p(h /h) of interfacial redox electrons in the adsorbed protons and hydrogens. The Fermi level e gc) th interface of semiconductor electrode depends on the potential /l< )sc of the space charge layer as shown in Eqn. 9-66 ... 

Such a migration of photoexdted electrons and holes induces in the electrode an inverse potential which reduces the potential across the space charge layer and retards the migration of electrons and holes in the opposite direction as shown in Fig. 10-5. This inverse potential, induced by photoexdtation, is called the photopotential. Since the photopotential, AE. = - he,/c, arises in a direction to reduce the potential across the space charge layer, the Fermi level of the semiconductor interior rises by an energy of de j, (the electrode potential lowers... 

For p-type electrodes in the dark and in the photoexdted state, the concentration of majority charge carriers (holes) is sufficiently great that the Fermi level eptso of the electrode interior nearly equals the quasi-Fermi level of interfacial holes hence, the overvoltage Up sc for the generation and transport of holes in the space charge layer is zero even as the transfer of anodic holes progresses as expressed in Eqn. 10-30 ... 

Fig. 10-26. Energy diagrams of a cell for photoelectrolytic decomposition of water consisting of a metal cathode (M) and an n-type semiconductor anode (n-SC) of which the Fermi level is higher than the Fermi level of hydrogen redox reaction ( R8o>ep(H /H2)) (a) cell circuit is open in the dark, (b) cell circuit is closed in the daric, (c) cell circuit is closed in a photoezdted state (cell reaction proceeds.). potential hairier of a space charge layer.

When the cell circuit is closed in the dark, as shown in Fig. 10-25(b), the Fermi level is equilibrated between the metallic cathode and the n-lype semiconductor anode. As a result, a depletion layer of space charge (potential barrier, is formed in the semiconductor anode, thereby shifting the potential of the anode from the flat band potential to a more anodic (more positive) potential (= + ). In the dark, however, the anodic hole transfer... 

Figure 10-30(a) applies to an open circuit cell in the dark Fig. 10-30(b) applies to a short circuit cell in the dark. After the cell circuit is closed in the dark, the Fermi level is equilibrated between the two electrodes thereby forming a space charge layer both in the n-tyi>e anode and in the p-type cathode. The overall potential, AE, generated in the two space charge layers nearly equals the difference of the flat band potential between the n-type anode and the p-type cathode as eiqpressed in Eqn. 10-57 ... 

Upon immersion of the CdSe semiconductor into the electrolyte, electron exchange at the interface occurs until equilibrium is attained. At equilibrium, the Fermi level of the semiconductor is adjusted by the presence of a space charge layer at the semiconductor surface. This layer is due to the difference between the Fermi level of the semiconductor and the Fermi level of the electrolyte which is measured at the redox couple (X) The potential drop at the space charge layer and the amount of band bending also depend on the degree of Fermi level mismatch at the semiconductor-... 

An example of the difficulties encountered when trying to fabricate an ohmic electrode, able to sustain a space-charge-limited current, is the recent work of the Neher group [179]. The authors deposited barium as an electron injection cathode on top of an electron transporting polymer based on a naphthalene diimide core whose LUMO is as low as 4 eV below vacuum level. Although the Fermi level of barium should be above the LUMO of the polymer, the electron current is. 

An externally applied potential controls the Fermi level of the semiconductor with respect to the reference electrode in the solution. Changes of the potential affect the potential drop across the semiconduc-tor/electrolyte interface. In most situations of electrochemical reactions, the potential drop in the solution Helmholtz layer can be neglected, and thus a gradient of potential is generated in the space charge... 

For a semiconductor like Ge, the pattern of electronic interaction between the surface and an adsorbate is more complex than that for a metal. Semiconductors possess a forbidden gap between the filled band (valence band) and the conduction band. Fig. 6a shows the energy levels for a semiconductor where Er represents the energy of the top of the valence band, Ec the bottom of the conduction band, and Ey is the Fermi energy level. The clean Ge surface is characterized by the presence of unfilled orbitals which trap electrons from the bulk, and the free bonds give rise to a space-charge layer S and hence a substantial dipole moment. Furthermore, an appreciable field is produced inside the semiconductor, as distinct from a metal, and positive charges may be distributed over several hundred A. 

In p—n junctions a space-charge layer is formed in the transition region between an -type and a -type region by an exchange of charge carriers. This is a consequence of the overall tendency for the Fermi levels to become equal in height throughout the system (Fig. 2). 

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