9.2 Ohmic Contacts

In our discussion of metal-semiconductor contacts，we have thus far considered cases in which the semiconductor near the metal has a lower majority-carrier density than the bulk and in which there is a barrier to electron transfers from the metal. In such cases any applied voltage is dropped mainly across the junction region，and currents are contact limited. The inverse case，in which the contact itself offers negligible resistance to current flow when compared to the bulk，defines an ohmic contact. Although this definition of an ohmic contact may sound awkward，it emphasizes one essential aspect：when voltage is applied across a device with ohmic contacts，the voltage dropped across the ohmic contacts is negligible compared to voltage drops elsewhere in the device. Thus，no power is dissipated in the contacts，and the ohmic contact can be described as being at thermal equilibrium even when currents are flowing. An important and useful consequence of this property is that all free-carrier densities at an ohmic contact are unchanged by current flow；the densities remain at their thermal-equilibrium values.

The metal-semiconductor contacts that we have considered in the previous section can，for example，be made ohmic if the effect of the barrier on carrier flow can be made negligible. In practice this is accomplished by heavily doping the semiconductor so that the barrier width xd is very small. The space-charge region，therefore，narrows as ND increases. When the barrier width approaches a few nanometers，a new transport phenomenon，tunneling through the barrier，can take place.

Fig.9.7（a）is a schematic illustration of the tunneling process through a very thin Schottky barrier. When the barrier is of the order of nanometers and the metal is biased negatively with respect to the semiconductor，electrons in the metal need not be energetic enough to surmount the barrier to enter the semiconductor. Instead，they can tunnel through the barrier into the conduction-band states in the semiconductor. Likewise，when the semiconductor is biased negatively with respect to the metal，electrons from the semiconductor can tunnel into electronic states in the metal ［Fig.9.7（b）］. Many electrons are available to take part in these processes and currents rise very rapidly as bias is applied. Hence，a metal-semiconductor contact at which tunneling is possible has a very small resistance. It is virtually always an ohmic contact. Ohmic contacts are frequently made in this way in practice. To assure a very thin barrier，the semiconductor is often doped until it is degenerate（i. e.，until the Fermi level enters either the valence or the conduction band）. In modem devices the conductivity of the semiconducting regions is higher than in previous device generations. Consequently，ohmic contacts must have even lower resistance so that no appreciable voltage drop occurs across them.

Fig.9.6 Measured values of current（plotted on a logarithmic scale）versus voltage for an aluminum-silicon Schottky barrier. Values for IS and n are obtained from an empirical fit of the data to Equation（9.1）.

Fig.9.7 Metal-semiconductor barrier with a thin space-charge region through which electrons can tunnel.（a）Tunneling from metal to semiconductor.（b）Tunneling from semiconductor to metal.

Fig.9.8（a）Idealized equilibrium energy diagram for a Schottky ohmic contact between a metal and an n-type semiconductor.（b）Charge at an ideal Schottky ohmic contact. A delta function of positive charge at the metal surface couples to a distributed excess-electron density n′（x）in the semiconductor.（c）Field，and（d）potential at an idealized Schottky ohmic contact. The Debye length LD is a characteristic measure of the extent of the charge and field.

Another method of obtaining an ohmic contact is to cause the majority carriers to be more numerous near the contact than they are in the bulk of the semiconductor. An ohmic contact of this type results if the semiconductor surface is not depleted when it comes into equilibrium with the metal，but rather has an enhanced majority-carrier concentration. Using the ideal Schottky theory，we see that this condition occurs in a metal-semiconductor junction between a metal and an n-type semiconductor with a larger work function than that of the metal. In this case，electrons are transferred to the surface of the semiconductor and the metal is left with a skin of positive charge. For an ohmic contact to a p-type semiconductor，the relative sizes of the work functions in the two regions need to be reversed to achieve a net positive charge in the semiconductor and，thereby，an enhanced hole density near the contact.

The relevant energy diagram is sketched in Fig.9.8（a），and the corresponding charge and field diagrams are given in Fig.9.8（b）and 9.8（c）. There is a qualitative similarity between these figures and Figs.9.4（a）to 9.4（c）that referred to a rectifying contact；the important distinction between the two situations is that the semiconductor charge consists of free electrons in the case of the ohmic contact，but it is fixed（on positive donor sites）in the barrier case. The charge distribution，field，and potential for such a contact can be calculated using techniques similar to those employed for the rectifying contact.