Semiconductor Work Function

May 16, 2020

Before discussing the behaviour of a metal-semiconductor boundary, it is first necessary to introduce the concept of the work function. The work function of a material is the energy required to remove an electron from the level of the chemical potential and give it enough energy to escape to infinity and arrive there with zero energy.

Albert Einstein first proposed the concept of the work function in his work on the in metals. It was for this work, rather than for his work on relativity, that Einstein was awarded the Nobel prize in 1921.

When a metal and a semiconductor are joined, two possible types of contact can result, depending on the combination of metal and semiconductor used. The contact may be rectifying, which only allows current to pass in one direction. Alternatively, it could be ohmic, in which case current can pass in either direction. Here we will discuss the rectifying contact, sometimes called the Schottky barrier contact.

The schematic below shows a metal and an n-type semiconductor. The work functions of the metal and semiconductor have been labelled ΦM and ΦS respectively. The dashed line at the top represents the zero of energy at infinity.

The chemical potential of the semiconductor is represented by the dotted line labelled μ. For the metal, the chemical potential is taken to be at the Fermi energy (also known as the Fermi level), labelled EF – at low temperatures the difference between the chemical potential and the Fermi energy can be neglected (see the section in this TLP entitled The Fermi-Dirac Distribution). The energy difference labelled as χ between the bottom of the conduction band and infinity is called the electron affinity of the semiconductor.

It should be noted that the above schematic is different to the illustrations showing energy bands plotted against momentum. Any horizontal change in a diagram such as that shown above represents a variation in the energy bands with position in real space.

In the metal the electron affinity is the energy which would be released if an electron were to fall in from infinity to the highest unoccupied energy level of the material. Hence, for a metal, the electron affinity is identical to the chemical potential or Fermi energy. This is also why the electron affinity for a semiconductor is the energy difference between the bottom of the conduction band and infinity.

The schematic above is an interactive animation. By clicking the button in the bottom right hand corner, the metal and the semiconductor are brought into contact. When contact is made, electrons can move between the metal and the semiconductor. The chemical potential can be interpreted as the free energy per electron, so because ΦM > ΦS, the electrons in the conduction band of the n-type semiconductor can lower their energy by moving from the semiconductor into the metal.

Take a moment to navigate through the whole of the animation using the buttons at the bottom of the animation. As you read further through this section, keep returning to this animation as more is gradually explained.

The flow of electrons into the metal from the semiconductor on contact in this animation causes a slight negative charging of the metal, thereby repelling further flow of electrons from the conduction band of the semiconductor into the metal. The electric potential generated by the charging of the metal causes a deformation of the energy bands in the semiconductor close to the metal-semiconductor interface. The chemical potential in the semiconductor also falls in this region, as higher energy electrons in the region adjacent to the metal have moved into the metal. The Fermi level of the metal is not appreciably affected because there are over 1010 times more valence electrons in the metal then there are conduction electrons in the semiconductor before contact – the addition of a few extra electrons from the semiconductor clearly makes little difference to the Fermi level in the metal.

Far from the metal-semiconductor interface, the band structure in the semiconductor is unaffected except for an overall displacement down in energy. The way to think about this change in the band structure is to consider the bands in the semiconductor to be 'pinned' at the interface with the metal. The chemical potential in the region of semiconductor immediately adjacent to the metal moves in order to be in equilibrium with the metal. This has also to be true at distances in the semiconductor remote from the interface. At a large distance from the interface, the bands therefore have the same position relative to the shifted chemical potential as they did before the metal and semiconductor were joined.

The most immediate consequence of making contact with the metal is that a region near to the metal-semiconductor interface is produced in the semiconductor which has no conduction electrons in it – this region is depleted of electrons in the conduction band. This region is therefore called the depletion layer, or the space charge region. This region is labelled in one of the intermediate steps of the illustration.

The depletion layer acts like a potential barrier. This barrier has a height of ΦM − χ for electrons moving from the metal to the semiconductor. The size of the barrier for electrons moving from the semiconductor to the metal is ΦM − ΦS, which is also equal to the total downward displacement of the bands in the semiconductor remote from the metal-semiconductor interface.

Source: www.doitpoms.ac.uk
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