Fermi level temperature dependence

If the p-n junction is in thermal equilibrium, and no external voltage is applied, then the Fermi level must be equal in each region. Since the distance between band edges and Fermi level merely depends on temperature, close to the junction there must arise a band bending (Fig. 2.7, left). With a reverse biased voltage (Fig. 2.7, centre), the Fermi levels of N-type and P-type materials are different. Transfer of electrons is hampered additionally in comparison to a non-polarized state, since an additional barrier is formed. The opposite situation arises with forward biasing (Fig. 2.7, right). 

Evidence on this question may be taken by the behavior of the electrical conductivity CT as a function of temperature. A thermally activated process T dependence on log(CT), Arrhenius plot) is expected if doping takes place, whereas j -i/4 dependence, characteristic of a variable range hopping at the Fermi level is expected for a nondoping situation. 

Schematic representation of how doping can lead to (a) w-type and (b) / -type semiconductors. Note that the exact position of the Fermi level is temperature-dependent.

The final question we shall consider here has to do with the extrapolation of the solubility of hydrogen in silicon to lower temperatures. Extrapolation of a high-temperature Arrhenius line, e.g., from Fig. 11, would at best give an estimate of the equilibrium concentration of H°, or perhaps of all monatomic species, in intrinsic material the concentration of H2 complexes would not be properly allowed for, nor would the effects of Fermi-level shifts. Obviously the temperature dependence of the total dissolved hydrogen concentration in equilibrium with, say, H2 gas at one atmosphere, will depend on a number of parameters whose values are not yet adequately known the binding energy AE2 of two H° into H2 in the crystal, the locations of the hydrogen donor and acceptor levels eD, eA, respectively, etc. However, the uncertainties in such quantities are not so... 

Contrary to silicon, very little work has been done in germanium regarding quantitative hydrogen diffusion or electric field drift studies. Such experiments may be complicated by the fact that ultra-pure germanium becomes intrinsic already at temperatures near 200 K. It would be worthwhile to explore the possibility of using lightly doped germanium for such studies in order to explore Fermi level dependent effects. 

The position of the Fermi level in an extrinsic semiconductor depends upon the dopant concentrations and the temperature. As a rough guide the Fermi level can be taken as half way between the donor levels and the bottom of the valence band for n-type materials or half way between the top of the valence band and the acceptor levels for p-type semiconductor, both referred to 0 K. As the temperature rises the Fermi level in both cases moves toward the center of the band gap. 

As in the static case, the position of the Fermi level Sp is important, since whether Eq is greater than or less than 8p should determine the direction of charge transfer, i.e. to or from the surface. However, the situation is not quite as clear-cut as this suggests, because non-adiabaticity can come into play. Also, the effect of image forces means that Eq is not a constant but, rather, a function of the atom-surface separation distance and, hence, of time, so that the position of Eq relative to Ep can change as the atom approaches the surface. Further complications can arise if adsorbed atoms are present on the surface, since this can change Ep, or if temperature dependence is examined, since, with non-zero temperature, band levels above Ep begin to be occupied. 

We see that the adsorptivity of the surface with respect to molecules of a given kind, i.e., the total number of molecules of this kind N bound to unit surface under conditions of equilibrium with the gaseous phase (i.e., at given pressure P and temperature T) depends on the position of the Fermi level. By shifting the Fermi level (other conditions being fixed), one can vary the adsorptivity of the surface. 

Equation (4.20) has been verified by a large number of metals at various temperatures, from 77 K to 900 K. Most metals conform to this equation very well. However, Swanson and Grouser (1966) found that the field-emission spectra on W(IOO) deviates substantially from Eq. (4.20). There is a pronounced peak at about 0.35 eV below the Fermi level (Fig. 4,7). This phenomenon later acquired a nickname, the Swanson hump. A series of more extensive experiments were then conducted to investigate its nature (Swanson and Grouser 1967). The main results are (1) The spectrum depends dramatically on the crystallographic orientation. On W(310), W(211), W(lll), and W(611), the measured field-emission spectra agree well with the free-electron... 

Different is the case for a state (a resonant state) as 1 in Fig. 12 a, lying very close to the Fermi level (E very close to Ep). For this state, (13) gives a probability of occupation which is strongly temperature dependent. How much it determines the position of the Fermi level depends on the contribution which, together with the population of the broad (s-d) conduction band which we have super-imposed in Fig. 12b, it gives to (14). 

We depart briefly from our discussion of SI GaAs to consider an example that better illustrates some of the features of temperature-dependent Hall measurements. This example (Look et al., 1982a) involves bulk GaAs samples that have sc — F — 0-15 eV. We suppose, initially, that the impurity or defect controlling the Fermi level is a donor. Then any acceptors or donors above this energy (by a few kT more) are unoccupied and any below are occupied. Also, p n for kT eG. From Eq. (B34), Appendix B, we get... 

In Fermi-Dirac statistics, g is the Fermi energy Er, which is such that the probability, that a state of energy is occupied, is 1/2. States with energies higher than Et have a smaller probability of being occupied, those with lower energy, a higher probability. The position of the Fermi level in a semiconductor depends markedly on the temperature and on the concentration of impurities. The Fermi levels of two conductors in electrical contact and in thermal equilibrium are the same. 

Weisz assumption that, when equilibrium is attained, the energy of the adsorption traps is at the Fermi level, may not be valid in all cases. This assumption has the effect of removing the temperature dependence from the equilibrium adsorption. It is equivalent to the assumption that the number of empty adsorption traps is about equal to the number of ionized adsorption traps, and is invalid if empty adsorption traps are physically adsorbed atoms or molecules. For the latter case, A " in Fig. 5 will in general be greater than zero. 

Fig. 37. Band edge profile of a (In,Mn)As/GaSb heterostmcture. Eq. E. and Ep denote band edges of conduction band, valence band, and Fermi level, respectively, (b) Temperature dependence of the magnetization observed during cooldown in the dark (open circles) and warmup (solid circles) under a fixed magnetic field of 0.02 T. The effect of light irradiation at 5 K is also shown by an arrow, (c) Magnetization curves at 5 K observed before (open circles) and after (solid circles) light irradiation. Solid line shows a theoretical curve, (d) Hall resistivity />Hall observed at 5 K before (dashed line) and after (solid line) light irradiation (Koshihara...

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