Quasi fermi level

Fermi levels in soUds and redox energies in electrolytes represent electrochemical potentials given by the chemical potential /Xj and a term that describes the Galvani potential p related to the charge of the considered particles (electrons) [50, 51]  

In semiconductors, the Fermi level is given by the conduction or valence band edge, Fcb or Fvb, respectively, and by the effective density of states at the band edge, Ncb or Nvb, in relation to the carrier concentration in the bands, n and p  

For Stationary illumination and suffidently long lifetimes (or diffusion lengths) of the photogenerated electrons and holes, excess minority and majority carriers exist that are stationary at the respective band edges in addition to the concentration values without illumination. This change in the total concentration of carriers alters the energy of the system (cf. Equation 2.13) and can be viewed as a new quasi-equilibrium situation. Attempts have been made to describe the situation in the terminology of the equilibrium between the carrier concentrations n and p in the dark and the Fermi level [52]  

05/o the excess carrier profile Appq is shifted towards the interior of the semiconductor by the diffusion length and therefore differs from the generation profile (G = al exp -ax)). 

At present, several methods have been developed to describe photoelectrochemical phenomena. For example, the kinetics of photoelectrochemical processes can be studied theoretically by solving transport equations (with allowance for charge-carrier photogeneration) which are supplemented by certain boundary conditions accounting for the specific features of electrode reactions with electrons and holes involved. In the simplest cases, this approach enables one to obtain an exact solution of the problem but, with rather complicated boundary conditions, it appears to be somewhat difficult, and the expressions obtained are cumbersome. 

At the same time, another approach is quite efficient for qualitative understanding and sometimes for quantitative interpretation this approach is quasithermodynamic rather than kinetic and is based on the concept of quasi-Fermi levels.  

in the quasi-thermodynamic approximation considered here the occurrence of nonequilibrium electrons and holes in the bands can be described as the splitting of the initial Fermi level F into two quasi-levels F and Fp. 

Substitution of n and into Eqs. (21a)-(21c) yields the quasi-Fermi levels F and Fp. For example, in the case of an n-type semiconductor, Eq. (21b) gives F, if the condition An no is satisfied. At the same time, since po n the condition Ap po can be satisfied simultaneously for F - Fp we find then from Eq. (21c)  

(35) implies that a noticeable shift of the electrochemical potential level under photogeneration can only take place for minority carriers. For moderate illumination intensity, the shifts of the quasi-levels, F p, are proportional to the logarithm of the intensity as it increases, further growth of the shifts slows down due to enhancement of recombination processes. The ultimate shift of F p with increasing illumination intensity, so long as Eq. (35) (and a similar formula for a p-type semiconductor) is valid, is the edge of the corresponding band. 

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The distributions of excess, or injected, carriers are indicated in band diagrams by so-called quasi-Fermi levels for electrons, Ep or holes, These... 

Fig. 1. The energy levels in a semiconductor. Shown are the valence and conduction bands and the forbidden gap in between where represents an occupied level, ie, electrons are present O, an unoccupied level and -3- an energy level arising from a chemical defect D and occurring within the forbidden gap. The electrons in each band are somewhat independent, (a) A cold semiconductor in pitch darkness where the valence band levels are filled and conduction band levels are empty, (b) The same semiconductor exposed to intense light or some other form of excitation showing the quasi-Fermi level for each band. The energy levels are occupied up to the available voltage for that band. There is a population inversion between conduction and valence bands which can lead to optical gain and possible lasing. Conversely, the chemical potential difference between the quasi-Fermi levels can be connected as the output voltage of a solar cell. Fquilihrium is reestabUshed by stepwise recombination at the defect levels D within the forbidden gap.

Electrons excited into the conduction band tend to stay in the conduction band, returning only slowly to the valence band. The corresponding missing electrons in the valence band are called holes. Holes tend to remain in the valence band. The conduction band electrons can estabUsh an equihbrium at a defined chemical potential, and electrons in the valence band can have an equiUbrium at a second, different chemical potential. Chemical potential can be regarded as a sort of available voltage from that subsystem. Instead of having one single chemical potential, ie, a Fermi level, for all the electrons in the material, the possibiUty exists for two separate quasi-Fermi levels in the same crystal. 

The idea of having two distinct quasi-Fermi levels or chemical potentials within the same volume of material, first emphasized by Shockley (1), has deeper implications than the somewhat similar concept of two distinct effective temperatures in the same block of material. The latter can occur, for example, when nuclear spins are weakly coupled to atomic motion (see Magnetic spin resonance). Quasi-Fermi level separations are often labeled as Im p Fermi s name spelled backwards. 

