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It is claimed that Schottky type of contact between low work function p-type semiconductor and higher work function metal creates an ohmic contact in which current can flow both sides almost fluently with very low resistance. It is also claimed that electrons have tendency to flow (when no potential is applied) from low work function material to higher work function material. So, when low work function p-type semiconductor comes in contact with higher work function metal, a p-type semiconductor will not let electrons from metal enter into semiconductors and will push them back. It seems a bit counterintuitive to me, because how in this case could it be an ohmic contact? Doesn't p-type semiconductor with lower work function suppose to cause resistance when potential is applied to make electrons flow from metal to semiconductor? Furthermore, if electrons penetrate from metal to p-type semiconductor under applied potential, doesn't they suppose to recombine with holes and create a depletion zone poor for any charge carriers and further increase resistance?

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Doesn't p-type semiconductor with lower work function suppose to cause resistance when potential is applied to make electrons flow from metal to semiconductor?

If you apply a positive voltage to the p-type semiconductor, the holes in the semiconductor will want to flow into the metal. In this case, there isn't any barrier potential, since the electric field induced by diffusion opposes the flow of electrons into the metal, not holes, and thus the holes can flow freely. In the opposite case, where a negative voltage is applied to the semiconductor, the holes flow from metal to semiconductor and have to overcome the difference between the valence band energy and the fermi level, but this is small since the semiconductor is heavily p-type. This has the effect of making an Ohmic contact.

Furthermore, if electrons penetrate from metal to p-type semiconductor under applied potential, doesn't they suppose to recombine with holes and create a depletion zone poor for any charge carriers and further increase resistance?

When electrons leave the p-type semiconductor they don't leave behind ions because the p-type semiconductor is doped with acceptor ions. Instead, they leave behind a surface charge of mobile holes. This also induces a surface charge in the metal, which creates an electric field that opposes diffusion current. This causes band bending to match the fermi-levels of both materials. This happens as soon as the two materials make contact, and an applied potential isn't necessary.

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1) How electron holes can enter a metal? Does it mean that all the metals are hole conductors? Typicaly it is claimed that only some metals (for example bivalent metals) can conduct holes because they have energy overlap between s and p orbitals. What about monovalent metals such as Lithium and Sodium? There seem to be no such overlap in these metals. How can you inject a holes in the metal if it doesn't conduct holes?

2) If work function of P-type semiconductor is lower than that of a metal it contacts, it means that electrons will flow from semiconductor to metal when no potential is applied and recombination is energetically unfavorable. Does it mean there is some treshold potential you need to apply over which electrons start to fly from metal to semiconductor? For example, if work function of a metal is 2 eV higher than that of a P-type semiconductor, how many Volts do you need to make electrons flow from metal to semiconductor (what is energetically unfavorable under normal conditions)?

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A conventional example tells that when two pieces of metal come in contact, electrons will flow from the metal with lower work function to the metal with higher work function until the later will get negative charge and the electron flow will stop. In my understanding it is assumed that electron holes (positive charge carriers) take no significant role in this process.

But what if we will bring in contact a two pieces of an intrinsic (UN-doped) semiconductors with sufficient difference in work functions? Electrons will start to flow from a semiconductor A (with lower work function) to semiconductor B (with higher work function). But if semiconductor B will start to charge negatively doesn't it mean that the holes suppose start to flow in the same direction been attracted by charge imbalance? If yes, how long will this process of the electrons and the holes flow will continue and will there be a principal difference from two metals brought in contact?

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