Ph.D. thesis by Shabbir A Bashar

**2.2.1 Theory of Rectifying Metal Semiconductor Contacts**

The basic theory of these contacts is outlined in the following material. A more comprehensive version was recently reviewed by Rhoderick [19,20]. Figure 2.2 shows a schematic of the band structure of an unbiased metal semiconductor contact.

The Schottky-Mott theory is expressed as follows:

fbo = fm - cs | (eqn. 2.1) |

where,

fbo
= contact barrier height, at zero applied bias

fm
= work function of the metal

cs =
electron affinity of the semiconductor and is further expressed:

cs = fs - (Ec - Ef) | (eqn. 2.2) |

where,

fs
= work function of the semiconductor

Ec = conduction band energy, in eV

Ef = Fermi energy level, in eV

Figure 2.2: Unbiased band structure of a metal/n-type semiconductor contact

Vbi = fm - fs = fbo - (Ec - Ef) |
(eqn. 2.3) |

This theory is rather simplistic in the sense that it assumes ideal conditions where dipole surface contributions to the barrier height and the electron affinity are thought to be unchanged when the metal and the semiconductor are brought into contact. It also assumes that there are no chemical reactions or physical strains created between the two when they are brought into contact.

In practice, however, surface dipole layers do arise. This is because at the surface of a solid the atoms have neighbours on one side only. This causes a distortion of the electron cloud belonging to the surface atoms, so that the centres of the positive and negative charge do not coincide. It was discovered that fbo does not depend on fm in contradiction to (eqn. 2.1). Thus the assumption of constancy of the surface dipole cannot be justified.

One of the first explanations for the departure of experiment from this theory was given in terms of localised surface states or "dangling bonds". The surface states are continuously distributed in energy within the forbidden gap and are characterised by a neutral level, fo, such that if the surface states are occupied up to fo and empty above fo, the surface is electrically neutral.

In general, the Fermi level does not coincide with the neutral level. In this case, there will be a net charge in the surface states. If, in addition (and often in practice due to chemical cleaning of the semiconductor prior to processing) there is a thin oxide layer between the metal and the semiconductor the charge in the surface states together with its image charge on the surface of the metal will form a dipole layer. This dipole layer will alter the potential difference between the semiconductor and the metal. Thus the modification to the Schottky-Mott theory is expressed as follows [19]:

fbo = g(fm - cs) + (1 - g)(Eg - fo) | (eqn. 2.4) |

where,

Eg = bandgap of the semiconductor, in eV

fo
= position of neutral level (measured from the top of the valence band) and,

g = ei /(ei + qdDs) | (eqn. 2.5) |

where,

ei
= permittivity of oxide layer

d = thickness of oxide layer

Ds = density of surface states

Hence if there are no surface states, Ds = 0 and g = 1 and (eqn. 2.4) becomes identical to (eqn. 2.1) (the original Schottky-Mott approximation). But if the density of states is very high, g becomes very small and fbo approaches the value (Eg - fo). This is because a very small deviation from the Fermi level from the neutral level can produce a large dipole moment, which stabilises the barrier height by a negative feedback effect [19,20]. When this occurs, the Fermi level is said to be "pinned" relative to the band edges by the surface states.

**2.2.2 Current transport mechanisms in the Schottky diode**

The current transport through the device by emission over the barrier is essentially a two step process: first, the electrons have to be transported through the depletion region, and this is determined by the usual mechanisms of diffusion and drift; secondly, they must undergo emission over the barrier into the metal, and this is controlled by the number of electrons that impinge on unit area of the metal per second. This is expressed in (eqn. 2.6):

I = AA**T2 . exp(-qfbo/kT) . (exp{-qVeff/nkT} - 1) | (eqn. 2.6) |

where,

A = cross-sectional area of the metal/semiconductor interface

A** = Modified Richardson constant for metal/semiconductor interface

T = temperature in kelvins

k = Boltzmann constant

q = electronic charge

Veff = effective bias across the interface

n = ideality factor

The ideality factor, n, in (eqn. 2.6) gives a measure of the quality of the junction which is highly process dependent. For an ideal Schottky junction, n = 1. In practice, however, larger values are obtained due to the presence of non-ideal effects or components to the current through the junction. This mode of current transport is commonly referred to as the "thermionic emission" current [21,22]

*2.2.2.1 Other current transport mechanisms *

*2.2.2.2 *fbo
*Barrier Lowering due to Image Force Effects *

The application of an electric field causes the image-force-induced lowering of the potential energy for charge carrier emission [23]. Consider an electron, in vacuum, at a distance x from a metal surface. A positive charge will be induced on the metal at a distance -x from its surface and will give rise to an attractive force between the two, known as the image force. This force has associated with it an image potential energy which corresponds to the potential energy of an electron at a distance x from the metal. When an external field, Eext, is applied, together these two energy components have the effect of lowering the Schottky barrier. Thus at high fields, the Schottky barrier is considerably lowered.

These results can be applied to a metal/semiconductor junction. However, the field is now replaced by the maximum field, Emax, at the interface. The amount of reduction due to the induced-image-force, Df, is given by [23]:

Df = Ö(qEmax/4peoer) | (eqn. 2.7) |

*2.2.2.3 Generation-Recombination Effects *

Generation-recombination effects within the depletion region give rise to a parallel component to the thermionic emission current transport mechanism. This is particularly significant at moderately low temperatures (175K to 235K). The current contribution, Igr, due to this mechanism can be represented by [22]:

Igr = Igro (exp{qVapp/2kT} - 1) | (eqn. 2.8) |

where,

Igro = generation-recombination saturation current and is given by:

Igro = qnixdepA/2to | (eqn. 2.9) |

where,

ni = intrinsic carrier concentration of the semiconductor

xdep = depletion width

to
= effective carrier lifetime within the depletion width

*2.2.2.4 Current due to Quantum Mechanical Tunneling*

For a moderately to heavily doped semiconductor or for operation at low temperatures, the current due to quantum mechanical tunneling of carriers through the barrier may become the dominant transport process [21,24]. For all except very low biases, the tunneling current, Itn, can be represented by:

Itn = Itno (exp{qVapp/Eo} - 1) | (eqn. 2.10) |

where,

Itno = tunneling saturation current

Eo = tunneling constant

The tunneling saturation current is a complicated function of temperature, barrier height and semiconductor parameters. In the notation of Padovani and Stratton [21], Eo is given by :

Eo = Eoo coth(Eoo/kT) | (eqn. 2.11) |

where,

Eoo = a tunneling parameter inherently related to material properties of the
semiconductor and is expressed as:

Eoo = (qh/2p) . Ö(ND/m*neoer) | (eqn. 2.12) |

where,

h = Planck's constant

ND = impurity doping concentration

m*n = effective mass of electron

Ilk = Vapp/Rlk | (eqn. 2.13) |

© 1998: Shabbir A. Bashar (in accordance with paragraph 8.2d, University of London Regulations for the Degrees of M.Phil. and Ph.D., October 1997). The Copyright of this thesis rests with the author, and no quotation from it or information derived from it may be published without the prior written consent of the author.

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