One of the most remarkable and exciting discoveries of recent years has been the application of physics solid body to the technical development of a number of electrical devices, such as transistors. The study of semiconductors led to the discovery of their useful properties and to many practical applications. Things are changing so fast in this area that what you've been told today may not be true, or at least incomplete, in a year's time. And it is quite clear that by studying such substances in more detail, we will eventually be able to do much more amazing things. You won't need the material in this chapter to understand the following chapters, but you'll probably be interested in seeing that at least some of what you've learned still has something to do with practical matters.
A lot of semiconductors are known, but we will limit ourselves to those that are most used today in technology. In addition, they have been studied better than others, so that having understood them, we will understand many others to some extent. The most widely used semiconductor materials are silicon and germanium. These elements crystallize in a diamond-type lattice - in such a cubic structure in which atoms have a quadruple (tetrahedral) bond with their nearest neighbors. At very low temperatures(near absolute zero) they are insulators, although at room temperature they conduct electricity a little. These are not metals; they are called semiconductors.
If somehow we introduce an additional electron into a silicon or germanium crystal at a low temperature, then what will happen is what was described in the previous chapter. Such an electron will begin to wander around the crystal, jumping from the place where one atom stands to the place where another stands. We have considered only the behavior of an atom in a rectangular lattice, and for a real lattice of silicon or germanium, the equations would be different. But everything essential can become clear already from the results for a rectangular lattice.
As we saw in Chap. 11, these electrons can only have energies in a certain range of values, called the conduction band. In this zone, the energy is related to the wavenumber of the probability amplitude [see (11.24)] by the formula
Different are the amplitudes of jumps in the directions , and , and , , are lattice constants (intervals between nodes) in these directions.
For energies near the bottom of the zone, formula (12.1) can be written approximately as follows:
(see Ch. 11, §4).
If we are interested in the motion of an electron in some particular direction, so that the ratio of the components is the same all the time, then the energy is quadratic function wave number and hence the momentum of the electron. You can write
, (12.3)
where - some constant, and draw a graph depending on (Fig. 12.1). We will call such a graph an “energy diagram”. An electron in a certain state of energy and momentum can be represented on such a graph by a dot (in the figure).
Fig. 12.1. Energy diagram for an electron in an insulator crystal.
We have already mentioned in Chap. 11 that the same state of affairs would occur if we removed an electron from a neutral insulator. Then an electron from a neighboring atom can jump to this place. He will fill the “hole”, and he will leave a new “hole” in the place where he stood. We can describe this behavior by giving the amplitude that a hole will be near a given particular atom, and by saying that the hole can jump from atom to atom. (Moreover, it is clear that the amplitude for the hole to jump from atom to atom is exactly equal to the amplitude for the electron from the atom to jump into the hole from the atom.) The math for the hole is the same as for the extra electron, and again we find that The energy of a hole is related to its wave number by an equation that exactly coincides with (12.1) and (12.2), but, of course, with other numerical values of the amplitudes , and . A hole also has an energy associated with the wavenumber of its probability amplitudes. Its energy lies in a certain limited zone and, near the bottom of the zone, changes quadratically with an increase in the wave number (or momentum) in the same way as in Fig. 12.1. Repeating our reasoning in Chap. 11, § 3, we will find that the hole also behaves like a classical particle with some definite effective mass, with the only difference that in noncubic crystals the mass depends on the direction of motion. So, a hole resembles a particle with a positive charge moving through a crystal. The charge of the hole particle is positive because it is concentrated in the place where there is no electron; and when it moves in one direction, it is actually electrons moving in the opposite direction.
If several electrons are placed in a neutral crystal, then their movement will be very similar to the movement of atoms in a gas under low pressure. If there are not too many of them, their interaction can be neglected. If then applied to the crystal electric field, then the electrons will begin to move and flow electricity. In principle, they should end up at the edge of the crystal and, if there is a metal electrode, go to it, leaving the crystal neutral.
Similarly, many holes could be introduced into the crystal. They would start wandering around all over the place. If an electric field is applied, then they will flow to the negative electrode and then they could be "removed" from it, which happens when they are neutralized by electrons from the metal electrode.
Electrons and holes can be in the crystal at the same time. If there are not very many of them again, then they will wander independently. In an electric field, they will all contribute to total current. For obvious reasons, electrons are called negative carriers and holes are called positive carriers.
Until now, we have assumed that electrons are introduced into the crystal from the outside or (to form a hole) are removed from it. But you can also "create" an electron-hole pair by removing a bound electron from a neutral atom and placing it in the same crystal at some distance. Then we will have a free electron and a free hole, and their motion will be as we have described.
The energy needed to put an electron into a state (we say: to "create" a state) is the energy shown in Fig. 12.2. This is some energy that exceeds . The energy needed to "create" a hole in some state is the energy (Fig. 12.3) that is some fraction higher than . And to create a pair in the states and , you just need energy.
Fig. 12.2. The energy required for the "birth" of a free electron.
Fig. 12.3. The energy , required for the “birth” of a hole in the state .
