Modelos de Riesgo Crediticio

Surface electronic states distinct from those of the bulk were first considered the- oretically by Tamm [28], although substantial insight was added by Shockley [29] who showed that localized levels deriving from the bulk bands exist within the semiconductor band gap. Maue [30] considered the existence of evanescent surface states within the nearly-free electron model discussed above. Provided boundary conditions are fulfilled for matching of the wavefunction tails of the surface states into both the vacuum and the semiconductor bulk [10, 12], these surface states can therefore be seen to derive from the ViGS of the complex band structure.

The ViGS inherent to a clean surface with full translational symmetry in two-dimensions (in the plane of the surface) are termed intrinsic surface states. The microscopic origin for such state formation is the dangling bonds formed when a surface is generated by truncating the bulk structure – each atom at the surface has fewer nearest neighbours than those in the bulk. The breaking of bonds at the surface costs energy, and so a rearrangement of the surface atoms in order to minimize this energy, that is, minimize the energy associated with dangling bonds, almost invariably results in a surface that is different to an ideally truncated bulk. Intrinsic surface states are specific to such surface reconstructions [12, 31].

Extrinsic surface states can also be formed when imperfections (for example adatoms or defects) cause electronic states to become localized at the surface. In the case of adatom induced states, charge transfer will occur between the semicon- ductor and the adatom dependent on their difference in electronegativity, although M¨onch [12] incorporates this effect within the ViGS concept.

From the discussions in Section 1.3.1, the surface ViGS will either be donor- like or acceptor-like if they are below or above the CNL, respectively. These states can be occupied or unoccupied dependent on the Fermi level position at the sur-

1.3. Charge neutrality level 9

face and are either neutral (occupied donor-like or unoccupied acceptor-like states), positively charged (unoccupied donor-like states) or negatively charged (occupied acceptor-like states). In the presence of charged surface states, the carriers in the near-surface region of the semiconductor rearrange in order to screen the macro- scopic electric field induced by these states. This occurs over a distance determined by the Thomas-Fermi screening length. While this is very short in metals due to their extremely high free carrier concentration, it is appreciably longer in semiconductors leading to macroscopic regions of charge redistribution known as space-charge re- gions.

Consequently, the Fermi level shifts as a function of depth within the semi- conductor, which can equivalently be viewed as a bending of the conduction and valence bands with respect to the Fermi level. The position of the Fermi level at the surface is determined by the condition of charge neutrality within the semiconduc- tor: the total charge due to surface states, Qss, must be compensated by an equal

but opposite charge within the space-charge region,Qsc,

Qss=−Qsc. (1.11)

For an n-type material, when the surface states are negatively (positively) charged, a positive (negative) space-charge region is therefore required to maintain charge neutrality. This is achieved by an upward (downward) bending of the bands in order to decrease (increase) the electron concentration at the surface with re- spect to the bulk values, resulting in a depletion (accumulation) of electrons at the surface. If there are sufficient negatively charged surface states, the band bending required to maintain charge neutrality can be so severe that the Fermi level at the surface moves below the middle of the direct band gap and a p-type surface layer of holes exists, separated from the n-type bulk region by a depletion layer. This is termed an inversion layer. In the situation where the Fermi level at the surface is located exactly at the CNL, there are on average no charged surface states, and so there is no band bending. This is referred to as the flat-band case. The pre- ceding considerations are reversed when the bulk region has p-type conductivity. Schematic representations of the band bending and carrier concentration variation

1.3. Charge neutrality level 10 E n e r g y n ( z) Depth (z) E n e r g y n ( z) Depth (z) Depth (z) p ( z) E n e r g y E n e r g y n ( z) Depth (z) E n e r g y Depth (z) p ( z) Depth (z) p ( z) E n e r g y E n e r g y Depth (z) n ( z) , p ( z) N A - - N D + n(z) p(z) p - t yp e E n e r g y n - t yp e

Inversion Depletion Flat bands Accumulation

E C E mid E V E F Depth (z) n ( z) , p ( z) N D + - N A - p(z) n(z)

Figure 1.4: A schematic representation of the band bending and associated charge profiles for inversion, depletion, flat-bands and accumulation space-charge layers at the surface of n- and

p-type semiconductors. The variation of the conduction and valence band edges (EC and EV)

and the mid-gap energy (Emid, dotted line) are shown. A schematic Fermi level (EF) is also

represented (dashed line). In the carrier concentration plots, the background net ionized donor density (N+

D−NA−) and net ionized acceptor density (NA−−ND+) is shown forn-type andp-type

samples, respectively (dotted line).

as a function of depth in each of these space-charge regions is shown in Fig. 1.4. Allen and Gobeli [32] investigated the clean Si(111) surface prepared in the (2×1) reconstruction by cleavage in ultra-high vacuum (UHV), and observed a strong pinning of the Fermi level at the surface by intrinsic states for a wide range of bulk Fermi level positions, with upward (downward) band bending observed for n-type (p-type) samples. This results in electron (hole) depletion layers, consistent with the CNL lying only a little below the middle of the direct band gap [24]. Similar space-charge layers result for the Si(100)−(2×1) and Si(111)−(7×7) surfaces [31, 33], consistent with the position of the CNL, but with differences in surface state distributions for the different crystal surfaces and reconstructions leading to small differences in the Fermi level pinning position. Meanwhile, a much larger tendency for upward band bending for n-type material than downward band bending for p- type material was observed at the cleaved Ge(111)−(2×1) surface [34], consistent

1.3. Charge neutrality level 11

with the CNL lying approximately at the VBM in Ge [24].

The situation is a little more complex for III-V compound semiconductors. While a pinning of the Fermi level a little below mid-gap was observed at the (001) surface of as-grown GaAs [35], consistent with the CNL position [24], flat-bands were observed at the perfectly cleaved (110) surface [36, 37]. However, the pinning of the Fermi level was recovered upon adsorption of a small quantity of O, or in the presence of step edges on the cleaved surface [37]. This was subsequently understood by theoretical predictions that a relaxation of the surface layer, where the Ga–As zigzag chains tilt with the As atoms being pushed outwards at the cleaved (110) surface, results in the intrinsic surface states related to dangling bonds being pushed out of the band gap. Thus, the pinning of the Fermi level in as-grown or ex-situ prepared GaAs(110) can be attributed to extrinsic surface states. This confirms that the CNL still represents the relevant energy level for discussing the electronic properties of surfaces where extrinsic, rather than intrinsic, surface states dominate. A final interesting example is the case of InAs, which has been observed to exhibit an accumulation of electrons at the clean surface ofn-type material [38, 39]. This is because the CNL actually lies outside of the fundamental band gap in InAs, above the CBM [24]. A pinning of the Fermi level close to this energy leads to an increase in electron density approaching the surface. This example is of particular relevance for the materials considered in this thesis.