Introduction to Semiconductors
This is a continuation from the previous tutorial - optical fiber lasers.
Semiconductors are important materials. Because of their unique electronic properties. they are the materials of choice for modern electronic devices. Silicon, in particular, has become the most important material for the electronics industry.
Besides their unique properties for electronics applications, semiconductors also have many other important properties that are very useful for photonic device applications.
In earlier tutorials, we have already seen that III-V semiconductors are useful materials for optical waveguides and electro-optic devices. Many semiconductors are also used for acousto-optic devices and nonlinear optical devices. In such applications, which are based solely on the dielectric properties of semiconductors, semiconductors are nothing but another group of dielectric optical materials.
Nevertheless, semiconductors do have many optoelectronic properties that are not shared by other dielectric materials. These optoelectronic properties make semiconductors once again, beyond their unique position in the electronics industry, the key materials for many important optoelectronic devices, such as light-emitting diodes, semiconductor lasers, and photodetectors.
In this tutorial, we review the basic properties of semiconductors that are relevant to their optoelectronic device applications.
In the tutorials covering optical transitions, optical absorption and amplification, population inversion and optical gain, laser amplifiers, and rare-earth ion-doped fiber amplifiers, optical transitions between discrete atomic or molecular energy levels are considered, though the atoms or molecules may be embedded in a host solid-state material as dopants.
In a semiconductor, however, the allowed states of the electrons of its constituent atoms form continuous energy bands rather than discrete levels. The optical processes associated with such electrons are a strong function of the characteristics of the energy bands.
For a solid material in thermal equilibrium at a temperature \(T\), the probability of any electronic state at an energy \(E\) being occupied by an electron is given by the Fermi-Dirac distribution function:
\[\tag{12-1}f(E)=\frac{1}{\text{e}^{(E-E_\text{F})/k_\text{B}T}+1}\]
where \(E_\text{F}\) is the Fermi level of the material and \(k_\text{B}\) is the Boltzmann constant.
At a very low temperature approaching \(0\text{ K}\), all of the states below the Fermi level are occupied while those above it are empty.
If the Fermi level lies within an energy band, that band is partially filled. A solid with one or more partially filled bands is a metal because electrons in a partially filled band can move under an electric field to conduct an electric current.
If the Fermi level lies between two separate energy bands, as illustrated in Figure 12-1, all of the energy bands are either completely filled or completely empty at \(T\rightarrow0\text{ K}\).
The filled bands are known as the valence bands, whereas the empty bands are known as the conduction bands.
The lowest conduction band and the highest valence band are separated by an energy gap. The energy separation between the bottom of the lowest conduction band and the top of the highest valence band is called the bandgap, \(E_\text{g}\).
A material is an insulator if all of its bands are either completely filled or completely empty because, due to the Pauli exclusion principle, electrons in a completely filled energy band cannot move under an electric field.
At a nonzero temperature, however, electrons in high valence bands have a probability of being thermally excited to low conduction bands. The probability of this thermal excitation is a function of \(E_\text{g}/k_\text{B}T\); it increases with rising temperature but decreases with increasing bandgap.
The electrons that are excited to conduction bands become conduction electrons. The removal of electrons from valence bands results in electron deficiencies, known as holes, in the form of unoccupied electron states among occupied states.
An electron in a conduction band is a carrier of negative charge, whereas a hole in a valence band behaves like a carrier of positive charge. Both contribute to the electrical conductivity of a semiconductor. A semiconductor is an insulator at \(T\rightarrow0\text{ K}\), but has appreciable conductivity as the temperature rises.
The energy of an electron is a function of its quantum-mechanical wavevector, \(\mathbf{k}\). In a semiconductor, this dependence of electron energy on its wavevector forms the band structure of the semiconductor.
The illustration in Figure 12-1 shows the band structures of Si and GaAs.
In the case of Si, the minimum of conduction bands and the maximum of valence bands do not occur at the same \(\mathbf{k}\) value. A semiconductor that has such a band characteristic is called an indirect-gap semiconductor, and its bandgap is referred to as an indirect bandgap.
