9.1 Schottky Contacts

One of the first practical semiconductor devices used in the early 1900s was the metal-semiconductor diode. This diode，also called a point contact diode，was made by touching a metallic whisker to an exposed semiconductor surface. Now，the Schottky barrier diode is actually a variation of the point-contact diode in which the metal semiconductor junction is a surface rather than a point contact. In fact，a large contact area between the metal and the semiconductor in a Schottky barrier diode provides some advantages over the point-contact diode. Lower forward resistance and lower noise generation are the most important advantages of the Schottky barrier diode. The applications of a Schottky barrier diode are similar to those of the point-contact diode. The low noise level generated by Schottky diodes makes them especially suitable for uses in microwave receivers，detectors，and mixers. The Schottky barrier diode is sometimes called the hot electron or hot carrier diode because the electrons flowing from the semiconductor to the metal have a higher energy level than electrons in the metal. The effect is the same as it would be if the metals were heated to a higher temperature than normal.

9.1.1 Schottky Contacts in Equilibrium

The most distinctive characteristic of the electronic energy states of the metal and the semiconductor is the relative positions of the Fermi levels within the densities of allowed states g（E）. In the metal，the Fermi level is immersed within a continuum of allowed states，while in a semiconductor，under usual circumstances，the density of states is negligible at the Fermi level. Plots of g（E）versus energy for idealized metals and semiconductors are shown in Figs.9.1（a）and 9.1（b）.

Fig.9.2 shows the energy levels for the metal gold and the semiconductor silicon. The difference between vacuum level E0 and metal Fermi level EF is called the work function，usually given the symbol qWin energy units and often listed as W in volts for particular materials. The work function for various metals is shown in Fig.9.3. The minima of the work function exist for group-I elements. Since the electron density in the metal conduction band is very high，the position of the metal Fermi level does not change considerably when charge is exchanged between the metal and the semiconductor.

In the case of the semiconductor，the difference between E0 and EF is a function of the dopant concentration of the semiconductor，because EF changes position within the gap separating Ev and Ec as the doping is varied. The difference between the vacuum level and the conduction-band edge is，however，a constant of the material. This quantity is called the electron affinity，and is conventionally denoted by qχin energy units. Tables of χ in volts exist for many materials.（The symbol χ is a Greek capital letter chi .）

The choice of E0 as a common energy reference makes clear that if Wm is less than Ws and the materials do not interact，an electron in the metal has，on average，a total energy that is higher than the average total energy of an electron in the semiconductor. On the contrary，if Wm is greater than Ws，the average total energy of an electron in the semiconductor is higher than it is in the metal. For the sake of discussion，we consider the latter case where Wm > Ws. When an intimate contact is established，the disparity in the average energies can be expected to cause the transfer of electrons from the semiconductor into the metal.

To construct a proper band diagram for the metal and the semiconductor in thermal equilibrium，we need to note two additional facts. The first is that the vacuum level E0 must be drawn as a continuous curve. This is because E0 represents the energy of a "just-free" electron and thus must be a continuous，single-valued function in space. Second，we note that electron affinity is a property associated with the crystal lattice like the forbidden-gap energy. Hence，it is a constant in a given material. Considering these three factors：constancy of EF，continuity of E0，and constancy of χ in the semiconductor，we can sketch the general shape of the band diagram for the metal-semiconductor system. The sketch is given in Fig.9.4（a）for an n-type semiconductor for which Wm > Ws.

Fig.9.4（a）indicates that electrons at the band edges（Ec and Ev）in the vicinity of the junction in the semiconductor are at higher energies than are those in more remote regions. This is a consequence of the transfer of negative charge from the semiconductor into the metal. Because of the charge exchange，there is a field at the junction and a net increase in the potential energies of electrons within the band structure of the semiconductor. The free-electron population is thus depleted near the junction，as indicated by the increased separation between Ec and EF at the surface compared to that in the bulk.

Before considering the electrical properties of the metal-semiconductor junction we note that our development thus far has relied on the important idealization that the basic band structures of the two materials are unchanged near their surfaces.