Pairing is, as we shall see later, a very common process, and many people prefer to place figs. 12.2 and 12.3 on one drawing, and the energy of holes is laid down, although, of course, this energy is positive. In FIG. 12.4 we have combined these two graphs. The advantage of such a schedule is that the energy 
required to form a pair (an electron in and a hole in ) is simply given by the vertical distance between and , as shown in FIG. 12.4. The smallest energy required to form a pair is called the energy width, or gap width, and is equal to .
Fig. 12.4. Energy diagrams for an electron and a hole.
Sometimes you may come across a simpler diagram. It is drawn by those who are not interested in the variable, calling it an energy level diagram. This diagram (shown in Fig. 12.5) simply indicates the allowable energies of electrons and holes.
Fig. 12.5. Energy level diagram for electrons and holes.
How is an electron-hole pair created? There are several ways. For example, light photons (or X-rays) can be absorbed and form a pair, as long as the energy of the photon is greater than the energy width. The rate of pair formation is proportional to the light intensity. If you press two electrodes to the ends of the crystal and apply a "bias" voltage, then electrons and holes will be attracted to the electrodes. The current in the circuit will be proportional to the intensity of the light. This mechanism is responsible for the phenomenon of photoconductivity and for the operation of photocells.
Electron-hole pairs can also be formed by high-energy particles. When a fast-moving charged particle (such as a proton or pion with an energy of tens or hundreds of MeV) flies through a crystal, its electric field can pull the electrons out of their bound states, forming electron-hole pairs. Hundreds and thousands of similar phenomena occur on every millimeter of the track. After the particle has passed through, the carriers can be collected and thereby induce an electrical impulse. Here is the mechanism of what is played out in semiconductor counters, recently used in experiments in nuclear physics. Semiconductors are not needed for such counters; they can also be made from crystalline insulators. And so it was in fact: the first of these counters was made of diamond, which is an insulator at room temperature. But we need very pure crystals if we want electrons and holes to get to the electrodes without fear of capture. This is why silicon and germanium are used because samples of these semiconductors of reasonable size (on the order of a centimeter) can be obtained in high purity.
So far, we have dealt only with the properties of semiconductor crystals at temperatures near absolute zero. At any non-zero temperature, there is another mechanism for creating electron-hole pairs. The thermal energy of the crystal can supply energy to the steam. The thermal vibrations of the crystal can transfer their energy to the pair, causing the "spontaneous" creation of pairs.
The probability (per unit time) that the energy reaching the energy gap will be concentrated at the location of one of the atoms is proportional to 
, where is the temperature, and is the Boltzmann constant [see ch. 40 (issue 4)]. Near absolute zero, this probability is hardly noticeable, but as the temperature rises, the probability of the formation of such pairs increases. The formation of pairs at any final temperature must continue indefinitely, giving all the time at a constant rate more and more positive and negative carriers. Of course, this will not actually happen, because after a moment, the electrons will accidentally meet the holes again, the electron will roll into the hole, and the released energy will go to the lattice. We will say that an electron with a hole "annihilated". There is a certain probability that a hole will meet an electron and both of them will annihilate each other.
Speaking of a constant, we mean its approximate constancy. A more complete theory, taking into account various details of how electrons and holes "find" each other, shows that the "constant" also slightly depends on temperature; but the main dependence on temperature still lies in the exponential.
Take, for example, a pure substance that was originally neutral. At a finite temperature, one can expect that the number of positive and negative carriers will be the same, . This means that each of these numbers should change with temperature as . The change in many properties of a semiconductor (for example, its conductivity) is determined mainly by an exponential factor, because all other factors depend much less on temperature. The gap width for germanium is approximately equal to 0.72 eV, and for silicon 1.1 eV.
At room temperature is about 1/40 eV. At such temperatures, there are already enough holes and electrons to provide appreciable conduction, while, say, at 30°K (one-tenth of room temperature) the conduction is imperceptible. The gap width of a diamond is 6-7 eV, so diamond is a good insulator at room temperature.
electrons and holes in crystal lattice semiconductor
When a certain amount of energy is communicated to the crystal lattice, individual electrons can leave the valence bonds and turn into free charge carriers.
However, the departure of an electron from its atom violates its electrical neutrality, the positive charge of the nucleus turns out to be uncompensated by one unit charge (electron charge) and the atom turns into a positively charged ion (Fig. 2.1, a).
Strictly speaking, since this electron was common to two atoms, it cannot be said that one of these atoms is ionized. The departure of an electron will lead to partial ionization of two neighboring atoms. Therefore, the single positive charge that appears in this case, equal in absolute value to the charge of the electron, will be attributed not to this or that atom, but to the defective bond left by the electron. This positive charge is called hole .
Rice. 2.1 Model of breaking the valence bond and the appearance of an electron as a free charge carrier:
a) in a planar image; b) in the band energy diagram.
So, with the departure of an electron in one of the valence bonds, a “vacant” place appears, which can be occupied by one of the valence electrons of neighboring bonds. On the band model, such an electron transition from a filled bond to a defective one is represented by an electron transition inside the valence band to a vacant level.