In contrast, a semiconductor like GaAs is a direct-gap semiconductor because its band structure is characterized by a direct bandgap, with the minimum of the conduction bands and the maximum of the valence bands occurring at the same value of \(\mathbf{k}\), which in this particular instance is \(\mathbf{k}=0\).
The minimum of the conduction bands is called the conduction-band edge, \(E_\text{c}\), and the maximum of the valence bands is called the valence-band edge, \(E_\text{v}\). The bandgap, \(E_\text{g}\), is the energy difference between \(E_\text{c}\) and \(E_\text{v}\):
\[\tag{12-2}E_\text{g}=E_\text{c}-E_\text{v}\]
The bandgap of a semiconductor is typically less than 4 eV. With the exception of some IV-VI compound semiconductors, such as lead salts, the bandgap of a semiconductor normally decreases with increasing temperature. The bandgaps and other properties of many important semiconductors are listed in Table 12-1.
In this table, \(\lambda_\text{g}\) is the free-space optical wavelength of a photon that has an energy equal to the bandgap of a given material: \(\lambda_\text{g}=hc/E_\text{g}\).
A semiconductor can be an elemental material or a compound material. The group IV elements Si and Ge are elemental semiconductors. Crystalline C can take the form either of diamond, which is more an insulator than a semiconductor because of its large bandgap of 5.47 eV at room temperature, or graphite, which is a semimetal.
Though C is not a semiconductor, Si and C can form the IV-IV compound semiconductor SiC, which has many different structural forms with different bandgaps. Si and Ge can be mixed to form the IV-IV alloy semiconductor SixGe1-x. These group IV crystal and IV-IV compounds are indirect-gap materials.
The most important semiconductors for photonic devices, however, are the III-V compound semiconductors, which are formed by combining group III elements, such as Al, Ga, and In, with group V elements, such as N, P, As, and Sb.
A binary compound consists of two elements. There are more than ten binary III-V semiconductors, such as GaAs, InP, AlAs, and InSb. Different binary III-V compounds can be alloyed with varying compositions to form mixed crystals of ternary compound alloys and quaternary compound alloys.
A ternary III-V compound consists of three elements, two group III elements and one group V element, such as AlxGa1-xAs, or one group III element and two group V elements, such as GaAs1-xPx.
A quaternary III-V compound consists of two group III elements and two group V elements, such as In1-xGaxAs1-yPy.
A III-V compound can be either a direct-gap or an indirect-gap material. A III-V compound with a small bandgap tends to be a direct-gap material, whereas one with a large bandgap tends to be an indirect-gap material.
Among the III-V compounds, the nitrides are quite unique. The binary nitride semiconductors AlN, GaN, and InN, as well as their ternary alloys such as InGaN, are all direct-gap semiconductors.
These direct-gap semiconductors form a complete series of materials that have bandgap energies ranging from 1.9 eV for InN to 6.2 eV for AlN, corresponding to the spectral range from 650 to 200 nm.
Therefore, the nitride compounds and their alloys cover almost the entire visible spectrum and extend to the ultraviolet region. They are particular important for the development of semiconductor lasers, light-emitting diodes, and semiconductor photodetectors in the blue, violet, and ultraviolet spectral regions.
Another unique property of nitride semiconductors is that they crystallize preferentially in hexagonal wurtzite structure, which has uniaxial optical properties. However, nitride semiconductors can also crystallize in cubic zinc blend structure, which is the common structure of all III-V compounds.
Group II elements, such as Zn, Cd, and Hg, can also be combined with group VI elements, such as S, Se, and Te, to form binary II-VI semiconductors. Among such compounds, the Zn and Cd compounds, ZnS, ZnSe, ZnTe, CdS, CdSe, and CdTe, are direct-gap semiconductors with large bandgaps ranging from 1.5 eV for CdTe to 3.78 eV for ZnS, whereas the Hg compounds HgSe and HgTe are semimetals with negative bandgaps and HgS has two forms: \(\alpha\)-HgS being a large-gap semiconductor and \(\beta\)-HgS being a semimetal.
The II-VI compounds can be further mixed to form mixed II-VI compound alloys, such as the ternary alloys HgxCd1-xTe and HgxCd1-xSe. The ternary II-VI alloys that include Hg can have a wide range of bandgaps covering the visible to the mid infrared spectral regions.