The charge and field diagrams for an ideal metal-semiconductor junction are sketched in Figs.9.4（b）and 9.4（c）. To the extent that the metal is a perfect conductor，the charge transferred to it from the semiconductor exists in a plane at the metal surface. In the idealized n-type semiconductor，positive charge can consist either of ionized donors or of free holes while electrons make up the negative charge. We have made several assumptions about the semiconductor charge in drawing Figs.9.4（b）and 9.4（c）. First，the free-hole population is assumed to be everywhere so small that it need not be considered；second，the electron density is much less than the donor density from the interface to a plane at x=xd . Beyond xd，the donor density ND is taken to be equal to n. These assumptions make up what is usually called the depletion approximation. Although they are not precisely true，they are generally sufficiently valid to permit the development of very useful relationships.

9.1.2 Schottky Contacts Under Applied Bias

Up to this point，we have been considering thermal equilibrium conditions at the metal-semiconductor junction. Now we add an applied voltage and consider the resulting nonequilibrium condition. We saw in Fig.9.4（a）that there is an abrupt step in allowed electron energies at the metal-semiconductor interface. This step makes it more difficult to cause a net transfer of free electrons from the metal into the semiconductor than it is to obtain a net flow of electrons in the opposite direction. There is a barrier of qφB electron volts between electrons at the Fermi level in the metal and the conduction band states in the semiconductor near its surface ［Fig.9.4（a）］. To first order this barrier height is independent of bias. Referring to Fig.9.4（c），we see that the voltage drop across the near delta function of space charge in the metal（equivalent to the area between the E-field curve and the axis）is effectively zero in equilibrium；that is，no voltage drop can be sustained across the metal.

The total voltage drop across the space-charge region φi occurs within the semiconductor，as can be seen in Fig.9.4（a）. An applied voltage is similarly dropped entirely within the semiconductor and alters the equilibrium-band diagram ［Fig.9.4（a）］ by changing the total curvature of the bands，modifying the potential drop from φi. Thus，electrons in the bulk of the semiconductor at the conduction-band edge are impeded from transferring to the metal by a barrier that can be changed readily from its equilibrium value qφi by an applied bias. The barrier is reduced when the metal is biased positively with respect to the semiconductor，and it is increased when the metal is more negative.

Energy-band diagrams for two cases of bias are shown in Fig.9.5（a）and 9.5（b）. Because these diagrams correspond to nonequilibrium conditions，they are not drawn with a single Fermi level. The Fermi energy in the region from which electrons flow is higher than is the Fermi energy in the region into which electrons flow. Currents，of course，move in the direction opposite to the electron flow. To investigate bias effects on the barrier，we consider the semiconductor to be grounded and take forward bias to correspond to the metal electrode being made positive. The applied voltage is called Va＞ and the bias polarity is indicated in Fig.9.5（c）. Under reverse bias the metal is negatively biased（Va＜0）.

The basic dependence of current on voltage in a Schottky-barrier diode can be deduced from qualitative arguments. These arguments provide fundamental insight into the nature of the equilibrium behavior of the metal-semiconductor system.

The band diagram at thermal equilibrium shown in Fig.9.4 is the starting point for the derivation. At equilibrium，the rate at which electrons cross over the barrier into the semiconductor from the metal is balanced by the rate at which electrons cross the barrier into the metal from the semiconductor. From the discussion of diffusion in previous lessons，we know that free carriers in crystals are constantly in motion because of their thermal energies. For example，this fact was used to show that a density n0 of free carriers in thermal motion can be considered to cause a current density equal toqn0 Vth/4 in an arbitrary direction. At thermal equilibrium，of course，this current density is balanced by an equal and opposite flow，and there is zero net current. Applying this concept to the boundary plane of the band diagram in Fig.9.4，we see that there is a tendency of electrons to flow from the semiconductor into the metal and an opposing balanced flux of electrons from the metal into the semiconductor. These currents are proportional to the density of electrons at the boundary. When a bias Va is applied to the junction as in Fig.9.5，the potential drop within the semiconductor is changed，and we can expect the flux of electrons from the semiconductor toward the metal to be modified. We can，therefore，approximate the current-voltage characteristic by

where J′S is independent of voltage and n is taken to be a parameter having a value that is usually found experimentally to be between 1. 02 and 1.15. Experimental measurements for a forward-biased，aluminum-silicon Schottky barrier are shown in Fig.9.6. The good fit between the measured data in Fig.9.6 and Equation（9.1）with n=1. 07 is typical.