2 ATOMS, MOLECULES, AND SOLIDS
2.7 SEMICONDUCTORS
between the molecules. Because of the weakness of the van der Waals interaction, molecular solids are much easier to deform or compress than covalent or ionic solids.
Of course, electrical technology as we know it would be impossible withoutmetallic solids, which are good conductors of electricity (copper, silver, etc.). In a metallic solid the electrons are not all tightly bound at crystal lattice sites. Some of the electrons are free to move over large distances in the metal, much as atoms move freely in a gas.
This occurs because metals are formed from atoms in which there are one, two, or occasionally three outer electrons in unfilled configurations. The binding is associated with these weakly held electrons leaving their parent ions and being shared by all the ions, and so we can regard metallic binding as a kind of covalent binding. We can also think of the positive ions as being held in place because their attraction to the “electron gas” exceeds their mutual repulsion.
It is sometimes a useful approximation to regard the conduction electrons of a metal as completely free to move about. Of course, conduction electrons are not really completely free, as evidenced by the fact that even the very best conductors—copper, silver, and gold—have a finite resistance to the flow of electricity.
It should be emphasized that many solids do not fit so neatly into the covalent, ionic, molecular, or metallic categories. Furthermore many important properties of various solids are determined by imperfections such as impurities and dislocations in the crystal lattice. Steel, for instance, is much harder than pure iron because of the small amount of carbon that was mixed into the iron melt. Impurities can also determine the color of a crystal, as in the case of ruby (Section 3.1). We will shortly discuss how the addition of certain impurities in semiconductors is responsible for modern electronic technology.
in an electric circuit is controlled by the intensity of incident light have applications similar to photoelectric cells (exposure meters in photography, automatic door openers, etc.), except that they require an auxiliary voltage supply to move electrons that have been put into the conduction band.
By far the most important means of producing conduction electrons in a semicon- ductor is by doping it with a certain type of impurity. Tiny junctions of differently doped semiconductors are the basis not only for transistors but also for light-emitting diodes and diode lasers. To understand the operation of such devices, however, we must first discuss the concept of aholein the valence band of a semiconductor.
The basic idea is very simple. If an electron somehow goes from the valence band to the conduction band, it leaves a hole—the absence of an electron—in the valence band (Fig. 2.15). That is, a hole corresponds to the absence of an electron from an otherwise filled valence band. It turns out to be very useful to think of a hole as a particle like an electron. The removal of an electron increases the charge of the valence band, so clearly a hole must be a positively charged “particle,” with charge opposite to that of an electron.
Consider a piece of a semiconductor in which electron – hole pairs have been created in some way (i.e., electrons have been put into the conduction band, leaving holes in the valence band). If we connect it with wires to the terminals of a battery, there will be a flow of current since there are “free” electrons in the conduction band ready to respond to an externally applied field. These electrons drift in the direction shown in Fig. 2.16, from the negative electrode to the positive. The net effect, as seen from the outside, is that electrons enter the semiconductor from the right and exit to the left. However, this is not the whole story, for electrons in the valence band are also affected by the potential difference. Specifically, an electron to the right of the hole indicated in Fig. 2.16 can fall into the hole; that is, it will go into the state previously occupied by another electron. In doing so, it leaves a hole at the site it left, which can now be filled by another electron.
This electron drift in the valence band constitutes a current in the same direction (left to right, by convention, in Fig. 2.16) as the current of the electrons in the conduction band. Equivalently, we can view the situation as one in which electrons in the conduc- tion band are moving from right to left, while holes in the valence band are moving from left to right. In other words, we can describe the charge motion in the conduction band in terms of electrons, and that in the valence band in terms of holes, andboth electrons and holes contribute to the total current.
Conduction band electron
hole
Valence band
Figure 2.15 In going from the valence band to the conduction band, an electron leaves a hole in the valence band.
By doping a semiconductor with a certain kind of impurity, we can arrange for a current in the semiconductor to be due predominantly to either electrons or holes. In the former case the semiconductor is calledn type(because electrons are negatively charged) and in the latter it is calledp type(because holes have positive charge). To see how this works, we will consider the example of silicon doped with phosphorus.