These two groups of excited carriers are not in equilibrium with each other. Each of them corresponds to a particular value of electrochemical potential we shall call these values pf and Often, these levels are called the quasi-Fermi levels of excited electrons and holes. The quasilevel of the electrons is located between the (dark) Fermi level and the bottom of the conduction band, and the quasilevel of the holes is located between the Fermi level and the top of the valence band. The higher the relative concentration of excited carriers, the closer to the corresponding band will be the quasilevel. In n-type semiconductors, where the concentration of elec-ttons in the conduction band is high even without illumination, the quasilevel of the excited electrons is just slightly above the Fermi level, while the quasilevel of the excited holes, p , is located considerably lower than the Fermi level. 

The electrons produced in the conduction band as a result of illumination can participate in cathodic reactions. However, since in n-type semiconductors the quasi-Fermi level is just slightly above the Fermi level, the excited electrons participating in a cathodic reaction will almost not increase the energy effect of the reaction. Their concentration close to the actual surface is low hence, it will be advantageous to link the n-type semiconductor electrode to another electrode which is metallic, and not illuminated, and to allow the cathodic reaction to occur at this electrode. It is necessary, then, that the auxiliary metal electrode have good catalytic activity toward the cathodic reaction. 

For a more detailed description of the semiconductor/electrolyte interface, it is convenient to define the quasi-Fermi levels of electrons, eFyC and holes, p p,... 

Quasi-Fermi levels, 9 728-729, 730 Quasifullerenes, 12 232-233 Quasi-iso tropic laminates, 26 754 26 782 Quasi-Monte Carlo sampling methods, 26 1005, 1011-1015, 1024 parallelization with Monte Carlo sampling, 26 1016... 

Quasi-Fermi Level of Excited Electrons and Holes... 

Fig, 10-1. Splitting of Fermi level, cnsci, into both quasi-Fermi level of electrons, bCp, and quasi-Fermi level of holes, pSp, in photoezcited semiconductors (a) in the dark, (b) in photon irradiation. SC = semiconductor hv = photon energy. 

In the case in which the photoexcited pairs of electrons and holes are relatively stable so that thermal equilibrium is established between phonons and electrons in the conduction band as well as between phonons and holes in the valence band, we can define individually the electrodiemical potentials of photoexcited electrons and of holes in the photostationary state. Here, thermal equilibrium is not established between the photoexcited electrons in the conduction band and the holes in the valence band. The electrochemical potential, thus defined, for the photoexcited electrons and holes is caUed the quasi-Fermi level of electrons nCp, and the quasi-Fermi level of holes,[Schockley, 1950 Gerischer, 1990]. 

In the dark, thermal equilibrium is established between electrons in the conduction band and holes in the valence band so that both the quasi-Fermi level of electrons and the quasi-Fermi level of holes equal the oiiginal Fermi level of the semiconductor (nCr = pC, = ep). Under the condition of photoexdtation, however, the quasi-Fermi level of electrons is higher and the quasi-Fermi level of holes is lower than the original Fermi level of the semiconductor (nSp > cp > pCp). Photoexdtation consequently splits the Fermi level of semiconductors into two quasi-Fermi levels the quasi-Fermi level of electrons for the conduction band and the quasi-Fermi level of holes for the valence band as shown in Fig. 10-1. 

Such a concept of quasi-Fermi level is valid under the condition that the time constant for the establishment of thermal equilibrium of electrons or holes in the conduction or valence band (the redprocal of the rate of establishing equilibrium between electrons and phonons) is much smaller than the time constant for the recombination of exdted electron-hole pairs (the redprocal d the recombination... 

The quasi-Fermi levels of electrons and holes are then expressed, respectively, in Eqns. 10-3 and 10-4 ... 

For n-type semiconductors (n p, and n An ), the quasi-Fermi level of electrons (Eqn. 10-3) approximately equals the original Fermi level (Eqn. 10-2) whereas, the quasi-Fermi level of holes (Eqn. 10-4) is lower than the original Fermi level (Eqn. 10-2) because the concentration of photoexcited holes, Ap , exceeds the concentration of holes, p, in the dark (p Ap ). In general, under the condition of photoexcitation, the quasi-Fermi level of mq ority charge carriers remains close to the original Fermi level but the quasi-Fermi level of minority charge carriers shifts away from the originEd Fermi level. 

Fig. 10-2. Splitting of Fenni level of electrode, cnsci. into quasi-Fermi levels of electrons, ep, and of holes, pCp, respectively, in a surface layer of photoexcited n-type and p-type semiconductors a shift of quasi-Fermi levels from original Fermi level is greater for minmity charge carriers than for majority charge carriers.

Since photoexcited electron-hole pairs are formed only within a limited depth from the semiconductor surface to which the irradiating photons can penetrate, the photon-induced split of the Fermi level into the quasi-Fermi levels of electrons and holes occurs only in a surface layer of limited depth as shown in Fig. 10-2. 