In addition to III-V and II-VI compounds, the IV-VI lead-salt compound semiconductors, PbS, PbSe, and PbTe, as well as their alloys like PbxSn1-xTe and PbSxSe1-x, are also useful. These lead-salt compounds are direct-gap semiconductors with small bandgaps in the range of 0.145-0.41 eV. These lead-salt compounds have the unusual property that their bandgaps increase with increasing temperature, whereas the bandgaps of other semiconductors decrease with increasing temperature.
By examining the data listed in table 12-1, an important trend regarding the bandgaps and the refractive indices of semiconductors can be observed: as the atomic weight of a component in a semiconductor increases by moving down a particular column of the periodic table, the bandgap decreases while the refractive index at a given optical wavelength corresponding to a photon energy below the bandgap increases.
These characteristics can be seen in the Group IV elemental semiconductors as the bandgap decreases, while the refractive index increases, from C through Si to Ge.
For the III-V compounds, these characteristics can be seen by comparing those compounds of the same group III element but different group V elements or those of the same group V element but different group III elements. For example, the bandgap decreases but the refractive index increases from AlP through AlAs to AlSb as the atomic weight increases from P to As to Sb among the group V elements. As another example, the bandgap decreases but the refractive index increases from AlAs through GaAs to InAs as the atomic weight increases from Al through Ga to In among the group III elements.
Similar characteristics exist for II-VI compounds. Therefore, one can expect that among CdS, CdSe, and CdTe, for example, CdS has the largest bandgap while CdTe has the smallest bandgap. As another example, ZnSe has a larger bandgap than CdSe while HgSe has a negative bandgap.
Lattice-Matched Compounds
Two crystals that have the same lattice structure and the same lattice constant are lattice matched. Figure 12-2 shows the lattice constant as a function of bandgap for III-V compounds at \(300\text{ K}\).
In this plot, a ternary compound lies on the curve connecting the two binary compounds that form the ternary alloy. For example, the lattice constant and the bandgap of AlxGa1-xAs for \(0\le{x}\le1\) can be found on the curve connecting AlAs and GaAs.
A quaternary compound lies in the area defined by the four binary compounds that form the quaternary alloy. For example, the parameters of In1-xGaxAsyP1-y for \(0\le{x}\le1\) and \(0\le{y}\le1\) are found within the area bounded by the curves connecting InAs, GaAs, InP, and GaP.
In Figure 12-2, all compositions that are lattice matched to a given binary compound lie on the horizontal line passing through the binary compound.
Because the lattice constants of different compounds vary with temperature at different rates, two compositions that are lattice matched at a particular temperature are normally not matched at other temperatures.
As can be seen from Table 12-1 and Figure 12-2, the lattice constants of AlAs and GaAs have a very small mismatch of only \(0.13\%\) at \(300\text{ K}\). They are the most closely lattice matched among all pairs of III-V binary compounds. In fact, they are perfectly lattice matched at \(900^{\circ}\text{C}\). Consequently, the lattice constant of the ternary compound AlxGa1-xAs varies very little with \(x\), and AlxGa1-xAs is closely lattice matched to AlAs and GaAs over the entire composition range.
Similar characteristics exist for the AlP - AlxGa1-xP - GaP system and the AlSb - AlxGa1-xSb - GaSb system as well, but the lattice constants of AlxGa1-xP and AlxGa1-xSb have larger variations with \(x\) than that of AlxGa1-xAs.
Other than these systems, a ternary compound can be lattice matched to a binary compound or to another ternary compound only at a particular composition but not over the entire composition range. Such lattice-matched compositions can be found using Figure 12-2.
The quaternary compound alloys normally have a large flexibility for lattice matching to other compounds because each quaternary system has two variable composition parameters and, consequently, occupies an area, rather than a curve as for a ternary system, in Figure 12-2. For example, the In1-xGaxAsyP1-y system, which lies in the lower left shaded area of Figure 12-2, can be lattice matched to InP over a composition range of \(0\le{y}\le1\) and \(x\approx0.47y\), and it can lattice matched to GaAs over another composition range of \(0\le{y}\le1\) and \(1-x\approx0.49(1-y)\).
