Lecture 1 - Introduction to Semiconductors
Introduction
This lecture aims to explain what a miconductor is by contrasting its properties with tho of conductors and insulators. Basic physical arguments will then be ud to provide an understanding of the properties. Contrasting properties of conductors, insulators and miconductors The ability to be able to conduct or not conduct electricity is an important property of all materials. By 'conducting electricity' we mean the flow of an electrical current, which results from the transport of electrically charged particles (usually electrons) through the material. The ability of a material to conduct electricity is therefore dependent upon the number of charged particles it contains and, in addition, how easy it is for the particles to move through the material. It is found experimentally that some materials (e.g. metals) are very good conductors of electricity; the are hence known as conductors. On the other hand other materials (e.g. rubber, most plastics) are extremely poor conductors and are hence known as insulators. In between the two limits are a class of materials which conduct electricity considerably less well than conductors but much better than insulators. The are the miconductors. The aim of this lecture is to understand why different materials fall into one of the three categories.
Quantitative description of electrical conduction - electrical condu ctivity
A, and length L, and apply a voltage V between the ends of the bar (e.g. using a battery) a current I will flow along the bar. For the majority of materials V and I are proportional to each other and are hence related by an equation of the form
V=IR
This equation is known as Ohm's Law and a material which obeys this equation (R is a constant) is said to be an ohmic material.
The constant in the above material, R, is known as the resistance of the bar and, for a given material, is related to the physical size of the bar by
R=L/(σA)
σ is the electrical conductivity of the material. The larger the value of σ the easier it is for the material to conduct electricity. σ is a property of the material from which the bar is constructed (but not the shape or size of the bar) although for some materials σ can vary quite rapidly with temperature.
Material dependence of electrical conductivity
Material Conductivity at 20°C (Ωm)-1
Conductors
Aluminium 3.8x107
Copper 5.8x107
Gold 4.5x107
Iron 1.0x107
Nichrome 1.0x106
Platinum 9.4x106
猫能吃狗粮吗Silver 6.3x107
Semiconductors酵母双杂
Carbon (graphite) 6.7x104
Germanium (pure) 2
Silicon (pure) 3.3x10-4
Insulators
Glass 10-7-10-10
Quartz 1.3x10-18
The above table shows values for the electrical conductivity of a range of materials. The are grouped into the three class: conductors, miconductors and insulators. Immediately apparent is the huge variation in σ, a factor of ~1025 occurring between copper and quartz. As a comparison the
ratio between the size of the Earth's orbit around the sun to that of a hydrogen atom is only ~1021. Any model we propo to explain the reason for the three class of material will also have to explain this huge variation of σ. Other differences between Semiconductors and Insulators/Conductors
In addition to their intermediate electrical conductivity, miconductors display other important differences in comparison to conductors and/or insulators. The include:
•Sensitivity to impurities. When certain impurities are added to a miconductor, even at extremely low concentrations, the conductivity
can increa by many orders of magnitude. This effect, which is known
as doping and will be discusd in detail in Lecture 4, is not obrved
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for insulators or conductors.
•Sensitivity to light. When illuminated by light of certain wavelengths the conductivity of a miconductor can increa dramatically.
如何设置wifi密码•Sensitivity to temperature. Increasing the temperature of a miconductor results in a rapid increa in its conductivity. In contrast
the conductivity of a conductor decreas rather weakly with increasing
temperature.
Chemical trends of miconductors
If we look at the periodic table we e that the common miconductors, silicon (Si), germanium (Ge) and certain forms of carbon (C) and tin (Sn) occur in the same column. In addition to the so-called elemental miconductors, it is possible to form miconductors by combining an element from column-III with an element from column-V. The III-V miconductors include GaAs, InP, InSb etc. Similarly II-VI miconductors
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The fact that all the miconductors are found in the same region of the periodic table suggests that there is an underlying physical explanation for miconducting behaviour. It is important to note that the number of electrons per atom increas with increasing atomic number Z. Hence the ability of a material to conduct electricity is not simply dependent upon the number of electrons it contains. Instead it reflects the ability of the electrons to move through the material.
The physics of isolated atoms - the Bohr model of the atom
Quantum mechanics tells us that for very small entities we can no longer describe their properties wholly in terms of a particle model but that they also display wave-like properties. If a particle has momentum p (given by mass x velocity mv) then the wavelength, λ, associated with the particle is
λ=h/p
where p is Planck's constant (6.6x10-34Js).
If we now confine the particle within a well defined region of space (e.g. within a box) then the particle can only have wavelengths such that an integer number (or whole number) of the wavelengths 'fit' into the region in which the particle is confined. This is very similar to the waves whi
ch occur when we pluck a stretched string. Only wavelengths which fit in an integer number of times into the string length are allowed.
