Semiconductor theory

Introduction

Semiconductors are a special class of materials with properties in between those of insulators and conductors.
They typically have narrow bandgaps of < 2 eV
Semiconductors are generally covalently bonded substances.
The elements Si and Ge are intrinsic semiconductors.
Si has a band gap of 1.1 eV while that for Ge is 0.55 eV.
Modern synthetic semiconductors include the III-V compounds such as GaAs, GaP, InSb, InAs and InP. These have a range of different band gaps
Carbon, silicon, germanium and tin are atoms in ascending order of atomic number from column IV A of the period table. Each is characterised by having four valence electrons in its outermost shell of electrons, and requires a further four to make up the full complement of the shell. All can solidify to form elemental, covalently bonded crystals where the four valence electrons of one atom are shared between its four nearest neighbours so that every atom effectively gains eight electrons in its valence shell. A group IV atom and its four nearest neighbours from a tetrahedron as shown in Figure 1.

Figure 1: Schematic diagram to show the orientation of covalently bonded group 4 atoms. A tetrahedron is formed by the nearest neighbours, with the principal atom located at its centre.

Taking a larger scale perspective of the arrangement of the atoms, or crystal lattice , it is found that they organise themselves into two interpenetrating face centred cubic (fcc) sub-lattices, one displaced from the other by 1/4(a 0 , a 0 , a 0 ) along a diagonal of the unit cell . a 0 is called the lattice constant or lattice parameter and is a measure of the size of the unit cell, often expressed in Angstrom (A) units (1A=0.1nm or 1x10 -10 m). It is determined by techniques such as X-ray diffractometry. Figure 2 shows a complete unit cell for a group 4 crystal covalently bonded with the diamond structure.

Figure 2: Unit cell of a crystal such as silicon or germanium

This structure is of course difficult to visualise and draw, hence it is usually represented by an equivalent 2-D, "square" arrangement shown in figure 3.

Figure 3: 2-D representation of a covalently bonded crystal at 0K, eg Si. Note that the heavy lines between adjacent atoms depict the covalent bonds which contain TWO electrons and are all completely filled.

In the example shown, it is assumed that the solid, Si say, is both crystallographically perfect and pure. At 0 K, all the covalent bonds are complete and there are no free charge carriers moving around randomly through the lattice; the crystal is an insulator.
Before commenting further on the elemental "semiconductors", it is worth mentioning another group of technologically important solids which possess semiconducting properties to varying degrees, namely the III-V compounds . These are formed when equal numbers of group III and group V elements combine with the same basic arrangement as the group IV elemental solids. The difference lies in the fact that whereas the elemental solids contain only one type of atom such that every atom in the (perfect) lattice is bonded to four identical nearest neighbour atoms, in the III-V compounds a group III atom is bonded to four group V nearest neighbours, and a group V atom is bonded to four group III nearest neighbours. The two interpenetrating fcc sub-lattices now contain either all group III atoms or all group V atoms. Figures 4 and 5 show the 3-D and 2-D representations, respectively, for these materials.

Figure 4 : Diagram to show the 3D unit cell of a III-V semiconductor compound (eg. GaAs, gallium arsenide) with the zinc blende lattice.

Figure 5 : 2-D representation of a III-V semiconductor. Note the way in which the group III and V alternate through the lattice on their individual fcc sub-lattices.

Although the Silicon Devices and Technology 3 course will concentrate on Si, it is important to remember that the electronic band structure of Si (and Ge) makes it unsuitable for certain applications; a prime example is light emitting devices. Most semiconductor lasers and LEDs are made from the III-V materials; it is NOT possible to get efficient light emission from Si. (There is now a great deal of interest world-wide in the II-VI compounds because it has been shown that blue and green LEDs and laser diodes can made from them, but that's yet another story!).

Silicon doped with group V impurity atoms � n type semiconductor

As a first example of the effect of impurities, assume that a small concentration of group V atoms is added to the host Si crystal at the manufacturing stage. These atoms will substitute for Si atoms in the lattice. Since they have five rather than four valence electrons, one electron for each impurity atom is unused in the bonding. At 0 K the spare electron remains in the vicinity of its parent atom by virtue of Coulombic attraction to the one remaining uncompensated proton in the nucleus of the impurity. The single, unbonded electron effectively orbits the impurity in much the same way that an electron orbits an isolated hydrogen nucleus. This is shown in schematic form in figure 7; it is easily shown that the radius of the electron orbit is large, extending over many unit cells. As a consequence, the binding energy is low and only a small amount of "external" energy is required to liberate the electron from its parent impurity atom. Therefore, the electron becomes a free, conduction band electron at relatively low temperatures as depicted in figure 8. However, this time note that the release of the electron by the impurity does NOT result in the generation of a hole. The impurity, called a donor , becomes ionised (positive charge) but is locked into the lattice and unable to move. Nonetheless, it can still effect the motion of electrons when an electric field is applied to the semiconductor and a current flows. This will be discussed in more detail later in the course. Generally, it can be assumed that the electron concentration at room temperature in n -type semiconductor is just equal to the total concentration of all donors since the overwhelming majority will be ionised;

and the concentration of donors which remain neutral (un-ionised) at 300 K , N 0 D = 0.

Figure 7 : Silicon at T = 0K containing a trace concentration of group V impurity atoms. There are no free charges so the crystal is still an insulator.

Figure 8: Silicon at T > 0K, with the group V impurities ionised and free electrons available for conduction.

Silicon doped with group III impurity atoms � p-type semiconductor

Consider now the case when a small concentration of group III atoms is added to the host Si crystal at the manufacturing stage. These atoms will also substitute for Si atoms in the lattice. Since they have three rather than four valence electrons, one of the four covalent bonds associated with the impurity is unfilled. At 0 K an unfilled bond remains in the vicinity of the impurity atom through Coulombic attraction between the negative charge acquired by the impurity on completing its complement of four valence bonds and the subsequent appearance of a positive charge on a nearby Si atom which must sacrifice an electron from one of its covalent bonds. Rather than thinking in terms of unfilled covalent bonds, it is usual to treat the problem as one of a positive charge, called a hole , orbiting the negatively charged acceptor impurity to form a hydrogenic-like system. The two alternative pictures are shown in figure 9. Again, the radius of the hole orbit is large, extending over many unit cells. As a consequence, the binding energy is low and only a small amount of "external" energy is required to liberate the hole from its acceptor impurity atom. Therefore, the hole becomes a free, valence band hole at relatively low temperatures as depicted in figure 10. Note that the release of the hole by the acceptor impurity does NOT result in the generation of an electron. Rather, the acceptor becomes ionised (negative charge) but is locked into the lattice and unable to move. It too can influence the motion of charge when an electric field is applied to the semiconductor and a current flows. Generally, it can be assumed that the hole concentration at room temperature in p -type semiconductor is just equal to the total concentration of all acceptors since the overwhelming majority will be ionised;

and the concentration of acceptors which remain neutral (un-ionised) at 300 K , N 0 A = 0

Figure 9: Silicon at T = 0K containing a trace concentration of group III impurity atoms. The hole is bound to the impurity atoms. The hole is bound to the impurity by Coulombic attraction.