As can be seen from Figure 12-2, the bandgap of a ternary or a quaternary compound varies with the composition of the compound when the lattice constant of the compound is, for the purpose of lattice matching, kept at a fixed value as the composition varies. As the bandgap varies, the refractive index of the compound also varies and, as discussed above, they vary in opposite directions.
A semiconductor waveguide or a semiconductor heterostructure device generally consists of layers of compound semiconductors of different compositions that are all lattice matched to a binary compound substrate, such as GaAs, InP, InAs, or GaSb, on which the structure is fabricated.
The bandgaps and refractive indices of lattice-matched compounds are very important factors to be considered in the design and fabrication of semiconductor optical waveguides and heterostructure devices. In the following, the properties of two important systems, namely, AlxGa1-xAs lattice matched to a GaAs substrate and In1-xGaxAsyP1-y lattice matched to an InP substrate, are summarized.
AlxGa1-xAs/GaAs
Over the entire composition range of \(0\le{x}\le1\), AlxGa1-xAs is closely, though not perfectly, lattice matched to GaAs. Because this ternary compound is an alloy of indirect-gap AlAs and direct-gap GaAs, it is a direct-gap semiconductor for small values of \(x\) in the range \(0\le{x}\lt0.45\) but becomes an indirect-gap semiconductor for large values of \(x\) in the range \(0.45\lt{x}\le1\). Its bandgap in electron volts at \(300\text{ K}\) as a function of the composition parameter \(x\) can be described by
\[\tag{12-3}E_\text{g}(x)=1.424+1.247x\qquad\text{direct gap for }0\le{x}\lt0.45\]
\[\tag{12-4}E_\text{g}(x)=1.900+0.125x+0.143x^2\qquad\text{indirect gap for }0.45\lt{x}\le1\]
The direct bandgap ranges from 1.424 to 1.985 eV, corresponding to an optical wavelength \(\lambda_\text{g}\) in the range between 870 and 625 nm. The indirect bandgap covers a range from 1.985 to 2.168 eV, corresponding to \(\lambda_\text{g}\) in the range between 625 and 572 nm.
The bandgap of GaAs decreases with increasing temperature. It has a value of 1.5216 eV at \(0\text{ K}\). The temperature dependence of the bandgap of GaAs is
\[\tag{12-5}E_\text{g}=1.5216-\frac{5.405\times10^{-4}T^2}{T+204}(\text{eV})\]
The refractive index of AlxGa1-xAs is a function of \(x\) as well as of the optical wavelength because of dispersion. At an optical wavelength of \(\lambda\) = 900 nm, corresponding to a photon energy of 1.38 eV, which is below the bandgap of AlxGa1-xAs over the entire composition range, the refractive index as a function of \(x\) can be approximated by
\[\tag{12-6}n(x)=3.593-0.710x+0.091x^2\qquad\text{at }\lambda=900\text{ nm for }0\le{x}\le1\]
We see, by examining the variations of \(n(x)\) and \(E_\text{g}(x)\) with the parameter \(x\), that as the value of \(x\) increases, the bandgap of AlxGa1-xAs increases but its refractive index at the fixed wavelength of 900 nm decreases.
The refractive index of GaAs at \(300\text{ K}\) as a function of optical wavelength in the spectral range of \(\lambda\ge870\text{ nm}\) for photon energies below the GaAs bandgap is given by the following Sellmeier equation:
\[\tag{12-7}n^2=8.950+\frac{2.054\lambda^2}{\lambda^2-0.390}\]
where \(\lambda\) is in micrometers.
The refractive index of GaAs varies with temperature approximately as
\[\tag{12-8}\frac{1}{n}\frac{\text{d}n}{\text{d}T}=4.5\times10^{-5}\text{ K}^{-1}\]
In1-xGaxAsyP1-y/InP
The In1-xGaxAsyP1-y quaternary compounds that are lattice matched to InP are direct-gap semiconductors over the entire lattice-matched composition range of \(0\le{y}\le1\) and \(x=0.47y\).