In pure silicon each atom shares its four valence electrons in the unfilled 3s3psubshell (see inside cover) to form covalent bonds with its four nearest neighbors. Each silicon atom needs four more electrons to complete thespconfiguration, and by sharing elec- trons in this way it comes closer to having a filled outer subshell. The crystal structure is that of diamond, with each silicon atom at the center of a regular tetrahedron (pyramid) and its four nearest neighbors at the vertices. This structure is a consequence of the fact that the bonds associated with shared electrons are spaced as far from each other as possible at equal angles from each atom. It is useful to represent the situation in the schematic, two-dimensional form of Fig. 2.17.
In its pure form silicon has a very low conductivity at room temperature because so few electrons can be thermally excited across the 1.12-eV energy gap (eEg=kT e1 eV=1=40 eV¼ e40). Under ordinary circumstances the current passed is so small as to be practically useless. To pass useful current, we must find a way to get more electrons into the conduction band or holes into the valence band.
Suppose that one of the silicon atoms in the crystal is replaced by an atom of phosphorus, which has five electrons in the unfilled 3s3p subshell. Four of these can contribute to the covalent bonding of the crystal, as indicated in Fig. 2.18, but there is one electron left over that cannot take part in the bonding. This fifth electron is very loosely bound, and so is free to move through the crystal when an electric field is applied. In other words, if we add a small amount of phosphorus to a silicon melt, the crystal that forms will be ann-type semiconductor.7 We can also make an n-type semiconductor by doping silicon with other pentavalent elements such as arsenic and antimony.
+
+ –
e
Figure 2.16 When a potential difference is applied to two ends of a semiconductor, electrons in the valence and conduction bands drift from the negative side to the positive. In the valence band, the effect is equivalent to the drift of positively charged holes from the positive side to the negative. The total current can therefore be attributed to electrons in the conduction band and holes in the valence band.
7The proportion of dopant must be small in order to preserve the integrity of the host crystal lattice, since the dopant by itself forms its own crystal lattice structure.
2.7 SEMICONDUCTORS 41
Imagine instead that we replace a silicon atom by an atom of boron, which hasthree electrons in an unfilled outer shell. In this case there is one electron short of the four needed to join in complete covalent bonding in the host silicon lattice. Thus, if boron shares its three valence electrons with neighboring silicon atoms, there will be a missing bond in the crystal, as indicated in Fig. 2.19. This missing electron is a hole that can be filled by an electron that happens to be nearby. But when that electron fills the hole, it leaves another hole, which can be filled by another electron, and so in an electric field we get a migration ofholes (or equivalently, of course, a migration of electrons in the opposite direction). In other words, by doping silicon with boron we can create
Si Si
Si Si
Figure 2.17 Schematic illustration of covalent bonding in silicon, in which each atom shares its four valence electrons with its four nearest neighbors.
Si Si
Si
Extra electron
Si
Figure 2.18 If a silicon atom is replaced by a phosphorus atom in Fig. 2.17, there is an extra electron left over that cannot take part in the covalent bonding. This electron is very loosely bound and therefore available for conduction of electric current.
ap-type semiconductor. Other trivalent elements such as aluminum or gallium are also suitable dopants for this purpose.
As a matter of terminology, a dopant that produces ann-type semiconductor is called adonorbecause it donates electrons to the conduction band. A dopant that produces a p-type semiconductor is called anacceptorbecause it puts holes in the valence band, that is, it accepts electrons to fill the missing slots. In either case, of course, the crystal remains charge-neutral. Note also that the added impurities produce either electrons or holes, but no electron – hole pairs as in, for instance, photoconductivity. Thus, in ann-type semiconductor any current is due predominantly to electrons, whereas in p-type material it is due to holes.
Figure 2.20 shows an experiment that can distinguish between n-type and p-type semiconductors. Two ends of the material are connected to battery terminals to produce a current, and we also apply a magnetic fieldBat right angles to the current. The mag- netic field exerts a force qvB on particles of charge q moving with velocity v.
