Band Theory of Solid The energy trand structure of a solids determines another it is a conductor, an insulator or a semiconductor. No property of solid varies as widely as their ability to conduct alelctric current, Copper, a good conductor, has a relitivity of at room temperature whereas for quarty, a good insulator, , more than 25 powers of ten greater. The existence of electron energy band in solid makes it possible to understand this remarkable span. There are two ways to consider how energy band rise. (2 cara utk ketahui "energy band") The simplest is to took at what happens to the energy band of isolated atom as they are thought close and close together to form a solid. Later we will consider the origin of energy band in term of teh restrictions the periodicity of a crsytal lattice imposes on the motion of electon. The atoms in every solid, not give in metals, are so near one another that their valence electron warefunctions overlap. The greater the number of miteracting atoms, the greater the number of level produced by the mixing of their respective valence wave functions.
Figure 1
a) When two solid atoms come together closely, their 3s levels because of the overlaps of the corresponding electron wavefunctions. b) The number of new level equal the number of interesting atoms, here 5. c) When the number of the interacting atoms is very large, as in solid sodium, the result is an energy band of very closely spaced level.
The energy brand of a solid, the gaps betweeen them, and the extent to which tehy are filled by electron not only govern the electrical behaviour of the solid but also have important bearing on others of it properties. An electron in a solid can only have energies that fall within it energy brand. The various outer energy brand in a solid may overlap is in Fig.1 below, in which case it valence electrons have available a continuous distribution of permitted energies
Figure 2
in other solid the band may not overlap as in Fig.2, and the internal between them represent energies their electrons cannot have. Such internal are called forbidden band Figure 3 below shows the distribution of electron energies in a metal (eg. sodium) at various temperature. At OK all levels in the band are filled by electrons use to the Fermi energy , and those above are empty. At temperature above OK, electrons with energies below can move into higher states.
Figure 3
A sodium atom has s single 35 valence electron. Each S atomic level (l = 0) can hold electrons, so each s band formed by N atom can hold 2N electrons. Thus the 3 s band in in solid sodium is only half. Filled by electrons (See Fig.4 below) and the
Figure 4
Fermi energy EF lies in the middle of the band. When a potential difference is applied across a piece of sodium atom, 3 s electron can pick up additional energy while remaining in their original band . The additional energy is in the form of Kinetic energy and the drift of the electrons constitutes an electrical current. Sodium i therefore a good conductor, as are other solid with partly filled energy band. Magnesium atom have filled 3 s shell. Never***** Mg is a metal. How come? What actually happens is that the 3 s and 3 p bands overlap as Mg atoms come close closer together to give the structure shown in Fig 5. atomic the level can hold electrons, so a band formed by N atoms can hold 6N electrons. Together with the 2N electrons the 3 s band can hold, the band in magnesium can hold 8N electron in all. With only 2N electrons in band, it is only one-quarter filled and so Mg is a conducter. In a carbon atom the 2p shell contains only two electrons. Because a p shell can hold six electrons, we might think that the carbon is a conductor, just as sodium. What actually happens is that, although the and band that form when carbon atom come together overlap at first (as the and band in magnesium do), at smaller separatime the combined band splits into two band (Fig.5), each able to contain 4N electrons.
Figure 5
Because a carbon atom has two 2 s and two 2 p electrons, in diamond there are 4N valence electrons that completely fills the lower (or valence) band, as in Fig.6. Hence Ep for C is at the top of valence band. At least 6 eV of additional anergy must be provided to an electron in diamond of it is to climb to the conduction band where it can move about freely. With at room temperature, valence electrons in diamond do not have enough thermal energy to jump the 6eV gap.
Semiconductors Silicon has a crystal structure like that of diamond and, as in diamond, a gap separate the eop of it filled valence band above it. The forbiden band in silicon, however, is only about IeV wide. at low temperature Si is little better than diamond as a conductor, but at room temperature (~300K) a small number of it valence electrons have enough thermal energy to jump the forbidden band as enter the conduction band.
Figure 6
There electrons, through few, are still enough to allow a small amount of current to flow when an electric field is applied. Thus silicon has a resistivity intermediate between these of conductors and those y insulators, Si and others solid with similar band structure are classed as semiconducters.
Impurity Semiconductors Small amounts of impurity can drastically change the concudtivity of a semiconductors. Suppose we incorporate a few arsenic atoms in asillicon crystal. Arsenic atom have five electron in their outer shell, silicon atom have four (These shell have the configurations 4s2 4p2 and 3s2 3p2 respectively) When an arsenic atom replace a silicon atom in a silicon crystal, four of it electron participate in covalent band with it nearest neighbours. The fifth electron needs very little energy-only about 0.05eV in silicon, about 0.01eV in germanium - to be detached and move about freely in the crystal. As shown in Fig. below, arsenic as an impurity in silicon provides energy level just below the conduction band. Such band are called donor level, and the substance is called an a ntype semiconductor became electric current in it is carried by negative charge.