At \(300\text{ K}\), the bandgap in electron volts as a function of the composition parameter \(y\) is given by
\[\tag{12-9}E_\text{g}(y)=1.350-0.72y+0.12y^2\qquad\text{direct gap for }0\le{y}\le1\text{ and }x=0.47y\]
Therefore, the direct bandgap of In1-xGaxAsyP1-y that is lattice matched to InP covers the range from 0.75 to 1.35 eV, corresponding to \(\lambda_\text{g}\) in the range between 919 nm and 1.65 μm.
The bandgap of InP decreases with increasing temperature. It has a value of 1.4206 eV at \(0\text{ K}\). The temperature dependence of the bandgap of InP is
\[\tag{12-10}E_\text{g}=1.4206-\frac{4.906\times10^{-4}T^2}{T+327}\text{ (eV)}\]
The refractive index is a function of the composition parameter \(y\) and optical wavelength. Two optical wavelengths, 1.3 and 1.55 μm, are of particular interest for lasers and LEDs based on the In1-xGaxAsyP1-y/InP system because they lie in the windows of minimum dispersion and minimum loss, respectively, in silica fibers.
The 0.954 and 0.8 eV photon energies of 1.3 and 1.55 μm wavelengths are below the bandgap of In1-xGaxAsyP1-y for \(0\le{y}\le0.6\) and \(0\le{y}\le0.9\), respectively, and \(x=0.47y\) for lattice matching to InP. At these wavelengths for the respective composition ranges, the refractive index of In1-xGaxAsyP1-y that is lattice matched to InP can be approximated by
\[\tag{12-11}n(y)=3.205+0.34y+0.21y^2\qquad\text{ at }\lambda=1.3\text{ μm for }0\le{y}\le0.6\]
\[\tag{12-12}n(y)=3.166+0.26y+0.09y^2\qquad\text{ at }\lambda=1.55\text{ μm for }0\le{y}\le0.9\]
From the relations in (12-9), (12-11) and (12-12), we see that as the value of the composition parameter \(y\) increases, the bandgap of In1-xGaxAsyP1-y lattice matched to InP decreases, but its refractive index at a fixed wavelength of 1.3 and 1.55 μm increases.
The refractive index of InP at \(300\text{ K}\) as a function of optical wavelength in the spectral range of \(\lambda\ge920\text{ nm}\) for photon energies below the InP bandgap is given by the following Sellmeier equation:
\[\tag{12-13}n^2=7.255+\frac{2.316\lambda^2}{\lambda^2-0.3922}\]
where \(\lambda\) is in micrometers.
The refractive index of InP varies with temperature approximately as
\[\tag{12-14}\frac{1}{n}\frac{\text{d}n}{\text{d}T}=2.7\times10^{-5}\text{ K}^{-1}\]
Example 12-1
An InGaAsP quaternary compound that is lattice matched to InP at \(300\text{ K}\) has a bandgap optical wavelength of \(\lambda_\text{g}\) = 1.223 μm. Find the energy of its bandgap. Find its refractive indices at 1.3 and 1.55 μm wavelengths, respectively. What is the composition of this quaternary compound?
For \(\lambda_\text{g}\) = 1.223 μm, the bandgap
\[E_\text{g}=\frac{hc}{\lambda_\text{g}}=\frac{1.2398}{1.223}\text{ eV}=1.014\text{ eV}\]
According to (12-9), the composition parameter \(y\) can be found by solving
\[0.12y^2-0.72y+1.35=1.014\]
which yields \(y=0.51\). Using (12-11) and (12-12), we then find the refractive indices are
\[n=3.21+0.34\times0.51+0.21\times0.51^2=3.438\]
at \(\lambda\) = 1.3 μm and
\[n=3.17+0.26\times0.51+0.09\times0.51^2=3.326\]
at \(\lambda\) = 1.55 μm.
Because \(x=0.47y\) for an InGaAsP compound that is lattice matched to InP, we find that \(x=0.47\times0.51=0.24\) for \(y=0.51\). Therefore, the composition of this quaternary compound is
\[\text{In}_{0.76}\text{Ga}_{0.24}\text{As}_{0.51}\text{P}_{0.49}\]
The next tutorial covers the topic of electron and hole concentrations.