Semiconductors (Electrical Properties of Materials) Part 2

Extrinsic Semiconductors

Donors and Acceptors

We learned in the previous section that in intrinsic semiconductors only a very small number of electrons (about 109 electrons per cubic centimeter) contribute to the conduction of the electric current. In most semiconductor devices, a considerably higher number of charge carriers are, however, present. They are introduced by doping, i.e., by adding small amounts of impurities to the semiconductor material. In most cases, elements of group III or V of the periodic table are used as dopants. They replace some regular lattice atoms in a substitutional manner. Let us start our discussion by considering the case where a small amount of phosphorus (e.g., 0.0001%) is added to silicon. Phosphorus has five valence electrons, i.e., one valence electron more than silicon. Four of these valence electrons form regular electron-pair bonds with their neighboring silicon atoms (Fig. 8.6). The fifth electron, however, is only loosely bound to silicon, i.e., the binding energy is about 0.045 eV (see topic 4 and Problem 10.)

Two-dimensional representation of the silicon lattice. An impurity atom of group V of the periodic table (P) is shown to replace a silicon atom. The charge cloud around the phosphorus atom stems from the extra phosphorus electron. Each electron pair between two silicon atoms constitutes a covalent bond (electron sharing). The two electrons of such a pair are indistinguishable, but must have opposite spin to satisfy the Pauli principle.


Figure 8.6. Two-dimensional representation of the silicon lattice. An impurity atom of group V of the periodic table (P) is shown to replace a silicon atom. The charge cloud around the phosphorus atom stems from the extra phosphorus electron. Each electron pair between two silicon atoms constitutes a covalent bond (electron sharing). The two electrons of such a pair are indistinguishable, but must have opposite spin to satisfy the Pauli principle.

At slightly elevated temperatures this extra electron becomes disassociated from its atom and drifts through the crystal as a conduction electron when a voltage is applied to the crystal. Extra electrons of this type are called "donor electrons." They populate the conduction band of a semiconductor, thus providing a contribution to the conduction process.

It has to be noted that at sufficiently high temperatures, in addition to these donor electrons, some electrons from the valence band are also excited into the conduction band in an intrinsic manner. The conduction band contains, therefore, electrons from two sources, the amount of which depends on the device temperature (see Section 8.3.3). Since the conduction mechanism in semiconductors with donor impurities (P, As, Sb) is predominated by negative charge carriers (electrons) these materials are called n-type semiconductors. The electrons are the majority carriers.

A similar consideration may be done with impurities from the third group of the Periodic Chart (B, Al, Ga, In). They possess one electron less than silicon and, therefore, introduce a positive charge cloud into the crystal around the impurity atom. The conduction mechanism in these semiconductors with acceptor impurities is predominated by positive carriers (holes) which are introduced into the valence band. They are therefore called p-type semiconductors.

Band Structure

The band structure of impurity or extrinsic semiconductors is essentially the same as for intrinsic semiconductors. It is desirable, however, to represent in some way the presence of the impurity atoms by impurity states.

 (a) Donor and (b) acceptor levels in extrinsic semiconductors.

Figure 8.7. (a) Donor and (b) acceptor levels in extrinsic semiconductors.

It is common to introduce into the forbidden band so-called donor or acceptor levels (Fig. 8.7). The distance between the donor level and the conduction band represents the energy that is needed to transfer the extra electrons into the conduction band. (The same is true for the acceptor level and valence band.) It has to be emphasized, however, that the introduction of these impurity levels does not mean that mobile electrons or holes are found in the forbidden band of, say, silicon. The impurity states are only used as a convenient means to remind the reader of the presence of extra electrons or holes in the crystal.

Temperature Dependence of the Number of Carriers

At 0 K the excess electrons of the donor impurities remain in close proximity to the impurity atom and do not contribute to the electric conduction. We express this fact by stating that all donor levels are filled. With increasing temperature, the donor electrons overcome the small potential barrier (Fig. 8.7(a)) and are excited into the conduction band. Thus, the donor levels are increasingly emptied and the number of negative charge carriers in the conduction band increases exponentially, obeying an equation similar to (8.9). Once all electrons have been excited from the donor levels into the conduction band, any further temperature increase does not create additional electrons and the Ne versus T curve levels off (Fig. 8.8). As mentioned before, at still higher temperatures intrinsic effects create additional electrons which, depending on the amount of doping, can outnumber the electrons supplied by the impurity atoms.

Similarly, the acceptor levels do not contain any electrons at 0 K. At increasing temperatures, electrons are excited from the valence band into the acceptor levels, leaving behind positive charge carriers. Once all acceptor levels are filled, the number of holes in the valence band is not increased further until intrinsic effects set in.

