ABSTRACT
Origin of semiconductivity in materials• Differences between elemental versus compound, direct versus indirect bandgap, and • intrinsic versus extrinsic semiconductors Band diagrams for n-type and p-type semiconductors• Conductivity of semiconductors in relation to the majority and minority carrier • concentrations Factors that affect the conductivity of semiconductors• Device applications such as light-emitting diodes (LEDs)• Changes in the bandgap with temperature, dopant concentrations, and crystallite size • (quantum dots) Semiconductivity in ceramic materials•
Semiconductors are defined as materials with resistivity (ρ) between ∼10−4 and ∼103 Ω · cm An approximate range of sensitivities exhibited by silicon (Si)-based semiconductors is shown in Figure 31 Elements showing semiconductivity are called elemental semiconductors (eg, silicon), and the compounds that show semiconducting behavior are known as compound semiconductors (eg, gallium arsenide [GaAs]) A band diagram of a typical semiconductor is shown in Figure 32 For most semiconductors, the bandgap energy (Eg) is between ∼01 and 40 eV If the bandgap is larger than 40 eV, we usually consider the material to be an insulator or a dielectric In this chapter, we will learn that the composition of such dielectric materials can be altered so that they exhibit semiconductivity (see Section 321) As shown in Figure 32, the top of the valence band is known as the valence bandedge (Ev), and the bottom of the conduction band is known as the conduction bandedge (Ec) Recall from Chapter 2 that the band diagram shows the outermost part of the overall electron energy levels of a solid The vertical axis shows the increasing electron energy Thus, the magnitude of the bandgap energy (Eg) is given by
E E Eg c v= − (31)
Let us consider the origin of the semiconducting behavior in semiconductors such as silicon Silicon has covalent bonds; each silicon atom bonds with four other silicon atoms, which leads to the formation of a three-dimensional network of tetrahedra arranged in a diamond cubic crystal structure (Figure 33)
Figure 34 shows a two-dimensional representation of the covalent bonds between silicon atoms When the temperature is low (∼0 K), the valence electrons shared between the silicon atoms remain in the bonds and are not available for conduction Thus, at low temperatures, silicon behaves as an insulator (Figure 34) As the temperature increases, the electrons gain thermal energy Some
electrons can gain sufficient energy to break away from the covalent bonds These electrons are now free to move around and impart semiconductivity to the material (Figure 34)
The electrons breaking away from the bonds (Figure 34) can also be shown on a band diagram (Figure 35) At low temperatures, the valence electrons are in the covalent bonds, that is, the valence band is completely filled As the temperature increases to ∼>100 K, a small fraction of electrons gain enough thermal energy to make a jump across the bandgap (Eg) and into the conduction band
When an electron breaks away from a covalent bond, it leaves behind an incomplete bond A hole is an imaginary particle that represents an electron missing from a bond (Figure 36) On a band
diagram, a hole is an energy state left empty by an electron that moved to the conduction band If a hole is present at site X, then another electron from a neighboring bond at site Y can move into site X This creates a hole at site Y Movement of an electron from site Y to X is equivalent to the movement of a hole from site X to site Y Thus, the movement of holes also contributes to a semiconductor’s electrical conductivity
In materials such as silicon and germanium (Ge), the bandgap energy at room temperature is relatively low (for germanium and silicon, Eg is ∼067 and ∼11 eV, respectively) When we say the bandgap is small, we are comparing the bandgap energy with the thermal energy given by kBT, where kB is the Boltzmann’s constant (8617 × 10−5 eV/K or 138 × 10−23 J/K) and T is the temperature At T = 300 K (∼room temperature), the thermal energy kBT is ∼0026 eV
Promoting an electron into the conduction band creates an electron-hole pair (EHP; Figures 35 and 36) Electrons promoted into the conduction band because of thermal energy and the resulting holes that are created in the valence band are known as thermally generated charge carriers
A semiconductor in which charge carriers created as a result of thermal energy are the only source for creating conductivity is known as an intrinsic semiconductor Note that even if EHPs are created, the material still remains electrically neutral In an intrinsic semiconductor, the concentration of the electrons available for conduction (ni) is equal to that of the holes created (pi)
n pi i= (32)
Therefore, Figures 32 and 35 show the concentrations of conduction electrons and holes to be equal The conductivity of intrinsic semiconductors is controlled by the material itself and by the temperature that changes the carrier concentrations The word “intrinsic” emphasizes that no other “extrinsic,” or foreign elements or compounds are present in significant enough concentrations to have any effect the electrical properties of an intrinsic semiconductor Appropriately, a semiconductor whose conductivity and other electrical properties are controlled by foreign elements or compounds is known as an extrinsic semiconductor (see Section 38)
The conductivity of an intrinsic semiconductor is given by the following equation:
σ µ µ= +qn qpi i pn (33)
In this equation, q is the magnitude of the charge on the electron or hole (16 × 10−19 C), and ni and pi are the concentrations of electrons and holes in an intrinsic material, respectively The terms μn and μp are the mobilities of electrons and holes, respectively Since ni = pi for an intrinsic semiconductor, Equation 33 can be rewritten as
σ µ µ= +qni p( )n (34)
For a given intrinsic semiconductor, the electron or hole concentrations depend mainly on the temperature
Example 31 illustrates the calculation of the resistivity of an intrinsic semiconductor
SOLUTION We make use of Equation 3.4
As we can expect, at any given temperature the larger the bandgap (Eg) of a semiconductor, the lower the concentration of valence electrons (ni) that can pass across the bandgap The relationship among the carrier concentration, the bandgap, and the temperature is given by
n T E
k Ti ~ 3 2
2 / exp −
(35)