Naturally, when an electron passes from a filled bond to a defective bond, the defective bond is filled, and the filled bond becomes defective. The transition of an electron corresponds to the movement of a hole in the opposite direction. The electron transfer process will continue. The defect (hole) will then move from bond to bond. Along with this, a positive charge will also move from bond to bond. This process will be of a random nature, the trajectory of the hole will obey the laws of chaotic motion. However, this will only take place if there is no electric field in the crystal. If we place the crystal in an electric field, then the transitions of electrons from bond to bond, in which the hole (positive charge) would move along the lines electric field, become more likely
Directional movement of a positive charge - a hole - in an electric field there is already an electric current flow. Strictly speaking, the charge carriers in this case are also electrons. The transfer of current is carried out due to the successive transition of electrons from one bond to another, i.e., due to the successive displacement of valence electrons in the valence band. However, in practice it is much more convenient to consider the continuous movement of a positive charge formed in a defect bond than the sequential movement of electrons from bond to bond.
A hole should not be mixed with an ion, for example, in an electrolyte. In an electrolyte, an ionized atom moves in space. In a crystal lattice, atoms do not move and are stationary at the lattice sites. The motion of a hole is the successive ionization of immobile atoms.
Thus, the violation of the valence bond due to thermal energy leads to the appearance in the semiconductor crystal of two free charge carriers: a negative unit charge - an electron, and a positive unit charge opposite to it in sign - a hole. The electrical conductivity that occurs in a semiconductor crystal due to the violation of valence bonds is called own electrical conductivity .
Topics of the USE codifier: semiconductors, intrinsic and extrinsic conductivity of semiconductors.
Until now, speaking about the ability of substances to conduct electric current, we divided them into conductors and dielectrics. The specific resistance of ordinary conductors is in the range of Ohm m; the resistivity of dielectrics exceeds these values on average by orders of magnitude: Ohm m.
But there are also substances that, in their electrical conductivity, occupy an intermediate position between conductors and dielectrics. it semiconductors: their resistivity at room temperature can take on values in a very wide range of ohm m. Semiconductors include silicon, germanium, selenium, some other chemical elements and compounds (Semiconductors are extremely common in nature. For example, about 80% of the mass earth's crust are substances that are semiconductors). Silicon and germanium are the most widely used.
The main feature of semiconductors is that their electrical conductivity increases sharply with increasing temperature. The resistivity of a semiconductor decreases with increasing temperature approximately as shown in Fig. one .
Rice. 1. Dependence for a semiconductor
In other words, at low temperatures, semiconductors behave like dielectrics, and at high temperatures, they behave like fairly good conductors. This is the difference between semiconductors and metals: the resistivity of the metal, as you remember, increases linearly with increasing temperature.
There are other differences between semiconductors and metals. Thus, illumination of a semiconductor causes a decrease in its resistance (and light has almost no effect on the resistance of a metal). In addition, the electrical conductivity of semiconductors can change very strongly with the introduction of even a negligible amount of impurities.
Experience shows that, as in the case of metals, when current flows through a semiconductor, there is no transfer of matter. Therefore, the electric current in semiconductors is due to the movement of electrons.
A decrease in the resistance of a semiconductor when it is heated indicates that an increase in temperature leads to an increase in the number of free charges in the semiconductor. Nothing like this happens in metals; therefore, semiconductors have a different mechanism of electrical conductivity than metals. And the reason for this is the different nature of the chemical bond between the atoms of metals and semiconductors.
covalent bond
The metallic bond, remember, is provided by a gas of free electrons, which, like glue, holds the positive ions at the lattice sites. Semiconductors are arranged differently - their atoms are held together covalent bond. Let's remember what it is.
Electrons located in the outer electronic level and called valence, are weaker bound to the atom than the rest of the electrons, which are located closer to the nucleus. In the process of forming a covalent bond, two atoms contribute "to the common cause" one of their valence electrons. These two electrons are socialized, that is, they now belong to both atoms, and therefore are called common electron pair(Fig. 2).
Rice. 2. Covalent bond
The socialized pair of electrons just holds the atoms near each other (with the help of electrical attraction forces). A covalent bond is a bond that exists between atoms due to common electron pairs.. For this reason, a covalent bond is also called pair-electron.
Crystal structure of silicon
We are now ready to take a closer look at the internals of semiconductors. As an example, consider the most common semiconductor in nature - silicon. The second most important semiconductor, germanium, has a similar structure.
The spatial structure of silicon is shown in fig. 3 (image by Ben Mills). Silicon atoms are depicted as balls, and the tubes connecting them are channels of covalent bonding between atoms.
Rice. 3. Crystal structure of silicon
Note that each silicon atom is bonded to four neighboring atoms. Why is it so?
The fact is that silicon is tetravalent - on the outer electron shell of the silicon atom there are four valence electrons. Each of these four electrons is ready to form a common electron pair with the valence electron of another atom. And so it happens! As a result, the silicon atom is surrounded by four docked atoms, each of which contributes one valence electron. Accordingly, there are eight electrons around each atom (four own and four alien).
We see this in more detail on a flat diagram of the silicon crystal lattice (Fig. 4).
Rice. 4. Crystal lattice of silicon
Covalent bonds are shown as pairs of lines connecting atoms; these lines share electron pairs. Each valence electron located on such a line spends most of its time in the space between two neighboring atoms.
