Abstract
The classical Schottky barrier is introduced. The Schottky approximation is initially used with parameters listed and F(x) and Ψ(x) are given. The zero current solution for n(x). Diffusion potential and junction field is given. The Debye length and barrier width are defined. The accuracy of the Schottky approximation is discussed. n(x) for non vanishing currents are evaluated. The Dobson integral is given. The Boltzmann term is shown to be independent of the current. Current voltage characteristics are calculated. A modified Schottky barrier is introduced. Schottky barrier with current dependent Interface density is identified. Metal/semiconductor boundary conditions are given. Richardson-Dushman emission is identified. Current voltage characteristic in modified Schottky barriers are computed. The ideal diode equation is given. The shape factor is given. DRO and DO ranges are identified. A modified Boltzmann range is shown. Electrostatic and electrochemical potential in the Schottky barrier are identified.
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Notes
- 1.
Even though the electron density inside a metal is much higher than in the semiconductor, at its boundary to the semiconductor this density is substantially reduced according to its effective work function. It is this electron density which causes a reduction of n in the semiconductor at the interface.
- 2.
A similar Schottky barrier appears in p-type semiconductors near a metal electrode with low work function, again when the hole density near the electrode is much smaller than in the bulk. Here the space-charge region is negatively charged and the resulting field is positive.
- 3.
This is slightly different from N c within the semiconductor bulk (see Eq. (25.26)) because of a different effective mass at the interface.
- 4.
The error encountered at the boundary of this range (8⋅10−6 cm) seem to be rather large (factor 2) when judging from the plot in linear scale of Fig. 26.1. The accumulative error, when integrating from the metal/semiconductor interface, however, is tolerable, as shown in Fig. 26.2. The substantial simplification in the mathematical analysis justifies this seemingly crude approach.
- 5.
A comparison with the previously discussed example of majority carrier injection, in which n≫N d , presents the other alternative for the two cases for which the discussion of this one-carrier space-charge distribution can be drastically simplified.
- 6.
We have rewritten the first, the transport equation as a function of dn/dx to identify this set as a set of three differential equations that need to be solved.
- 7.
That is, the maximum field which lies in this approximation at the metal/semiconductor boundary (neglecting image forces).
- 8.
However, at higher doping densities, especially close to the metal interface, tunneling fields may be reached when N d >1018 cm−3. This often is desired to make a contact “ohmic” and such increased defect density can be reached, e.g. by gas discharge treatments (Butler 1980).
- 9.
We have introduced here a shifted coordinate system (x 1,n). The amount of the shift in x is determined by the boundary condition, as will be discussed later in this section.
- 10.
- 11.
Since its pre-exponential factor is the drift current, which for a large reverse bias (i.e. for a vanishing exponential) is the limiting current.
- 12.
The formalism used here is similar to the one used to develop the expression for the diffusion currents inside a semiconductor with gradually varying carrier density. However, the rather abrupt (in less than a mean free path) change in carrier density at both sides of the surface interlayer justifies the use of the Richardson-Dushman electron emission relation here.
- 13.
We assume that n c (at the metal side of the junction) remains constant and is given by Eq. (26.1).
- 14.
This approach is mathematically correct; however, one should recognize that, even though the drift velocity is limited to approximately the rms velocity in bulk semiconductors (Böer 2002, Chap. 26) resulting in a factor 1/2 in Eq. (26.50), conditions at the thin boundary layer are more complex, and need detailed studies to also become physically appropriate.
- 15.
Here we have used a general velocity v and a general field to indicate the type of relationship rather than the specific one explained in this section.
- 16.
Therefore this range is also referred to as the square root range.
- 17.
Since F(x) increases linearly with decreasing x, the product n(x)F(x) must remain constant in the DRO-range; namely nF=j n /eμ n and j n =j=const.
- 18.
We neglect here pre-breakdown effects which cause a steep increase of the current at still higher reverse bias.
- 19.
This tilting is too small to be visible in Fig. 26.6.
Bibliography
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functional (Dover, New York, 1972)
K.W. Böer, Ann. Phys. 42, 371 (1985a)
K.W. Böer, Phys. Status Solidi A 87, 719 (1985b)
K.W. Böer, Survey of Semiconductor Physics, vol. II, 2nd edn. (Wiley, New York, 2002)
K.H. Butler, Fluorescent Lamp Phosphors (University of Pennsylvania Press, University Park, 1980)
F. Seitz, The Modern Theory of Solids (McGraw-Hill, New York, 1940)
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Böer, K.W. (2013). The Schottky Barrier. In: Handbook of the Physics of Thin-Film Solar Cells. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36748-9_26
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