Schematic representation of the number of electrons per cubic centimeter in the conduction band versus temperature for an extrinsic semiconductor with low doping.

Figure 8.8. Schematic representation of the number of electrons per cubic centimeter in the conduction band versus temperature for an extrinsic semiconductor with low doping.

Conductivity

The conductivity of extrinsic semiconductors can be calculated, similarly as in the previous section (8.13), by multiplying the number of carriers by the mobility, m, and electron charge, e. Around room temperature, however, only the majority carriers need to be considered. For electron conduction, for example, one obtains

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where Nde is the number of donor electrons and me is the mobility of the donor electrons in the conduction band. As mentioned above, it is reasonable to assume that, at room temperature, essentially all donor electrons have been excited from the donor levels into the conduction band (Fig. 8.8). Thus, for pure n-type semiconductors, Nde is essentially identical to the number of impurities (i.e., donor atoms), Nd. At substantially lower temperatures, i.e., at around 100 K, the number of conduction electrons needs to be calculated using an equation similar to (8.8).

Figure 8.9 shows the temperature dependence of the conductivity. We notice that the magnitude of the conductivity, as well as the temperature dependence of s, is different for various doping levels. For low doping rates and low temperatures, for example, the conductivity decreases with increasing temperature (Fig. 8.9(b)). This is similar to the case of metals, where the lattice vibrations present an obstacle to the drifting electrons (or, expressed differently, where the mobility of the carriers is decreased by incoherent scattering of the electrons). However, at room temperature intrinsic effects set in, which increase the number of carriers and therefore enhance the conductivity. As a consequence, two competing effects determine the conductivity above room temperature: an increase of s due to an increase in the number of electrons, and a decrease of s due to a decrease in mobility. (It should be mentioned that the mobility of electrons or holes also decreases slightly when impurity atoms are added to a semiconductor.)

Conductivity of two extrinsic semiconductors, (a) relatively high doping and (b) low doping. Nd = number of donor atoms per cubic centimeter.

Figure 8.9. Conductivity of two extrinsic semiconductors, (a) relatively high doping and (b) low doping. Nd = number of donor atoms per cubic centimeter.

For high doping levels, the temperature dependence of s is less pronounced due to the already higher number of carriers (Fig. 8.9(a)). The resistivity (1/s) of p-and n-doped silicon at room temperature is given in graphical form in topic 4.

Fermi Energy

In an n-type semiconductor, more electrons can be found in the conduction band than holes in the valence band. This is particularly true at low temperatures. The Fermi energy must therefore be between the donor level and the conduction band (Fig. 8.10). With increasing temperatures, an extrinsic semiconductor becomes progressively intrinsic and the Fermi energy approaches the value for an intrinsic semiconductor, i.e., —(Eg/2). [Similarly, the Fermi energy for a p-type semiconductor rises with increasing temperature from below the acceptor level to — (Eg/2).]

Fermi level of an n-type semiconductor as a function of temperature. Nd « 1016 (atoms per cubic centimeter).

Figure 8.10. Fermi level of an n-type semiconductor as a function of temperature. Nd « 1016 (atoms per cubic centimeter).

Effective Mass

Some semiconductor properties can be better understood and calculated by evaluating the effective mass of the charge carriers. We mentioned in Section 6.7 that m* is inversely proportional to the curvature of an electron band. We now make use of this finding.

Let us first inspect the upper portion of the valence bands for silicon near r (Fig. 8.2). We notice that the curvatures of these bands are convex downward. It is known from Fig. 6.8 that in this case the charge carriers have a negative effective mass, i.e., these bands can be considered to be populated by electron holes. Further, we observe that the curvatures of the individual bands are slightly different. Thus, the effective masses of the holes in these bands must likewise be different. One distinguishes appropriately between light holes and heavy holes. Since two of the bands, namely, those having the smaller curvature, are almost identical, we conclude that two out of the three types of holes are heavy holes.

We turn now to the conduction band of silicon and focus our attention on the lowest band (Fig. 8.2). We notice a minimum (or valley) at about 85% between the r and X points. Since the curvature at that location is convex upward, we expect this band to be populated by electrons. (The energy surface near the minimum is actually a spheroid. This leads to longitudinal and transverse masses m{* and m*.) Values for the effective masses are given in topic 4. Occasionally, average effective masses are listed in the literature. They may be utilized for estimates.

Hall Effect

The number and type of charge carriers (electrons or holes) that were calculated in the preceding sections can be elegantly measured by making use of the Hall effect. Actually, it is quite possible to measure concentrations of less than 1012 electrons per cubic centimeter in doped silicon, i.e., one can measure one donor electron (and therefore one donor atom) per 1010 silicon atoms. This sensitivity is several orders of magnitude better than in any chemical analysis.