However, valence electrons are by no means "tightly tied" to the corresponding pairs of atoms. Electron shells overlap all neighboring atoms, so that any valence electron is the common property of all neighboring atoms. From some atom 1, such an electron can go to its neighboring atom 2, then to its neighboring atom 3, and so on. Valence electrons can move throughout the space of the crystal - they are said to belong to the whole crystal(rather than any single atomic pair).
However, silicon's valence electrons are not free (as is the case in metal). In a semiconductor, the bond between valence electrons and atoms is much stronger than in a metal; silicon covalent bonds do not break at low temperatures. The energy of the electrons is not enough to start an orderly movement from a lower potential to a higher one under the action of an external electric field. Therefore, at sufficiently low temperatures, semiconductors are close to dielectrics - they do not conduct electric current.
Own conductivity
If included in electrical circuit semiconductor element and begin to heat it, then the current in the circuit increases. Therefore, the semiconductor resistance decreases with an increase in temperature. Why is this happening?
As the temperature rises, the thermal vibrations of silicon atoms become more intense, and the energy of valence electrons increases. For some electrons, the energy reaches values sufficient to break covalent bonds. Such electrons leave their atoms and become free(or conduction electrons) is exactly the same as in metal. In an external electric field, free electrons begin an ordered movement, forming an electric current.
The higher the temperature of silicon, the greater the energy of the electrons, and the greater the number of covalent bonds does not withstand and breaks. The number of free electrons in a silicon crystal increases, which leads to a decrease in its resistance.
The breaking of covalent bonds and the appearance of free electrons is shown in fig. 5 . At the site of a broken covalent bond, a hole is a vacancy for an electron. The hole has positive charge, since with the departure of a negatively charged electron, an uncompensated positive charge of the nucleus of the silicon atom remains.
Rice. 5. Formation of free electrons and holes
Holes do not stay in place - they can wander around the crystal. The fact is that one of the neighboring valence electrons, "traveling" between atoms, can jump to the formed vacancy, filling the hole; then the hole in this place will disappear, but will appear in the place where the electron came from.
In the absence of an external electric field, the movement of holes is random, because valence electrons wander between atoms randomly. However, in an electric field directed hole movement. Why? It's easy to understand.
On fig. 6 shows a semiconductor placed in an electric field. On the left side of the figure is the initial position of the hole.
Rice. 6. Motion of a hole in an electric field
Where will the hole go? It is clear that the most probable are hops "electron > hole" in the direction against field lines (that is, to the "pluses" that create the field). One of these jumps is shown in the middle part of the figure: the electron jumped to the left, filling the vacancy, and the hole, accordingly, shifted to the right. The next possible jump of an electron caused by an electric field is shown on the right side of the figure; as a result of this jump, the hole took a new place, located even more to the right.
We see that the hole as a whole moves towards field lines - that is, where positive charges are supposed to move. We emphasize once again that the directed motion of a hole along the field is caused by hops of valence electrons from atom to atom, occurring predominantly in the direction against the field.
Thus, there are two types of charge carriers in a silicon crystal: free electrons and holes. When an external electric field is applied, an electric current appears, caused by their ordered counter motion: free electrons move opposite to the field strength vector, and holes move in the direction of the vector.
The occurrence of current due to the movement of free electrons is called electronic conductivity, or n-type conductivity. The process of orderly movement of holes is called hole conductivity,or p-type conductivity(from the first letters of the Latin words negativus (negative) and positivus (positive)). Both conductivities - electron and hole - together are called own conductivity semiconductor.
Each departure of an electron from a broken covalent bond generates a “free electron-hole” pair. Therefore, the concentration of free electrons in a pure silicon crystal is equal to the concentration of holes. Accordingly, when the crystal is heated, the concentration of not only free electrons, but also holes increases, which leads to an increase in the intrinsic conductivity of the semiconductor due to an increase in both electronic and hole conductivity.
Along with the formation of “free electron-hole” pairs, the reverse process also takes place: recombination free electrons and holes. Namely, a free electron, meeting with a hole, fills this vacancy, restoring the broken covalent bond and turning into a valence electron. Thus, in a semiconductor, dynamic balance: the average number of breaks of covalent bonds and the resulting electron-hole pairs per unit time is equal to the average number of recombining electrons and holes. This state of dynamic equilibrium determines the equilibrium concentration of free electrons and holes in a semiconductor under given conditions.
A change in external conditions shifts the state of dynamic equilibrium in one direction or another. The equilibrium value of the concentration of charge carriers naturally changes in this case. For example, the number of free electrons and holes increases when a semiconductor is heated or illuminated.
At room temperature, the concentration of free electrons and holes in silicon is approximately equal to cm. The concentration of silicon atoms is about cm. In other words, there is only one free electron per silicon atom! This is very little. In metals, for example, the concentration of free electrons is approximately equal to the concentration of atoms. Respectively, intrinsic conductivity of silicon and other semiconductors under normal conditions is small compared to the conductivity of metals.
Impurity conductivity
The most important feature of semiconductors is that their resistivity can be reduced by several orders of magnitude by introducing even a very small amount of impurities. In addition to its own conductivity, a semiconductor has a dominant impurity conductivity. It is thanks to this fact that semiconductor devices have found such wide application in science and technology.
