Ohmic Contact - Theory

Theory

The Fermi level (or strictly speaking, electrochemical potential) of any two solids in contact must be equal in thermal equilibrium. The difference between the Fermi energy and the vacuum level is termed the work function. A contact metal and a semiconductor can have different work functions, denoted and respectively. If so, when the two materials are placed in contact, electrons will flow from the one with the lower work function until the Fermi levels equilibrate. As a result, the material with the lower work function will take on a slight positive charge while that with the higher work function will become slightly negative. The resulting electrostatic potential is termed the built-in potential designated by . This contact potential will occur between any two solids and is the underlying cause of phenomena such as rectification in diodes. The built-in field is the cause of band bending in the semiconductor near the junction. Noticeable band-bending does not occur in most metals since their very short screening length means that any electrical field extends only a short distance beyond the interface.

In a classical physics picture, in order to surmount the barrier, a carrier in the semiconductor must gain enough energy to jump from the Fermi level to the top of the bent conduction band. The needed barrier-surmounting energy is the sum of the built-in potential and the offset between the Fermi level and the conduction band. Equivalently for n-type semiconductors, where is the semiconductor's electron affinity, defined to be the difference between the vacuum level and the conduction band (CB) level. For p-type materials, where is the bandgap. When the excitation over the barrier is thermal, the process is called thermionic emission. An equally important process in real contacts is quantum mechanical tunneling. The WKB approximation describes the simplest picture of tunnelling in which the probability of barrier penetration is exponentially dependent on the product of the barrier height and thickness. In the case of contacts, the thickness is given by the depletion width, which is the length scale that the built-in field penetrates into the semiconductor. The width of the depletion layer can be calculated by solving Poisson's equation and considering the presence of dopants in the semiconductor:

where in MKS units is the net charge density and is the dielectric constant. The geometry is one-dimensional since the interface is assumed to be planar. Integrating the equation once, approximating the charge density as being constant over the depletion width, we get

The constant of integration due to the definition of the depletion width as the length over which the interface is fully screened. Then

where the fact that has been used to fix the remaining integration constant. This equation for describes the dashed blue curves in the right-hand panels of the figures. The depletion width can then be determined by setting which results in

For 0 < x < W, is the net charge density of ionized donor or acceptors in the completely depleted semiconductor and is the electronic charge. and have positive signs for n-type semiconductors and negative signs for p-type semiconductors giving the positive curvature for n-type and negative curvature for p-type as shown in the figures.

Note from this crude derivation that the barrier height (dependent on electron affinity and built-in field) and barrier thickness (dependent on built-in field, semiconductor dielectric constant and doping density) can only be modified by changing the metal or changing the doping density. It can be noticed that the Fermi level pinning is roughly at the same energy within the forbidden gap for both n and p type Si (i.e. the sum of φbn and φbp is approximately Eg), suggesting that interface and structural factors pin the Fermi level because of a very high density of interface states (Fig. 11). Note that for ohmic contacts we never need worry about the occupancy of these states changing, because of very small potential drop across the contact. In general an engineer will choose a contact metal to be conductive, non-reactive, thermally stable, electrically stable and low-stress, and then will increase the doping density below the contact to narrow the width of the barrier region. The highly doped regions are termed or depending on the carrier type. Since the transmission coefficient in tunneling depends exponentially on particle mass, semiconductors with lower effective masses are more easily contacted. In addition, semiconductors with smaller bandgaps more readily form ohmic contacts because their electron affinities (and thus barrier heights) tend to be lower.

The simple theory presented above predicts that, so naively metals whose work functions are close to the semiconductor's electron affinity should most easily form ohmic contacts. In fact, metals with high work functions form the best contacts to p-type semiconductors while those with low work functions form the best contacts to n-type semiconductors. Unfortunately experiments have shown that the predictive power of the model doesn't extend much beyond this statement. Under realistic conditions, contact metals may react with semiconductor surfaces to form a compound with new electronic properties. A contamination layer at the interface may effectively widen the barrier. The surface of the semiconductor may reconstruct leading to a new electronic state. The dependence of contact resistance on the details of the interfacial chemistry is what makes the reproducible fabrication of ohmic contacts such a manufacturing challenge.