We assume for our discussion an n-type semiconductor in which the conduction is predominated by electrons. Suppose an electric current has a current density j, pointing in the positive x-direction (which implies by definition that the electrons flow in the opposite direction). Further we assume that a magnetic field (of magnetic induction B) is applied normal to this electric field in the z-direction (Fig. 8.11). Each electron is then subjected to a force, called the Lorentz force, which causes the electron paths to bend, as shown in Fig. 8.11. As a consequence, the electrons accumulate on one side of the slab (in Fig. 8.11 on the right side) and are deficient on the other side.

Schematic representation of the Hall effect in an n-type semiconductor (or a metal in which electrons are the predominant current carriers).

Figure 8.11. Schematic representation of the Hall effect in an n-type semiconductor (or a metal in which electrons are the predominant current carriers).

Thus, an electric field is created in the (negative) y-direction which is called the Hall field. In equilibrium, the Hall force

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balances the above-mentioned Lorentz force

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which is proportional to the velocity, vx, of the electrons, the magnetic induction Bz, and the electron charge, e. FH + FL = 0 yields, for the Hall field,

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Combining (8.18) with (7.4) (and knowing that the current is directed in a direction opposite to the electron flow; see above)

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yields for the number of conduction electrons (per unit volume)

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where Ax is the area perpendicular to the electron flow and Vy is the Hall voltage measured in the y-direction.

The variables on the right side of (8.20) can all be easily measured and the number of conduction electrons can then be calculated. Quite often, a Hall constant

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is defined which is inversely proportional to the density of charge carriers, N. The sign of the Hall constant indicates whether electrons or holes predominate in the conduction process. RH is negative when electrons are the predominant charge carriers. (The electron holes are deflected in the same direction as the electrons but travel in the opposite direction.)

There exists an anomalous Hall effect also called extraordinary Hall effect which is observed in ferromagnetic materials and induces an addition to the ordinary Hall effect. This contribution is larger than the ordinary Hall effect and is caused by the magnetization of the conductor. Its origin is still being debated.

The Hall conductivity, sy, has been found to be quantized at multiples of e2/h (where h, is as usual, the Planck constant). This quantum Hall effect (QHE) can be particularly observed for very clean Si or GaAs, at very low temperatures (around 3 K) and very high magnetic fields (e.g. 18 Tesla). If multiple integers of e2/h are found, one refers to it as the integer quantum Hall effect which is explained by postulating single particle orbitals of an electron in a magnetic field (Landau quantization). When rational fractions, u, of e2/h are observed (where u = 1/3, 1/5. 5/2, 12/5,… is called the filling factor), the phenomenon is termed the fractional quantum Hall effect which is explained by electron-electron interactions. Since the QHE can be measured to an accuracy of nearly one part per billion, it is used as a standard for the electrical resistance. (h/e2 = 25812.807557 O is called the von Klitzing constant or "quantum of resistance"). Interestingly enough, the QHE was first measured (by von Klitzing in 1980) in silicon field-effect transistors in which the electron concentration, N, can be continuously varied by changing the gate voltage, (see Section 8.7.9). The Hall effect is then measured by analyzing the Hall voltage as a function of the gate voltage. Plateaus are observed when the ratio between N and the number of flux quanta (the smallest unit of magnetic flux which can be enclosed in an electron orbit) is an integer. The Nobel Price in physics was awarded in 1985 to von Klitzing for discovering the integer QHE, whereas Tsui, Stormer, and Laughlin obtained it in 1998 for the fractional QHE.

Compound Semiconductors

Gallium arsenide (a compound of group III and group V elements of the Periodic Table) is of great technical interest, partially because of its large band gap,12 which essentially prevents intrinsic contributions in impurity semiconductors even at elevated temperatures, partially because of its larger electron mobility,12 which aids in high-speed applications, and particularly because of its optical properties, which result from the fact that GaAs is a "direct-band gap" material (see topic 12). The large electron mobility in GaAs is caused by a small value for the electron effective mass, which in turn results from a comparatively large convex upward curvature of the conduction electron band near r. (See in this context the band structure of GaAs in Fig. 5.24.) The electrons which have been excited into the conduction band (mostly from donor levels) most likely populate this high curvature region near r.

The atomic bonding in III-V and II-VI semiconductors resembles that of the group IV elements (covalent) with the additional feature that the bonding is partially ionic because of the different valences of the participating elements. The ionization energies12 of donor and acceptor impurities in GaAs are as a rule one order of magnitude smaller than in germanium or silicon, which ensures complete ionization even at relatively low temperatures. The crystal structure of GaAs is similar to that of silicon. The gallium atoms substitute for the corner and face atoms, whereas arsenic takes the places of the four interior sites (zinc-blende structure).