Suppose, for example, that a little pentavalent arsenic is added to the silicon melt. After crystallization of the melt, it turns out that arsenic atoms occupy places in some sites of the formed silicon crystal lattice.
The outer electronic level of an arsenic atom has five electrons. Four of them form covalent bonds with the nearest neighbors - silicon atoms (Fig. 7). What is the fate of the fifth electron not occupied in these bonds?
Rice. 7. N-type semiconductor
And the fifth electron becomes free! The fact is that the binding energy of this "extra" electron with an arsenic atom located in a silicon crystal is much less than the binding energy of valence electrons with silicon atoms. Therefore, already at room temperature, almost all arsenic atoms, as a result of thermal motion, remain without a fifth electron, turning into positive ions. And the silicon crystal, respectively, is filled with free electrons, which are unhooked from the arsenic atoms.
The filling of a crystal with free electrons is not new to us: we have seen it above when it was heated clean silicon (without any impurities). But now the situation is fundamentally different: the appearance of a free electron leaving the arsenic atom is not accompanied by the appearance of a mobile hole. Why? The reason is the same - the bond of valence electrons with silicon atoms is much stronger than with the arsenic atom on the fifth vacancy, so the electrons of neighboring silicon atoms do not tend to fill this vacancy. Thus, the vacancy remains in place; it is, as it were, "frozen" to the arsenic atom and does not participate in the creation of the current.
In this way, the introduction of pentavalent arsenic atoms into the silicon crystal lattice creates electronic conductivity, but does not lead to the symmetrical appearance of hole conductivity. The main role in creating the current now belongs to free electrons, which in this case are called main carriers charge.
The intrinsic conduction mechanism, of course, continues to operate even in the presence of an impurity: covalent bonds are still broken due to thermal motion, generating free electrons and holes. But now there are much fewer holes than free electrons, which are provided in large quantities by arsenic atoms. Therefore, the holes in this case will be minority carriers charge.
Impurities whose atoms donate free electrons without the appearance of an equal number of mobile holes are called donor. For example, pentavalent arsenic is a donor impurity. In the presence of a donor impurity in the semiconductor, free electrons are the main charge carriers, and holes are the minor ones; in other words, the concentration of free electrons is much higher than the concentration of holes. Therefore, semiconductors with donor impurities are called electronic semiconductors, or n-type semiconductors(or simply n-semiconductors).
And how much, interestingly, can the concentration of free electrons exceed the concentration of holes in an n-semiconductor? Let's do a simple calculation.
Suppose that the impurity is , that is, there is one arsenic atom per thousand silicon atoms. The concentration of silicon atoms, as we remember, is on the order of cm.
The concentration of arsenic atoms, respectively, will be a thousand times less: cm. The concentration of free electrons donated by the impurity will also turn out to be the same - after all, each arsenic atom gives off an electron. And now let's remember that the concentration of electron-hole pairs that appear when silicon covalent bonds are broken at room temperature is approximately equal to cm. Do you feel the difference? The concentration of free electrons in this case is greater than the concentration of holes by orders of magnitude, that is, a billion times! Accordingly, the resistivity of a silicon semiconductor decreases by a factor of a billion when such a small amount of impurity is introduced.
The above calculation shows that in n-type semiconductors, the main role is indeed played by electronic conductivity. Against the background of such a colossal superiority in the number of free electrons, the contribution of the motion of holes to the total conductivity is negligibly small.
It is possible, on the contrary, to create a semiconductor with a predominance of hole conductivity. This will happen if a trivalent impurity is introduced into a silicon crystal - for example, indium. The result of such implementation is shown in Fig. eight .
Rice. 8. p-type semiconductor
What happens in this case? The outer electronic level of the indium atom has three electrons that form covalent bonds with the three surrounding silicon atoms. For the fourth neighboring silicon atom, the indium atom no longer has enough electron, and a hole appears in this place.
And this hole is not simple, but special - with a very high binding energy. When an electron from a neighboring silicon atom enters it, it will “stuck forever” in it, because the attraction of an electron to an indium atom is very large - more than to silicon atoms. The indium atom will turn into a negative ion, and in the place where the electron came from, a hole will appear - but now an ordinary mobile hole in the form of a broken covalent bond in the silicon crystal lattice. This hole in the usual way will begin to wander around the crystal due to the "relay" transfer of valence electrons from one silicon atom to another.
And so, each impurity atom of indium generates a hole, but does not lead to the symmetrical appearance of a free electron. Such impurities, the atoms of which "tightly" capture electrons and thereby create a mobile hole in the crystal, are called acceptor.
Trivalent indium is an example of an acceptor impurity.
If an acceptor impurity is introduced into a crystal of pure silicon, then the number of holes generated by the impurity will be much greater than the number of free electrons that have arisen due to the breaking of covalent bonds between silicon atoms. A semiconductor with an acceptor dopant is hole semiconductor, or p-type semiconductor(or simply p-semiconductor).