Figure 7
If we instead incorporate gallium atom in a silicon crystal, a different effect occur outer shell, whose configuration is 4s2 4p1 and their presence leaves vacancies called holes in the electron structure of the crystal. An electron need relatively little energy to enter a hole, but as it does so, it leaves a new hole in it former location. When an electric field is aplied of gallium electrons move towards the anode by successful filling holes. The flow of current here is conveniently described with reference to the holes, whose behaviour is like that of positive charges since they move towards the negative electron. A substance of this kind is called a p-tyoed semiconductor. In the energy-band diagram below, we see that Ga as an impurity in Si provides energy levels, called acceptor levels, just above the valence band. Any elections that occupy these levels leave behind them vacancies in the valence band that permit electric current to flow.
Figure 8
Current in an n-type semiconductor is carried by surplus electrons that do not fit into the electronic structure of a pure crystal. Current in a p-type semiconductor is carried by the motion of "holes" which are sites of missing electrons. Holes move towards the megative electrodes as a succession of electrons move into them. The doped semiconductor is called as extrinsic semiconductor. An undoped semiconductor is called an intrinsic semiconductor. Transparent: Permitting the passage of radiation without significant deviation or absorption.
Adding an impurity to a semiconductor is called doping. Phosphorus, antimony, bismuth as well as arsenic atom with five valence electrons and so can be used as donor impurities in doping silicon and germanium to yield an n-type semiconductor. Similarly, indium, tellurium as well as gallium have atoms with three valence electrons and so can be used as acceptor impurities. A minute amount of impurity can produce a dramatic change in the conductivity of a semiconductor. As an example, I part of donor impurity per 109 part of germanium incerase it's
conductivity by a factor of nearly 103. Silicon and germanium are not the only semiconducting materials with practical applications. III - V compunds are also well known semiconductors.
Optical Properties of Solids The optical properties of solids are closely related to their energy-band structure. Photons of visible light have energies in a metal can readily absorb such an amount of energy without leaving its valence band, and metals are accordingly opaque. The characteristic luster of a metal in due to the radiation of light absorbed by it's free electrons. If the metal surface is smooth, the reradiated light appears as a reflection of the original incident light. For a valence electron in an, insulator to absorb a photon energy must be over 3eV if teh electron is to jump across the forbidden band to the conduction band. Insulators therefore cannot absorb photons of visible light and are transparent. Of course, most samples are transparent, but this is due to teh scattering of light by irregularities in their structures. Insulators are opaque to ultraviolet light, whose highes frequencies mean high enough photon energies to allow alectrons to cross the forbidden band. Because of the forbiden bands in semiconductors are about the same in width as the photon energies of visible light, they are usually opaque to visible light. However they are transparent to infrared light whose lower frequencies mean photon energies too low to be asorbed. For this reason infrared lenses can be made from the semiconductor germanium, whose appearance in visible light is that of an opaque solid. The conductivity of a semiconductor incerase with temperature because thermal disturbances excite some of the electrons into the conduction band, helping the semiconductor to carry a current. This is in sharp contrast to the behaviour of metals, whose conductivity always decreases with temperature, due to the increase of "friction" (in the form of photons) with temperature. In a semiconductor, this increase of friction is more than compensated by the increase of the number of electrons participating in the conduction of current. A semiconductor carries a current not only in the conduction band, but also in the valence band. The electrons that move from the valence band to the conduction band leave behind unoccupied states, and the electrons remaining in the valence band can them take advantage (ambil kesempatan) of these unoccupied states to carry a current. The unoccupied states are called holes. Since such a hole represent a deplicit of negative change, it can be regarded as an effective positive charge. Thus, the current carried by the electrons in the valence band can be associated with the motion of holes. Roughly, the electrons in the valence band play a game of musical chairs: the electrons move in one direction, by jumping into available holes, and the holes therefore move in the opposite direction. Mathematically, it is easier to describe the current
as a flow of holes than as a flow of electrons, because there are few holes but very many electronsin the valence band.
Figure 9
nenn = nins2 for Si at 350K n =/-5x1016/
nins : Density of holes in V.B Density of electron in C.B
(Density of electron hole pairs)
ne : Density of electrons in the conduction band. nn : Density of holes in the valence band.
Example : A sample of n-type Si (at 350K) 1021 donors per
Thus 7.0 holes?
3
. What is the number of electrons per
3
? What is the number of
Solution: The number of free electons matches the number of donors, is . e
2
3
According to Equation the number of holes is given by
Thus, the number of holes is relatively small, and then contribute only little to the conductivity.
Example: A sample of p-type Si has electrons in the conduction band?
acceptors per
? What is the number of free
Solution The number of holes matches the number of acceptors, ie.,
The number of free electrons is
Now the number of free electrons is relatively small, and they contribute noly little to the conductivity. In a pure semiconductor, often called an intrinsic semiconductor, the number of electrons in the conduction band necessarily matches the number of holes in the valence band - if the density of tree electrons in the conduction band is . The density is a function of temperature. It is zero at zero temperature, but it increase with temperature. For pure Si at 300k,
