Due to the quantum nature of electrons, one energy state can be occupied only by one electron.

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4 In crystalline solids, not all values of the electron energy are possible. The allowed intervals of energy are called allowed bands (shown as blue and chess-board blue). The forbidden intervals are called forbidden bands (or gaps) (the white strip between the blue strips). Due to the quantum nature of electrons, one energy state can be occupied only by one electron. At T = 0, the electrons occupy the states with the lowest possible energies. The energy up to which the states are occupied is called Fermi energy (Fermi level) and is denoted E F. Thus, at T = 0, the probability to find an electron in a state of energy E (distribution function of the electrons) is a step function centred at E F. In semiconductors, the number of electrons is just enough to completely fill a certain amount of allowed bands. So that in this case, at T = 0, one has only completely filled bands and absolutely empty allowed bands. The highest filled band is called valence band. The lowest empty band is called conduction band. At a finite temperature, the distribution function of electrons has the shape of a smeared step function. It is called Fermi - Dirac function. It is centred at E F which have the meaning of the energy at which the probability to find an electron is ½. At finite temperatures, due to the smearing of the distribution function there are some electrons in the conduction bands and there are some empty places (holes) in the valence band.

5 In a typical situation in semiconductors, the number of electrons in the conduction band is much smaller that the total amount of the available places. Thus, in the conduction band, all the electrons have energies which are very close to that of the bottom of this band E C. The density of electrons in the conduction band at a finite temperature can be calculated as the product: the probability to have a electron in a state with the energy E C times the number of available states per unit volume. The latter variable is called density of states. All holes are concentrated at the top of the valence band. They represent missing electrons of the energy corresponding to the top of this band E V. The number of holes is calculated as the number of missing electrons. As mentioned, the number of electrons in the conduction band and holes in the valence band is typically much smaller than the total amount of places in these band. Mathematically that means that the Fermi level is never too close neither to E V nor to E C so that E C E F >>kt and E F -E V >> kt. At room temperature kt = ev. ev is an energy unit equal to the energy that an electron acquires when passing a potential difference of 1 V. Typically in semiconductors E C E v is about 1 V.

6 A change of the Fermi level, E F, leads to a variation in the concentrations of the electrons and holes, n and p, however, the product np does not depend on E F. This makes the mass action law. The mass action law holds only when the concentrations of the electrons and holes correspond to the equilibrium distribution functions. It holds with a good accuracy as far as the Fermi level is not too close to the conduction or valence band which is usually the case.

7 Intrinsic semiconductor is a pure semiconductor which is absolutely stoichiometric and contains no impurities.

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9 In intrinsic semiconductors, the Fermi level is very close to the middle of the forbidden gap.

10 Substitution of some atoms of the semiconductor with atoms of other elements influences the amount of electrons and holes in the system. A substitution with an element having a more positive valence brings extra electrons. The substitution actually brings new states in the forbidden gap with electrons sitting on these. If these states are close enough to E C, the new electrons readily go to the conduction band. A doping donor atom is typically positive charged since it has given an electron to the conduction band. Such a substitution is called donor doping.

11 In donor doped semiconductors, the concentration of electrons in the conduction band, n, is much greater than that of intrinsic semiconductors, n i. The increase of n results in a strong reduction of the hole concentration, p (due to the mass action law). When a donor-impurity atom looses electrons (ionized impurity), it becomes positively charged. From the electroneutrality condition: the charge of electrons = the charge of ionized impurities (the charge of holes in this case is much smaller than that of the electrons and can be neglected). This increase of n with doping means that the Fermi level moves closer to E C.

12 A substitution with an element having a less positive valence brings about new states in the forbidden band. If these states are close enough to the valence band, they are readily occupied by electrons from this band. These electrons, when leaving the valence band, create extra holes in it. A doping acceptor atom is typically negatively charged since it has received an electron from the valence band. Such a substitution is called acceptor doping.

13 In the case of acceptor doped semiconductor, the hole concentration in the valence band, p, is much greater than that in the intrinsic semiconductor, n i. The increase of p results in a strong reduction of the concentration of n (due to the mass action law). Every acceptor-impurity atom with an electron sitting on it (ionized impurity) becomes negatively charged. Thus, the electroneutrality condition becomes: the charge of holes = the charge of ionized impurities (the charge of electrons being neglected). This increase of p with doping corresponds to a shift of the Fermi level towards to E V.

14 If the valence band is completely filled, no electric transport is possible in the system, since there are no empty places with energies close to those of the available electrons. Since, one place can be occupied only by one electron, without these empty places none of the electrons can change its state and no motion in the system is possible. For this reason, the electrical transport in semiconductors is possible only due to a small amount of electrons in the conduction band and a small amount of empty places (holes) in the valence band. Electric field, E, acting on the negative charge of an electron causes its movement in the opposite direction. A constant field corresponds to a force applied to the electron but, due to scattering against thermally vibrating host atoms and impurities, the electron does not accelerate but rather moves with a constant velocity. This regime of motion is called drift and the corresponding velocity drift velocity. If the electron concentration is inhomogeneous in space, the thermal motion tries to make it homogeneous. This leads to a motion of electrons called diffusion. The flux of electrons (number of electrons crossing an element of surface per second divided by the area of the element) is controlled by the gradient of electron concentration. The ratio drift velocity/field is called mobility,.( In the absence of the concentration gradient.) The ratio flux/concentration gradient is called diffusion coefficient, D. ( In the absence of the electric field.) The mobility and diffusion coefficients obey the Einstein relation. All these logic equally applies to holes. They can be considered as particles with a positive charge and a positive mass. These notions can be rationalized as follows: Under an electric field the electrons in the valence band move on average in the direction opposite to that of the field. This corresponds to a motion of the empty places (holes) in the direction of the field as expected from positively charged particles with positive mass.

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17 Strictly speaking, the amount of electrons in the conduction band is not fixed in time, rather it is fixed only on average. The electrons come to and go from the valence band. Actually any electron stays in the conduction band only during some typical times called electron life time, e. For the time intervals shorter than e, the electrons in the conduction band can be treated as normal particles which do not appear and diaper all the time. The same logic holds for the holes. The hole life time, h, is the typical time during which a hole can move in the valence band before it is filled by an electron coming to the valence band from the conduction band or from an impurity. The process of annihilation of an electron from the conduction band and a hole from the valence band is called recombination. In doped semiconductors, due to the mass action law, either n << p or p << n. That implies the terminology: majority carries and minority carriers. Since one electron and one hole participate in the recombination, the impact of the recombination is essential only on minority carriers. When an electron and a hole recombine, the electron loses the energy equal to E g. This energy is not lost. It can be transformed into light or heart. In the former case, a quantum of light - photon appears, with the energy equal to that lost by the electron.

18 Extra electrons can be brought to the conduction band with an electric current. At the same time holes form in the valence band. Electron-hole pairs can also be created by light. This phenomenon is called generation. Under illumination with photons (quanta of light), whose energies exceeded the width of the forbidden gap, the concentrations of electrons and holes differ form their equilibrium values.

19 Once the concentration of carries is changed compared to its equilibrium value, it tries to return back to is. It mainly returns back during the life time of particles (electrons or holes). During this time, the added particles diffuse away from the place where they were created. The distance on which they typically manage to diffuse during their lifetime is called diffusion length.