Fahlman B.D. Materials Chemistry

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CHAPTER 4

SEMICONDUCTORS

Our technologically advanced way of life would not be possible without the

semiconductor industry. The ﬁrst semiconductor device known as a transistor was

discovered at Bell Labs in the late 1940s, and was widely used shortly thereafter for

radio electronics. Today, transistors are still pervasive in every microelectronic

component such as CD/DVD players, cellular phones, modes of transportation

(e.g., planes, automobiles, etc.), and computers. In fact, modern computer chips

now contain over 2 billion individual transistors – all on a surface that is smaller than

a postage stamp!

This chapter will investigate the various types of semiconducting materials,

focusing on the inﬂuence of their structure on overall properties. We will also detail

the many applications for semiconductors, especially within the framework of

micro-electronic circuitry. It should be noted that nanostructural materials represent

a more recent realm of semiconduct ing materials, of use for next-generation solar

devices, sensors, batteries, etc. However, these materials will not be considered

in this chapter, but will instead be detailed in Chapter 6 that focuses solely on

nanotechnology.

4.1. PROPERTIES AND TYPES OF SEMICONDUCTORS

As their name implies, semiconductors possess electrical conductivity intermediate

between conductors such as metals and insulators such as ceramics. In Chapter 2,we

discussed the band structure for the molecular orbitals of inﬁnite lattices. The lack of an

energy gap between the ﬁlled (valence) and empty (conduction) bands indicated that no

thermal energy was required to facilitate electrical conductivity. That is, metals are able

to transport electrons through their lattices even at absolute zero. By contrast, semi-

conductors possess a bandgap between valence and conduction bands (Figure 4.1).

An appropriate amount of energy must be supplied to the material in order to promote

an electron from the valence band to the conduction band, wherein electron transport

may occur. For semiconductors, any temperature greater than 0 K is sufﬁcient for such

electron promotion; if the bandgap is too large, the material is unable to conduct

electricity, and is known as an insulator. The bandgap for semiconductors are typically

in the range 150–290 kJ mol

1

(i.e., 1–4 eV). Since the bandgap of Si is 1.12 eV, only

239

radiation with wavelengths ca. 1,100 nm (near IR, visible, UV, X-ray, etc.)will

be absorbed by silicon semiconductors, as the absorption of less-energetic radiation

will place the electron in the bandgap, where there are no available energy states.

The absorption of visible light by silicon results in its opacity, relative to transparent

glasses that do not absorb radiation within this wavelength regime.

Group 14 provides an interesting case-study for the most dramatic change in

electrical conductivity among its congeners. In fact, all types of conductors are

present in this group, from insulating carbon (diamond) to metallic tin and lead.

Since the electronic conﬁgurations o f these atoms are all {ns

}, the change in

electrical conductivities is related to the bond strength for the atoms comprising the

crystal lattice. Recall that the indivi dual atoms in diamond, Si, Ge, Sn, and Pb are

Figure 4.1. Electronic band diagrams for silicon. Shown is (a) bands resulting from overlap of sp

hybrid

orbitals and (b) bands resulting from overlap of molecular orbitals.

240 4 Semiconductors

tetrahedrally linked through the crystal lattice due to sp

hybridization. As the

individual s and p orbitals that comprise the hybrid orbital become more diffuse

(e.g., Sn, Pb), the bonding electrons are less tightly bound to individual atoms, and

become more polarizable. This results in delocalized metallic bonding in Sn and Pb

relative to very strong localized bonding in diamond. For intermediate Si and Ge, the

bonding between individual atoms is weaker than C, allowing the possibility for

thermal motion to break bonds in the solid-state lattice, by promoting bonding

electrons into the conduction band and propagating electrical conductivity.

There are two types of semiconductors: intrinsic and extrinsic (Figure 4.2).

Intrinsic semiconductors contain the same numbers of free bonding electrons (e



)

and holes (h

), created from the migration of electrons from the valence to conduc-

tion bands. The temperature-dependent concentration of e



carriers is known as

the intrinsic concentration ,n

(Figure 4.3). As we discusse d in Chapter 2, the

number of electrons that are able to traverse the bandgap is governed by the Fermi

function, which gives the probability of an electron occupying an available energy

state. The density of states (DOS), or numb er of available energy levels, is also

paramount for the promotion of electrons from valence-conduction bands. The DOS

begin at the bottom of the valence band and continue to the Fermi level – the highest

Figure 4.2. Band diagrams for intrinsic and extrinsic semiconductors. Shown are (a) an intrinsic

semiconductor, with an equivalent number of free electrons and holes, (b) an n-type extrinsic semiconductor,

with a greater number of electrons, and (c) a p-type extrinsic semiconductor, with an excess of holes.