Regardless of the sign ofq, this magnetic force is upward for the arrangement shown (Problem 2.4). Therefore, the top will become positively charged or negatively charged, depending on the sign of the charge carriers. This displacement of charge creates an electric force on the charge carriers, and this electric force opposes the magnetic force. This is called theHall effect. In equilibrium these vertical electric and magnetic forces exactly cancel each other and the current flows horizontally. By measuring with a voltmeter the potential difference between the top and bottom of the sample, we can determine whether the charge carriers are positive or negative, and therefore whether the semiconductor isptype orntype.
Actually, some metals, such as beryllium, also exhibit ananomalous Hall effectin which the dominant charge carriers are positive. This is because beryllium has a filled 2ssubshell in which the holes happen to be much more mobile than the 2pelectrons.
The important point, again, is that it is very convenient to think in terms of electrons and holes, even though the real charge carriers are, of course, the electrons.
Si Si
Si
B Hole
Si
Figure 2.19 If a silicon atom in Fig. 2.17 is replaced by a boron atom, there will be a missing bond because there is one electron short of the number necessary for complete bonding. The missing electron is represented by a hole.
2.7 SEMICONDUCTORS 43
† The extra electron indicated in Fig. 2.18 is not actually free but is attracted by the positively charged impurity ion. The electron – ion system is thus analogous to the hydrogen atom. It might be expected, therefore, that there are electron orbits about the ion with allowed energies given by the Bohr formula (2.2.11). In particular, the energy of the lowest energy state should be
E1¼ 1 4pe0
2
me4
2h2, (2:7:1)
and an energyjE1j¼13.6 eV should be required to free the extra electron and make it available for conduction.
This argument overestimates the binding energy of the extra electron for two reasons. First, the free-space permittivitye0in (2.7.1) should be replaced by the material dielectric constant e. Second, it turns out that the band theory ascribes to electrons (and holes) a certaineffective mass m; because the electron is in the periodic potential of the crystal and not in free space, it actsas ifits mass were smaller, saym (see the Appendix to this chapter). The value ofm depends on the energy of the electron within an energy band and can also vary with direction within the crystal. Thus, we should replacee0byeandmbymin (2.7.1):
E1¼ e0
e
2 m m
1 4pe0
2
me4
2h2 ¼ e0
e
2 m
m (13:6 eV): (2:7:2) Using the valuese¼11.8e0andm¼0.26mfor silicon, we obtain
E1¼ 0:025 eV: (2:7:3)
Therefore, the extra electron is in fact bound very weakly, requiring only an energy of about 0.025 eV to put it into the conduction band. This small energy is about equal to kT401 eV at room temperature, so that thermal excitation is enough to free some extra electrons inn-type doped silicon.
The Bohr levels of the extra electrons represent new energy levels not found in the pure semiconductor. These levels are calleddonor levels. Because they are small and negative, the donor levels lie just below the bottom of the conduction band.
+ + + + + + + + +
– – – – – – – – – I
B
I
Figure 2.20 Experimental arrangement to determine the sign of the charge carriers (Hall effect).
A magnetic field Bis applied in the direction into the page. The top and bottom of the sample become chargedþand 2, respectively, if the carriers are positive, and2and þ, respectively, if the carriers are negative.
The hole produced by an acceptor impurity as in Fig. 2.19 is likewise bound, and it also requires only a small amount of energy to be freed. Its Bohr energy levels, calledacceptor levels, lie just above the top of the valence band.
In summary, the doping of a semiconductor with a donor or acceptor does not by itself produce conduction-band electrons or valence-band holes, as assumed in our discussion based on Figs. 2.18 and 2.19. However, the energy required to “ionize” these donors or acceptors is so small that thermal excitation at moderate temperatures will do the job. † Another important semiconductor, germanium, is similar to silicon in that it is tetravalent. It may, therefore, be doped in the same ways to producen-type andp-type materials. In addition to such elemental semiconductors are “III – V” binary semi- conductors such as gallium arsenide or indium antimonide, in which trivalent and pentavalent atoms share in covalent bonding as a result of unfilled spsubshells (Problem 2.5).