Under the condition of photoexcitation, the quasi-Fermi level, instead of the original Fermi level, determines the possibility of redox electron transfer reactions. The thermodynamic requirement is then given, for the transfer of cathodic electrons to proceed from the conduction band to oxidant particles, by the inequality of Eqn. 10-7 ... 

The cathodic current of electron transfer is proportional to the concentration of interfadal electrons, n and the anodic current of hole transfer is proportional to the concentration of interfacial holes, p., in semiconductor electrodes as described in Sec. 8.3. Since the concentration of interfacial electrons or holes depends on the quasi-Fermi level of interfacial electrons or holes in the electrode as shown in Eqn. 10-3 or 10—4 (n, = n + dra and p, =p + 4P ), the transfer current of cathodic electrons or anodic holes under the condition of photoexdtation depends on the quasi-Fermi level of interfadal electrons, nCp, or the quasi-Fermi level of interfadal holes, pEp It also follows from Sec. 8.3 that the anodic current of electron transfer (the ipjection of electrons into the conduction hand) or the cathodic current of hole transfer (the ipjection of holes into the valence band) does not depend on the... 

Figure 10-3 juxtaposes the Fermi levels of the following redox reactions in aqueous solutions and the quasi-Fermi levels of interfacial electrons and holes in an n-type semiconductor electrode erhjo/Hj) of the hydrogen redox reaction F(0a/H20) of the oxj en redox reaction ersc) of the n- q)e semiconductor and... 

With n-type semiconductor electrodes, the anodic oiQ en reaction (Euiodic hole transfer) will not occur in the dark because the concentration of interfacial holes in the valence band is extremely small whereas, the same reaction will occur in the photon irradiation simply because the concentration of interfadal holes in the valence band is increased by photoexcitation and the quasi-Fermi level pEp of interfadal holes becomes lower than the Fermi level the o en redox... 

For metal electrodes, the anodic 03Q n reaction proceeds at electrode potentials more anodic than the equilibrium potential Bo of the reaction as shown in Fig. 10-14. For n-type semiconductor electrodes, the anodic photoexdted oxygen reaction proceeds at electrode potentials where the potential E of the valence band edge (predsely, the potential pEp of the quasi-Fermi level of interfadal holes, pCp = — CpEp) is more anodic than the equilibrium oxygen potential Eq, even i/the observed electrode potential E is less anodic than the equilibrium oxygen potential E03. The anodic hole transfer of the o Qgen reaction, hence, occurs at photoexdted n-type semiconductor electrodes even in the range of potential less anodic than the equilibriiun potential Eq of the reaction as shown in Fig. 10-14. 

Here, the reaction rate is proportional to the concentration of interfacial holes in the electrode as described in Sec. 8.3. Since the concentration of interfacial holes depends on the Fermi level er., of the electrode interface, the reaction current due to anodic holes depends on the Fermi level e ,. of the interface in the dark and on the quasi-Fermi level of interfacial holes pCp, in the photoexcited state. 

This conclusion is valid regardless whether the electrode is n-fype or p-fype. Consequently, if the quasi-Fermi level of interfacial holes in a photoexcited n-type semiconductor electrode equals the quasi-Fermi level of interfacial holes pEp, (eq ial to the Fermi level pEp., of the interface) in a p-type electrode of the same semiconductor in the dark, the current due to anodic holes will be the same on the two electrodes and, hence, the curves of the anodic reaction current as a function of the quasi-Fermi level of interfacial holes will be the same for the two electrodes as suggested in Fig. 10-21. The curves of the anodic reaction current represented as a function of the electrode potential (the Fermi level of the electrode), instead of the quasi-Fermi level of interfacial holes, are not the same for the two electrodes, however. 

The quasi-Fermi level of interfacial holes nearly equals the Fermi level pe , ( pEp,.) in photoexcited p-type electrodes, but the quasi-Fermi level pej. of interfacial holes is lower than the Fermi level aSp,. (> p p,) in photoexcited n-type electrodes as shown in Fig. 10-21. It then follows that the range of electrode potential, where the anodic reaction occurs on the photoexcited n-type electrode, shifts itself, from the range of potential where the same anodic reaction occurs on the dark p-type electrode, toward the caliiodic (more negative) direction by an energy equivalent to (nEp - p p,). 

Fig. 10-21. Quasi-Fermi levels of holes in transfer reaction of anodic holes, from the valence band (a) of a photoexcited n-type electrode and (b) of a dark p-type electrode of the same semiconductor, to redox particles pCp, = quasi-Fermi level of interfacial holes in a photoexcited n-type electrode where pCp, is lower than the Fermi level cp and in a dark n-type electrode where pCp, equals the Fermi level sp.

The overvoltage for ihe generation and transport of holes, ii p, ac, is the difTerence between the quasi-Fermi level of interfacial holes and the Fermi level enso of electrons in the electrode interior as defined in Eqn. 10-29 ... 

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