Holes play a major role in generating current in a p-semiconductor; holes - major charge carriers. Free electrons - minor carriers charge in a p-semiconductor. The motion of free electrons in this case does not make a significant contribution: the electric current is provided primarily by hole conduction.
p–n junction
The contact point of two semiconductors with various types conductivity (electronic and hole) is called electron-hole transition, or p–n junction. In the region of the p–n junction, an interesting and very important phenomenon arises - one-way conduction.
On fig. 9 shows the contact of p- and n-type regions; colored circles are holes and free electrons, which are the majority (or minor) charge carriers in the respective regions.
Rice. 9. Blocking layer p–n junction
By performing thermal motion, charge carriers penetrate through the interface between the regions.
Free electrons pass from the n-region to the p-region and recombine there with holes; holes diffuse from the p-region to the n-region and recombine there with electrons.
As a result of these processes, an uncompensated charge of the positive ions of the donor impurity remains in the electronic semiconductor near the contact boundary, while in the hole semiconductor (also near the boundary), an uncompensated negative charge of the acceptor impurity ions arises. These uncompensated space charges form the so-called barrier layer, whose internal electric field prevents further diffusion of free electrons and holes through the contact boundary.
Let us now connect a current source to our semiconductor element by applying the “plus” of the source to the n-semiconductor, and the “minus” to the p-semiconductor (Fig. 10).
Rice. 10. Turn on in reverse: no current
We see that the external electric field takes the majority charge carriers farther from the contact boundary. The width of the barrier layer increases, and its electric field increases. The resistance of the barrier layer is high, and the main carriers are not able to overcome the p–n junction. The electric field allows only minority carriers to cross the boundary, however, due to the very low concentration of minority carriers, the current they create is negligible.
The considered scheme is called turning on the p–n junction in the opposite direction. There is no electric current of the main carriers; there is only a negligible minority carrier current. In this case, the p–n junction is closed.
Now let's change the polarity of the connection and apply "plus" to the p-semiconductor, and "minus" to the n-semiconductor (Fig. 11). This scheme is called switching in forward direction.
Rice. 11. Forward switching: current flows
In this case, the external electric field is directed against the blocking field and opens the way for the main carriers through the p–n junction. The barrier layer becomes thinner, its resistance decreases.
There is a mass movement of free electrons from the n-region to the p-region, and holes, in turn, rush together from the p-region to the n-region.
A current arises in the circuit, caused by the movement of the main charge carriers (Now, however, the electric field prevents the current of minority carriers, but this negligible factor does not have a noticeable effect on the overall conductivity).
One-sided conduction of the p–n junction is used in semiconductor diodes . A diode is a device that conducts current in only one direction; in the opposite direction, no current passes through the diode (the diode is said to be closed). A schematic representation of the diode is shown in fig. 12 .
Rice. 12. Diode
In this case, the diode is open in the direction from left to right: the charges seem to flow along the arrow (see it in the figure?). In the direction from right to left, the charges seem to rest against the wall - the diode is closed.
| Hole | |
| Symbol: | h(eng. hole) |
|---|---|
 When an electron leaves a helium atom, a hole remains in its place. In this case, the atom becomes positively charged. |
|
| Compound: | Quasiparticle |
| Classification: | Light holes, heavy holes |
| Who and/or what is it named after? | Absence of an electron |
| Quantum0 numbers: | |
| Electric charge : | +1 |
| Spin : | Determined by the spin of electrons in the valence band ħ |
Definition according to GOST 22622-77: "An unfilled valence bond that manifests itself as a positive charge numerically equal to the electron charge."
Hole conduction can be explained with the following analogy: There are a number of people sitting in an auditorium where there are no spare chairs. If someone from the middle of the row wants to leave, he climbs over the back of a chair into an empty row and leaves. Here, the empty row is analogous to the conduction band, and the departed person can be compared to a free electron. Imagine that someone else came and wants to sit down. It is hard to see from an empty row, so he does not sit down there. Instead, a person sitting near an empty chair moves to it, and all his neighbors follow him. Thus, the empty space, as it were, moves to the edge of the row. When this place is next to a new viewer, he will be able to sit down.
In this process, each seated moved along the row. If the audience had a negative charge, such movement would be electrical conduction. If, in addition, the chairs are positively charged, then only the empty seat will have a non-zero total charge. it simple model showing how hole conduction works. However, in fact, due to the properties of the crystal lattice, the hole is not located in a certain place, as described above, but is spread over an area many hundreds of elementary cells in size.
To create holes in semiconductors, doping of crystals with acceptor impurities is used. In addition, holes can also arise as a result of external influences: thermal excitation of electrons from the valence band to the conduction band, illumination by light or exposure to ionizing radiation.
In the case of the Coulomb interaction of a hole with an electron, a bound state is formed from the conduction band, called an exciton.
heavy holes- the name of one of the branches of the energy spectrum of the valence band of the crystal.
Holes in quantum chemistry
The term hole is also used in computational chemistry, where the ground state of a molecule is interpreted as a vacuum state - there are no electrons in this state. In such a scheme, the absence of an electron in a normally filled state is called a hole and is treated as a particle. And the presence of an electron in normally empty space is simply called an electron.