4.1. Properties and Types of Semiconductors 241

occupied state at absolute zero (i.e., 273.15



C). For metals that readily exhibit

electrical conductivity, the Fermi level lies within the conduction band due to the

lack of a bandgap. However, unlike metals, the DOS for conduction electrons in a

semiconductor do not coincide with the Fermi level, but rather begin at the top of the

bandgap. Hence, the placement of the Fermi level represents the relative ease at

which an electron is promoted from the valance band to the conduction band in a

bulk semiconductor. Quantitatively, the density of occupied states per unit volume is

the product of the DOS and the Fermi function, f(E); the density of holes is the

product of the DOS and [1 – f(E)].

For extrinsic semiconductors, the Fermi level corr esponds to a level slightly above

or below conduc tion or valence bands, depending on the dopants introduced into the

lattice – either an excess of electrons or holes (Figure 4.2b, c). Since our frame of

reference for semiconducting ability is the Group 14 element of Si, the terms

electron-deﬁcient and electron-rich dopants refer to atoms possessing <4 valence

electrons (e.g., B, Al) and >4 electrons (e.g., N, P), respectively. By convention, if

additional electrons are introduced into the lattice, the sem iconductor is designated as

n-type, whereas the doping of additional holes yields a p-type semiconductor desig-

nation. From the convenient “n” and “p” notation, one may immediately recognize

whether there are excess negative or positive carriers in the lattice.

Figure 4.3. The intrinsic carrier concentration vs. temperature for Ge, Si, and GaAs. Reproduced with

permission from Kasap, S. O. Principles of Electronic Materials and Devices, 3rd ed., McGraw-Hill:

242 4 Semiconductors

Figure 4.4 illustrates the effect of temperature on the carrier concentration of a

semiconductor. At low temperatures, the e



concentration of extrinsic semicon-

ductors is governed by the ionization of the dopants (e.g.,B



,Al



for p-type;



,As



for n-type – Figure 4.4a). At the saturation temperature,T

, all donors

or acceptors have been ionized resulting in no further increase in the carrie r

concentration (Figure 4.4b). As the temperat ure is increased further, the intrinsic

temperature,T

, is reached when the thermal promotion of electrons across the

bandgap exceeds the concentration of acceptors/donors. Hence, at temperatures

above T

, an extrinsic semiconductor will exhibit an intrinsic carrier concentration,

with [e



] ¼ [h

] (Figure 4.4c). For a given carrier concentration, the lattice constant

will increase concomitantly with temperature. Accordingly, this will decrease the

energy required to break bonds thereby lowering the bandgap (Eq. 1 – the Varshni

equation, Figure 4.5).

¼ E



B+T

ð1Þ

where E

is the bandgap at T ¼ 0 K (GaAs ¼ 1.519 eV, Si ¼ 1.7 eV); A and B are

material-speciﬁc constants (e.g., Si: A ¼ 4.73  10

4

eV K

1

,B¼ 636 K; GaAs:

A ¼ 5.405  10

4

eV K

1

,B¼ 204 K), and T ¼ temperature (in K).

The temperature dependence is quite different for metals and semiconductors. At

a temperature of 0 K, a semiconductor will behave as a perfect insulator. However,

metals will exhibit electrical conductivity at absolute zero due to the delocalized

electron density and lattices described in Chapter 3. However, as the temperature is

increased, the resp ective conductivities of these materials will be reversed, with

Figure 4.4. Position of the Fermi level (E

) and relative ionization of n-type Si with increasing temperature.

Below the saturation temprature (T

), only a few As dopants are ionized; however, at increasing temperatures,

more As atoms become ionized until the intrinsic temperature is reached (T

). At that temperature, all dopants

have been ionized, and the electrical conductivity results from promotion of electrons across the bandgap.

Reproduced with permission from Kasap, S. O. Principles of Electronic Materials and Devices,3rded.,

4.1. Properties and Types of Semiconductors 243

metals showing a decrease and semiconductors an increase in conductivity

(Figure 4.6). The thermal motion of metal atoms causes less efﬁcient electron

mobility through the lattice, whereas a temperature increase causes the bandgap to

narrow for semiconductors, resulting in more effective electrical conductivity.

As the temperature continues to increase for semiconductors, the linear relationship

does not continue to hold, and the resistance begins to increase analogous to metals.

Instead, the atomic vibrations caused by the elevated temperature begin to outweigh

the thermally-induced decrease of the bandgap.

As Figure 4.7 illustrates, when thermal energy promotes a bonding electron from

the valence band to the conduction band, the released electrons are free to migrate

Figure 4.5. The effect of increasing temperature on the position of the donor/acceptor energy levels

within the bandgap for various doping levels, N

or N

. Reproduced with permission from Kasap, S. O.

The McGraw-Hill Companies.

Figure 4.6. Resistivity–temperature relationships for (a) metals and (b) semiconductors.

244 4 Semiconductors

throughout the lattice. However, the vacancies (i.e., holes) left behind are also free to

move – in the opposite direction as electrons. One may consider these holes as

positively charged species formed from loss of an electron. Thus, electrons and holes

represent the two types of carriers that correspond to electrical conductivity in

semiconductors.