The exponential term dominates, and hence the plot of ln(ni) with 1/T is essentially a straight line Note that even though the increase is exponential, only a very small fraction of the total number of valence electrons actually gets into the conduction band In Figure 37, a plot of ni on a logarithmic scale is shown as a function of the inverse of the temperature The slope of the line is essentially proportional to the bandgap (Eg) The bandgaps of germanium, silicon, and GaAs are ∼067, 11, and 143 eV, respectively From Equation 35, we expect the concentration of free electrons at a given temperature to be the highest for germanium, since it has the smallest bandgap of the three On the other hand, GaAs will have the lowest concentration of thermally generated conduction electrons, since it has the largest bandgap of these three materials (Figure 37) The bandgap values and some
of the other properties of semiconductors are shown in Table 31 The terms direct and indirect bandgap semiconductors, used in Table 31, are defined in Section 35
In the classical theory of conductivity, electrons are considered particles The kinetic energy (E) of an electron can be written as:
E mv= 1 2
2 (36)
where m is the mass of a free electron and v is the speed of the electron Momentum (p) is defined as mass × velocity
Therefore, we can also rewrite Equation 36 in terms of momentum (p) and mass as follows:
E p
m =
2 (37)
Note that we have also used the symbol “p” to designate the concentration of holes In a quantum mechanics-based approach, an electron is considered a plane wave with a propa-
gation constant, which is called the wave vector (k) The wave function for an electron is given by:
Ψ( , ) ( , ) exp( )kx k kx xx U x j x= (38)
where Ψ is the wave function that is related to the probability of finding an electron, U is the function that accounts for the periodicity of the crystal structure, j is the imaginary number, kx is the wave vector in x-direction along which the electron (now considered a wave) is traveling, and x is the distance
TABLE 3.1 Properties of Selected Semiconductors
Using quantum mechanics, we can show that the electron momentum (p) is related to wave vector (k) by the following equation:
p k= (39)
where , = h/2π = 1054 × 10−34 J · s, and h is the Planck’s constant (= 6626 × 10−34 J · s) From Equations 37 and 39, we can write the energy of an electron as:
E k
m =
2 (310)
Thus, for a free electron, that is, an electron that is not experiencing any other forces due to internal or external electric or magnetic fields, the relationship between its energy (E) and wave vector (k) is a parabola (Figure 38) The plot of electron energy as a function of the wave vector (k) is known as the band structure of a material
In a perfect crystal, if we assume that the electron is moving in a band and has a starting energy of V0, then the energy of the electron moving in this band is given by:
E k
m V= +
02 (311)
In semiconductors, electrons moving in different bands are not completely free They have built-in electric fields that are associated with other atoms, often periodically arranged in certain types of crystal structures Thus, the electrons move around in a material as if they have a different mass, known as the effective mass of an electron (me* or mn*) This is the mass that an electron would appear to have in a material and is different from the mass of a free electron in vacuum (m0 = 9109 × 10−31 kg) The concept of effective mass is illustrated in Figure 39 For convenience, we will express the effective mass as a dimensionless quantity m me*/ .0 This ratio is an indicator of the interactions of the electron with the atoms of the material Similarly, holes also have an effective mass
The significance of the effective mass is as follows: Smaller effective masses for carriers (electrons or holes) mean that the carriers can move faster, with a less apparent inertia This means materials with smaller apparent electron or hole masses are more useful for making faster semiconductor devices The concept of effective mass is quantum mechanical in nature, and analogies using classical mechanics must therefore be limited in scope
The effective masses of electrons and holes for some semiconductors are listed in Table 32 This table also lists the values of the dielectric constant (εr), which is a measure of the ability of a material to store a charge (Chapter 8) The dielectric constant is defined as the ratio of the permittivity of a material (ε) and the permittivity of the free space (ε0)
The effective mass of electrons is related to the E-k curvature as follows:
m d E
dk
2
(312)
Since a hole is an imaginary particle that represents a missing electron in the valence band, the effective mass of the hole is the negative of the mass of the missing electron The effective mass of holes is given by:
m d E
dk
h * = −
2
(313)
Note the negative sign in Equation 313 The effective mass of an electron and the E-k curvature (Equation 312) is derived as follows: The electron velocity is equal to the group velocity of the associated wave (vg), with which the
boundary of the wave propagates The group velocity is given by the following equation:
v d
dk vg = ( )2π (314)
where v is the frequency of the wave Rewrite Equation 314 as:
v d
dk
hv
hg =
2π (315)
Now, we replace hv with E and (h/2π) with , and we get:
v dE
dkg =
(316)
The acceleration (a) of an electron is given by:
a dv
dt =
g (317)
Substituting for vg from Equation 316 into Equation 317, we get:
a d E
dk
dk
dt =
1 2
2 (318)
Substituting for k with p/, we get:
a d E
dk
dp
dt =
1 2
2 (319)
Rewrite Equation 319 as:
a d E
dk
d mv
dt
d E
dk =
=
1 1 2
2
( ) F (320)
From classical mechanics, we know that F = m × a or a = F/m Comparing this with Equation 320, we get an expression for the effective mass of an electron as given in Equation 312 We can see from Equation 312 that the effective mass of an electron is inversely related to the E-k curvature The larger the curvature, the lesser the effective mass
For the conduction band of a semiconductor, we can write the relationship between the energy of an electron (E) and its wave vector k as follows:
E k E k
m ( )
* = +c
2 (321)
where Ec is the conduction bandedge energy and me* is the effective mass of the electron The effective mass of an electron depends strongly on the bandgap (Eg) The smaller the value of the bandgap (Eg), the smaller the value of me* (Table 32 and Figure 310)
TA B
LE 3
.2