Semiconductor crystals are formed from atoms arranged in a certain order. According to modern concepts, atoms consist of positively charged nuclei around which shells filled with electrons are located. In this case, each electron corresponds to a strictly defined level, at which there cannot be more than two electrons with different values spin characterizing the rotation of an electron. According to the laws of quantum mechanics, electrons can only exist in strictly defined energy states. A change in the energy of an electron is possible when a quantum of electromagnetic radiation is absorbed or emitted with an energy equal to the difference between the energies at the initial and final levels.
When two atoms, such as hydrogen, approach each other, their orbitals begin to overlap and a bond between them may occur. There is a rule according to which the number of orbitals in a molecule is equal to the sum of the numbers of orbitals in atoms, while the interaction of atoms leads to the fact that the levels of the molecule split, and the smaller the distance between the atoms, the stronger this splitting.
On fig. 1.6. the scheme of level splitting for five atoms with decreasing distance between them is shown. As can be seen from the graphs, when bonds are formed between atoms, valence electrons form zones allowed for electrons, and the number of states in these zones is the greater, the more interacting atoms. In crystals, the number of atoms is more than 10 22 cm -3 , approximately the same number of levels in the zones. In this case, the distance between the levels becomes extremely small, which makes it possible to assume that the energy in the allowed band changes continuously. Then an electron that has fallen into an unoccupied zone can be considered as classical, considering that under the action of an electric field it gains energy continuously, and not by quanta, i.e. behaves like a classical particle.
Rice. 1.6. Energy splitting of 1s and 2s levels for five atoms depending on the distance between them
During the formation of crystals, the bands formed by valence electrons can be partially filled, free, or completely filled with electrons. Moreover, if there is no band gap between the filled and free states, then the material is a conductor, if there is a small band gap, then it is a semiconductor, if the band gap is large and electrons do not enter it due to thermal energy, then this is an insulator. Figure 1.7. illustrates possible zone configurations.
For conductors, the allowed band is partially filled with electrons, so even when an external voltage is applied, they are able to gain energy and move around the crystal. Such a band structure is characteristic of metals. The F level separating the filled and unfilled part of the band is called the Fermi level. Formally, it is defined as a level whose probability of being filled with electrons is 1/2.
Rice. 1.7. Possible structure of energy bands created by valence electrons in crystals
For semiconductors and dielectrics, the band structure is such that the lower allowed band is completely filled with valence electrons, therefore it is called the valence band. The top of the valence band is designated Ev. In it, electrons cannot move under the action of the field (and, accordingly, gain energy), since all energy levels are occupied, and according to the Pauli principle, an electron cannot move from an occupied state to an occupied one. Therefore, electrons in a completely filled valence band do not participate in the creation of electrical conductivity. The upper zone in semiconductors and dielectrics in the absence of external excitation is free from electrons, and if an electron is somehow thrown there, then under the action of an electric field it can create electrical conductivity, therefore this zone is called the conduction band. The bottom of the conduction band is usually denoted Ec. Between the conduction band and the valence band there is a band gap Eg, in which, according to the laws of quantum mechanics, electrons cannot be (just as electrons in an atom cannot have energies that do not correspond to the energies of electron shells). For the band gap, we can write:
Eg = Ec - Ev (1.4.)
In semiconductors, unlike insulators, the band gap is smaller, this affects the fact that when the material is heated, much more electrons get into the conduction band of the semiconductor due to thermal energy than into the conduction band of the insulator and the conductivity of the semiconductor can be several orders of magnitude higher than the conductivity of the insulator , however, the boundary between a semiconductor and an insulator is conditional.
Since, in the absence of external excitation, the valence band is completely filled (the probability of finding an electron at Ev = 1), the conduction band is completely free (the probability of finding an electron at Ec = 0), then formally the Fermi level with a probability of filling ½ should be in the band gap. Calculations show that, indeed, in undoped, defect-free semiconductors and dielectrics (they are usually called intrinsic), it lies near the middle of the band gap. However, electrons cannot be there, because there are no allowed energy levels.
Rice. 1.7. Schematic representation of a defect-free silicon crystal.
Basic elementary semiconductors belong to the fourth group of the periodic table, they have 4 electrons on the outer shell. Accordingly, these electrons are in S (1 electron) and p (3 electrons). When a crystal is formed, the outer electrons interact and form a completely filled shell with eight electrons, as shown in the diagram in Fig. 1.7.
In this case, the atom can form chemical bonds with four neighbors, i.e. is four times coordinated. All bonds are equivalent and form a tetrahedral lattice (a tetrahedron is a figure with four identical surfaces).
The tetrahedral structure is characteristic of diamond crystals. Known semiconductors such as Si and Ge have a diamond type structure.
When an electron leaves for the conduction band, it delocalizes and can move along the band from one atom to another. It becomes a conduction electron and can create an electric current. Usually they say: a free charge carrier appeared, although in fact the electron did not leave the crystal, it only got the opportunity to move from one place in the crystal to another.
In the place from where the electron left, the electrical neutrality condition is violated and a positively charged electron vacancy appears, which is commonly called a hole (the positive charge is due to the uncompensated charge of the nucleus).