Since an electron that has been promoted to the conduction band will have a

greater energy than those left in the valence band, there is a possibility for the

electron to lose this excess energy. The spontaneous return of electrons in the

conduction band to the valance band is known as recombination, and is usually

accompanied by light emission and heat (Figure 4.8). This phenomenon happens all

the time for excited-state molecules. For instance, consider what happens when one

Figure 4.7. Illustration of creation/migration of electrons and holes created through Si–Si bond

thermolysis. Shown are (a) the release of an electron and concomitant formation of a hole and (b) the

migration of an electron from a nearby bond to ﬁll the vacancy.

Conduction Band

Valence Band

Energy

hn or D

Figure 4.8. Schematic of recombination of electron-hole pairs generating either a photon of energy or heat.

4.1. Properties and Types of Semiconductors 245

supplies sodium atoms with sufﬁcient ener gy to promote an electron from the 3s

energy level into an empty 3p, or 4s orbital. The ele ctron does not stay in the higher

energy level very long (typically ca.10

6

s) before it is spontaneously dropped to its

original ground state, releasing heat and/or light. There are three types of recombi-

nation

[1]

(i) Band-to-band: an electron moves from the conduction band into an empty state

in the valence band associated with the hole. This yields a radiative emission,

whose wavelength is inversely proportional to the bandgap.

(ii) Trap-assisted: an electron falls into an energy level within the bandgap that is

associated with a dopant atom or structural defect.

(iii) Auger recombination: when a conduction-band electron and valence-band hole

recombine, rather than emitting light the energy may kick off an outer-shell

electron known as an Auger electron.

To understand the band structure and recombination process in more detail, we

must revisit some concepts of quantum physics discussed in Chapter 2. Recall that

matter and light exhibit both wave-and particle-like behavior. This duality may be

expressed by the de Broglie equation, which equates the wavelength and momen-

tum, p, of a particle (Eq. 2). The potential energy of an electron in a crystal lattice

depends on its location, and will be periodic due to the regular array of lattice atoms.

The periodic wave functions that result from solving the Schr

€

odinger equation are

referred to as Bloch wave functions. Each wave function represents the energy of an

electron at a speciﬁc location in the lattice, referred to as k-space. Hence, an E-k

diagram may be constructed, with the potential energy of the electron on the y-axis,

and the wave vector, k, on the x-axis. The wave vector represents lattice directions of

the semiconductor crystal; changing values of k represents a change in momentum

of the electron. A comparison of the simpl iﬁed E-k diagrams for Si or Ge and a

compound semiconductor such as GaAs is shown in Figure 4.9. For GaAs, the

-k

Photon

Phonon

Direct bandgap

Figure 4.9. Comparison of (a) direct bandgap (e.g., GaAs) and (b) indirect bandgap (e.g., Si, Ge)

materials. Reproduced with permission from Kasap, S. O. Principles of Electronic Materials and

Devices, 2nd ed., McGraw-Hill: New York, 2002.

246 4 Semiconductors

minimum of the conduction band (CB) is directly above the maximum of the

valence band (VB). Accordingly, we refer to such solids as direct bandgap materi-

als. In contrast, for Si and Ge, the CB minimum and VB maximum are offset,

resulting in an indirect bandgap.

l ¼

ð2Þ

For Si, in order for an electron at the bottom of the CB to recombine with a hole from

the top of the VB, the momentum of the electron must shift from k

to k

(Figure 4.9b). However, this is not allowed by the Law of Conservation of Momen-

tum. Instead, an indirect recombination mechanism must take place, wherein the

electron is captured by an interstitial defect with energy E

, which facilitates its

relaxation to the top of the VB. This process is accompanied by the emission of

phonons, or lattice vibrations rather than light emission. In contrast, efﬁcient

electron-hole recombination may occur without any change in momentum for direct

bandgap materials, resulting in the emission of photons. We will describe some

important applications for dir ect bandgap semiconductors later in this chapter.

The band diagram for real solids is not as straight forward as what we have

illustrated thus far. Figure 4.10 presents the real band diagram s for indire ct and

direct bandgap semiconductors Si and GaAs, respectively, showing a great deal of

complexity associated with valence and conduction bands.

[2]

As noted in Chapter 2,

the Brillouin zones (BZ; primitive unit cell of the reciprocal lattice) for 3-D crystal

lattices are complex polyhedra (Figure 2.76). Hence, the band energy varies in 3-D

k-space and the resultant lines are slices through the BZ polyhedron in speciﬁc

Figure 4.10. Electronic energy bands for Si and GaAs. The bands below the gray zone (bandgap) are

completely ﬁlled (valence band – VB), and those above are completely empty (conduction band – CB) at

0 K. Reproduced with permission from Hofmann, P. Solid State Physics: An Introduction, Wiley:

4.1. Properties and Types of Semiconductors 247