The high expectations that have been set for GaAs as the semiconductor material of the future have not yet materialized to date. It is true that GaAs devices are two and a half times faster than silicon-based devices, and that the "noise" and the vulnerability to cosmic radiation is considerably reduced in GaAs because of its larger band gap. On the other hand, its ten-times higher price and its much greater weight (dSi = 2.3 g/cm3 compared to dGaAs = 5.3 g/cm3) are serious obstacles to broad computer-chip usage or for solar panels. Thus, GaAs is predominantly utilized for special applications, such as high-frequency devices (e.g., 10 GHz), certain military projects, or satellite preamplifiers. One of the few places, however, where GaAs seems to be, so far, without serious competition is in optoelectronics (though even this domain appears to be challenged according to the most recent research results).

We will learn in Part III that only direct band-gap materials such as GaAs are useful for lasers and light-emitting diodes (LED). Indirect-band gap materials, such as silicon, possess instead the property that part of the energy of an excited electron is removed by lattice vibrations (phonons). Thus, this energy is not available for light emission. We shall return to GaAs devices in Section 8.7.9.

GaAs is, of course, not the only compound semiconductor material which has been heavily researched or is being used. Indeed, most compounds consisting of elements of groups III and V of the periodic table are of some interest. Among them are GaP, GaN, InP, InAs, InSb, and AlSb, to mention a few.12 But also, group II-VI compounds, such as ZnO, ZnS, ZnSe, CdS, CdTe, or HgS are considered for applications. These compounds have in common that the combination of the individual elements possesses an average of four valence electrons per atom because they are located at equal distances from either side of the fourth column. Another class of compound semiconductors is the group IV-VI materials,12 which include PbS, PbSe, and PbTe. Finally, ternary alloys, such as AlxGa1—xAs, or quaternary alloys, such as AlxGa1—xAsySb1—y, are used. Most of the compounds and alloys are utilized in optoelectronic devices, e.g., GaAs1—xPx for LEDs, which emit light in the visible spectrum (see Part III). AlxGa1—xAs is also used in modulation-doped field-effect transistors (MODFET).

Finally, silicon carbide is the most important representative of the group IV-IV compounds. Since its band gap is around 3 eV, a-SiC can be used for very-high-temperature (700° C) device applications and for LEDs that emit light in the blue end of the visible spectrum. SiC is, however, expensive and cannot yet be manufactured with reproducible properties. Ga-N-In have now replaced SiC as blue-emitting LEDs, see p. 280.

Doping of GaAs is accomplished, for example, by an excess of Ga atoms (p-type) or an excess of As (n-type). Si acts as a donor if it replaces Ga atoms and as an acceptor by substituting for As atoms. The recently refined technique of molecular beam epitaxy (MBE) allows the production of the wanted compounds and dopings.

Semiconductor Devices

Metal-Semiconductor Contacts

If a semiconductor is coated on one side with a metal, a rectifying contact or an ohmic contact is formed, depending on the type of metal used. Both cases are equally important. Rectifiers are widely utilized in electronic devices, e.g., to convert alternating current into direct current. However, the type discussed here has been mostly replaced by p-n rectifiers. On the other hand, all semiconductor devices need contacts in which the electrons can easily flow in both directions. They are called ohmic contacts because their current-voltage characteristic obeys Ohm’s law (7.1).

At the beginning of our discussion let us assume that the surface of an n-type semiconductor has somehow been negatively charged. The negative charge repels the free electrons that had been near the surface and leaves positively charged donor ions behind (e.g., As+). Any electron that drifts toward the surface (negative x-direction in Fig. 8.12(a)) "feels" this repelling force. As a consequence, the region near the surface has fewer free electrons than the interior of the solid.

 (a) Band diagram for an n-type semiconductor whose surface has been negatively charged. (b) Band diagram for a p-type semiconductor, the surface of which is positively charged. X is the distance from the surface.

Figure 8.12. (a) Band diagram for an n-type semiconductor whose surface has been negatively charged. (b) Band diagram for a p-type semiconductor, the surface of which is positively charged. X is the distance from the surface.

This region is called the depletion layer (or sometimes space-charge region).

In order to illustrate the repelling force of an external negative charge, it is customary to curve the electron bands upward near the surface. The depletion can then be understood by stating that the electrons assume the lowest possible energy state (or colloquially expressed: "The electrons like to roll downhill"). The depletion layer is a potential barrier for electrons.

Similarly, if a p-type semiconductor is positively charged at the surface, the positive carriers (holes) are repelled toward the inner part of the crystal and the band edges are bent downward (Fig. 8.12(b)). This represents a potential barrier for holes (because holes "want to drift upward" like a hydrogen-filled balloon).

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