A neighboring electron can move to the place where the electron left, which will lead to the displacement of a positively charged hole. Thus, the movement of valence electrons filling the free electronic state (the Pauli prohibition is lifted) leads to the movement of a vacancy in which the charge compensation condition is violated, i.e. holes. Instead of considering the movement of valence electrons, which are extremely numerous in the valence band, they consider the movement of positively charged holes, which are few in number and which, like electrons, can transfer charge. This process is illustrated in Fig. 1.10.
Figure 1.10 shows a crystal in which, by some external excitation, for example, by a light quantum with hν > Eg, one of the electrons is transferred to the conduction band (becomes free), i.e. one of the atoms had one of its valence bonds broken. Then, in addition to the electron not bound to the atom, a positively charged ion appeared in the crystal. The ability of the ion itself to move under the action of the field is very small, so it should not be taken into account. Since atoms in a crystal are located close to each other, an electron from a neighboring atom can be attracted to this ion. In this case, a positive hole appears at the neighboring atom, from where the valence electron left, etc. For a perfect crystal free of impurities and defects, the electron concentration will be equal to the hole concentration. it own concentration of charge carriers n i = p i , the icon i means the concentration of carriers for the intrinsic semiconductor (intrinsic - intrinsic). For the product of the concentrations of electrons and holes, we can write:
np = n i 2 (1.5)
It should be noted that this relationship holds not only for intrinsic semiconductors, but also for doped crystals, in which the electron concentration is not equal to the hole concentration.
Rice. 1.10. Schematic representation of the appearance of an electron and a hole when light is absorbed
The direction of motion of the hole is opposite to the direction of motion of the electron. Each electron in a valence bond is characterized by its own level. All levels of valence electrons are located very close and form a valence band, so the movement of a hole can be considered as a continuous process similar to the movement of a classical free particle. Similarly, since the energy levels are very close in the conduction band, the dependence of energy on momentum can be considered continuous and, accordingly, the motion of an electron can be considered in the first approximation as the motion of a classical free particle.
1.2.3. Doping of crystals with a donor or acceptor impurity, "n" and "p" type semiconductors.
The presence of impurities and defects in a crystal leads to the appearance of energy levels in the band gap, the position of which depends on the type of impurity or defect. To control the electrical properties of semiconductors, impurities (doping) are specially introduced into them. So the introduction of an elementary semiconductor of group IV of a periodic system of elements, for example Si, impurities of elements of group V (donors) leads to the appearance of additional electrons and, accordingly, the predominance of electronic conductivity (n - type), the introduction of elements Group III leads to the appearance of additional holes (p-type).
Rice. 1.12. Scheme of the Formation of a Free Electron and a Charged Donor Atom in the Doping of Si with Group V Elements of the Periodic Table
On fig. 1.12 shows a diagram of a Si crystal in which phosphorus (V group) is introduced. An element of group V (donor) has 5 valence electrons, four of them form bonds with neighboring Si atoms, the fifth electron is bonded only with an impurity atom and this bond is weaker than the others, therefore, when the crystal is heated, this electron is detached first, while the phosphorus atom acquires a positive charge becoming an ion.
(1.7)
where E d is the ionization (activation) energy of the donor atom.
The ionization energy of donors, as a rule, is not high (0.005 - 0.01 eV) and, at room temperature, almost all of them donate their electrons. In this case, the concentration of electrons that appeared due to the ionization of donors is approximately equal to the concentration of the introduced impurity atoms and significantly exceeds the intrinsic concentration of electrons and holes n>>n i , therefore such materials are called electronic materials (n-type).
We will call the electrons in them the main carriers and designate n n , respectively, the holes will be called minor charge carriers and designate p n .
Consider what happens when an element of group III, for example B, is introduced into the same Si. An element of group III has 3 valence electrons that form bonds with neighboring Si atoms, a fourth bond can form if another electron from one of its nearest neighbors, see fig. 10. The energy of such a transition is not high, so the corresponding energy level receiving (acceptor) electron is located near the valence band. In this case, the boron atom is ionized by charging negatively, and in the place where the electron left, a positively charged hole is formed, which can participate in the charge transfer.
where e v is an electron from the valence band, E a is the energy of the acceptor level relative to the top of the valence band.
Rice. 1.13. Scheme of the Formation of a Free Hole and a Charged Acceptor Atom in the Doping of Si with Group III Elements of the Periodic Table
The number of additionally appeared holes approximately corresponds to the number of introduced acceptor atoms and, as a rule, significantly exceeds the number of electrons that arise due to transitions from the valence band, so the material doped with an acceptor impurity is a hole (p type).
The introduction of an acceptor impurity leads to an increase in the hole concentration and, accordingly, to a shift of the Fermi level to the valence band (the closer it is to it, the greater the hole concentration).
Test questions.
1. Why can electrons in a semiconductor crystal transfer charge if they are in the conduction band and cannot transfer charge if they are in a filled valence band?
2. Explain why crystals consisting of elements of the first group are good conductors?
3. What do you think, if it were possible to obtain crystalline hydrogen, would it be a conductor or a semiconductor?
4. Why does the introduction of impurity atoms belonging to the fifth group of the periodic system of elements into silicon (germanium) lead to the appearance of free electrons in the conduction band?
5. Why does the introduction of impurity atoms belonging to the third group of the periodic system of elements into silicon (germanium) lead to the appearance of free holes in the conduction band?