Becau of the above formula, if only certain wavelengths are allowed then, as a conquence, only certain values of momentum are allowed. However the momentum is related to the kinetic energy of the particle, T, by
T
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mv but p mv hence
包子的英语T p m =
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hence if only certain momentum values are allowed then as a conquence only certain energies are allowed. A particle confined within a given region of
space can therefore not have any arbitrary energy (as would be the ca for a free particle) but only certain allowed, discrete or quantid values. The energy of a confined particle is hence said to be quantid.
The simplest atom, hydrogen, is formed from a single electron which orbits the nucleus consisting of
a single proton. In this ca the electron is 'confined' within the region surrounding the nucleus and its energy will be quantid. As the electron orbits the nucleus is can only do so when it has a wavelength such that an integer number of the wavelengths fits exactly into the circumference of the orbit. However, becau the energy corresponding to a given orbit depends on the orbit's distance from the nucleus (due to the electrostatic energy between the oppositely charged electron and proton) and the wavelength of the electron in turn depends upon this energy, only certain orbits where an integer number of wavelengths equal the orbit length are possible.
The electron can hence only orbit the nucleus in well defined orbits, with each orbit corresponding to a quantid energy of the atom.
Standing waves on a stretched string
The allowed wavelengths and
quantid energies of a particle
confined in a box
Waves and energy levels for a hydrogen atom
The Pauli Exclusion Principle
Electrons belong to a class of particles known as Fermions. A basic law of quantum mechanics, known as the Pauli exclusion principle, is that 'no two identical Fermions may exist in the same energy state'. A slight complication is that electrons posss a property known as spin. Any electron can spin in one of two possible ns (e.g. clockwi or anti-clockwi - generally known as up or down spin and depicted as ↑ or ↓). When spin effects are included the exclusion principle is modified to 'not more that two electrons may exist in any energy state. Within a given state the two electrons must have opposite spins’.
One conquence of the exclusion principle is that as we add electrons to a system we must gradually fill higher and higher energy levels (all the electrons can not simply be put in the lowest energy level).
For example if we add 7 electrons to the following system then the lowest total energy state of the system is
of 8 electrons.
The construction of a solid from individual atoms
The periodic arrangement of atoms in a solid
(right). The formation of energy bands in a solid
from the individual states in an atom (below).
Examples are shown for the ca of two, six and
a very large number of atoms.
The Pauli exclusion
principle controls how
electrons occupy the
states in a system.
A solid is compod of a large number of identical atoms. However first we consider just two atoms, initially an infinite distance apart. In this state identical energy levels in each atom have the same energy. However as the atoms are brought clo together their orbits start to overlap resulting in the formation of two different energy levels in the coupled system ((a) above). For the ca of six atoms their mutual interaction results in the formation of six distinct states corresponding to each original atomic state. Becau the range of energy values over which the split states extend is a function only of the paration between the atoms and not their number, the paration between the levels in the six atom ca is smaller than in the two atom ca. For a typical solid the original discrete energy levels transform into a ries of bands of levels, each band resulting from one of the atomic levels. If the solid is compod of N atoms then each band will contain N states. Although the states are still discrete there are so many of them (N is very large) that it is impossible to experimentally probe this discreteness. Each band hence appears as a continuous distribution of allowed states. For each band of states, containing N levels, each state can hold two electrons of opposite spin. The total band can therefore contain a maximum of 2N electrons.
The band structure of solids
We have en that the electronic structure of a solid consists of a ries of bands, within which there are a very large number of allowed energy levels, parated by energy regions where there are no allowed states. An electron cannot have a given energy if it falls in one of the 'gaps' between the bands. Consider now a solid in which there are sufficient electrons to completely fill a number of bands with all higher energy bands completely empty (we will e below that this requires an even number of electrons per atom). We now apply a voltage across a piece of the solid and consider if it is possible for the solid to conduct an electrical current. To conduct a current some of the electrons have to be able to move through the solid. In doing so they gain kinetic energy due to their motion. Hence electrons are only able to move if they can gain the energy required by this motion. However typical energies associated with electron motion due to current conduction are very small (a very small fraction of 1eV) whereas the energy difference between the bands is typically ~1eV. Hence for an electron in a completely filled band the motion required by electrical conduction is impossible as there are no suitable, higher energy states into which the electron can jump. The electrons are unable to gain the energy required for electrical conduction and hence cannot contribute to electrical conduction. This leads to the important conclusion: electrons in a completely filled energy band are unable to contribute to electrical conduction. As an empty band contains no electrons, it too cannot contribute to electrical conduction. Hence a solid with only completely filled or completely empty bands cannot conduct electricity, it will be an insulator.
However if a solid contains one or more bands that are only partially filled then the situation is different. In, for example, a half filled band only the lowest energy states will be filled, the top half of the states will be empty. For the electrons in the bottom half of the band there will be a large number of empty states to slightly higher energy. Electrons can jump into the levels by gaining only a very small amount of energy and hence can contribute to
