© Springer Nature Switzerland AG 2020

K. Y. ChengIII–V Compound Semiconductors and DevicesGraduate Texts in Physicshttps://doi.org/10.1007/978-3-030-51903-2_4

4. Compound Semiconductor Crystals

Keh Yung Cheng^{1, 2}  

(1)

Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

(2)

Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan

 

 

Keh Yung Cheng

Email: kycheng@ee.nthu.edu.tw

Abstract

As we learned from previous chapters, silicon has a diamond crystal structure with pure covalent bonding. Due to its nature as a single element, the Si crystal structure is highly symmetric. Since there are two different elements in binary III–V compounds, the basic diamond structure turns into zinc-blende crystal structure, which destroys some of the symmetry observed in Si. This lower symmetry in crystal structure leads to band structures with different features which, in turn, give the unique properties not available to Si. Using GaAs as an example, the main advantages over Si are (a) Larger bandgap: The larger bandgap leads to a low intrinsic carrier concentration (n_i) suitable for high-temperature operation. (b) Smaller effective mass: The effective mass is inversely proportional to the carrier mobility. The higher mobility contributes to high-speed operation. (c) Direct bandgap: The location of the lowest conduction band minimum also varies with a decreasing degree of crystal symmetry in III–V compounds. Some of them have both the conduction band minimum and the valence band maximum located at the zone center. These are called direct bandgap semiconductors. This allows for their use in photonic device and transfer electron device applications. (d) Heterostructures: Additionally, binary III–V compounds can form ternary and quaternary compounds, which leads to an expanded selection of lattice constants and bandgap energies. New heterostructure and quantum effect devices then become possible.

In this chapter, the basic structural, transport, doping, and surface properties of III–V binaries are introduced first. The variation of lattice constant and energy gap in ternary and quaternary alloys are discussed next.

The history of preparation of III–V compounds goes back to the late nineteenth century. Since no naturally occurring III–V crystals have been found, the early work was to prepare these compounds. During the early twentieth century, before WWII, a wide range of the compounds, such as AlSb, AlN, AlP, AlAs, InP, InAs, InSb, and GaN, was studied for their crystal chemistry and structure without realizing the semiconducting nature of these compounds. After the war, work continued in both Russia and the Western world. In 1950, N. A. Goryunova, a Ph.D. student at Leningrad State University, came to the conclusion that compounds with the zinc-blende structure are not only structural analogs of group IV elements, but they are also semiconductors. She substantiated the idea with exploratory experiments and reported the results in her Ph.D. thesis in 1951. These ideas are contained in a summary of intermetallic compounds reported by A. F. Ioffe, where he pointed out that InSb, HgSe, and HgTe are semiconductors like silicon. Ioffe’s work was included in papers by A. I. Blum, N. P. Mokrovski, and A. R. Ragel [1].

At about the same time, in 1952, H. Welker in West Germany published his classical paper on new semiconducting compounds, including the important III–V compound semiconductor GaAs and applied for patents on III–V compound semiconductor devices and their method of manufacturing [2]. He was recognized in the Western world as the pioneer of III–V compound semiconductors. In the USA, the first discussion of the unique properties of III–V compound semiconductors was held at the American Physical Society March Meeting at Durham, North Carolina, in 1953. The materials discussed at this meeting were mainly Sb-related compounds, e.g., InSb and AlSb. Research on III–V compounds got more attention after the demonstration of semiconductor injection lasers in 1962 and the discovery of coherent microwave oscillation in transfer electron devices in 1963. These important early inventions precisely highlight the strength of III–V compound semiconductors over silicon—their light-emitting and high-speed properties.

4.1 Structural Properties

4.1.1 Lattice Constant

The lattice constant of a III–V compound zinc-blende crystal can be estimated by using the tetrahedron covalent radii defined by L. Pauling. Assuming the atoms pack in as hard spheres, the lattice constant (a) is simply related to the tetrahedron covalent radii (r_III and r_V) by

(4.1)

For all III–V compounds, except AlSb, as shown in Table 4.1, the error between the calculated and experimental results is less than 1%. Although this is a remarkably close fit, the accuracy is not good enough for predicting lattice constants of new compounds. In heterostructures, a lattice-mismatch of ≤0.1% is required to avoid the formation of misfit dislocations. In most III–V compounds, this tolerance is less than 0.0005 Å. Therefore, we have to rely on experimental lattice constant values.

Table 4.1

Measured lattice constant (in Å) of III–V compounds as compared to calculated values from covalent radii (value in parenthesis) of each pair of III and V elements

rIII	rV
	P (1.10Å)		As (1.18Å)		Sb (1.36Å)		N (0.70Å)
	Cal.	Exp.	Cal.	Exp.	Cal.	Exp.	Cal.	Exp.
Al (1.26Å)	5.450	5.464	5.635	5.660	6.051	6.136	a: 3.20 c: 5.227	a: 3.112 c: 4.982
Ga (1.26Å)	5.450	5.451	5.635	5.653	6.051	6.096	a: 3.20 c: 5.227	a: 3.189 c: 5.185
In (1.44Å)	5.866	5.869	6.051	6.058	6.466	6.479	a: 3.495 c: 5.706	a: 3.54 c: 5.70

For nitride compounds, a and c indicate the lattice constants in the (0001) plane and along the c-axis, respectively

For III-nitride compounds, their crystal structure is wurtzite. The lattice constants a and c are related to the tetrahedron covalent radii by

(4.2)

4.1.2 Cleavage Properties

The side views and 3D structures of a zinc-blende crystal along the [111], [110], and [100] directions are shown in Fig. 4.1. For silicon, the cleavage plane is in {111} due to its slightly larger separation between the two planes. For compound semiconductors, {111} is not the favored cleavage plane. This is because the two opposite surfaces are filled with different atoms where an electrostatic attraction force is formed upon cleaving. The (111) surface occupied by group III atoms is called the (111)A surface. The one filled with group V atoms is the (111)B surface. Because of this chemical difference, these two surfaces show very different characteristics in etching and epitaxial growth processes. On each side of the (110) cleaved plane, there are equal numbers of group III and group V atoms. Because there is no electrostatic attraction between the two sides, this is the easy cleavage plane among III–V compounds. The atoms on the [100] plane have double bonds and have the strongest bonding among three directions discussed here.

Fig. 4.1

(Top row) Side views of bond structures of (111), (110), and (100) surfaces of a zinc-blende crystal. The dashed line indicates a plane parallel to the surface and d is the bond length. The thicker lines in (111) side view represent double bonds. (Second row) The 3D structures of [111], [110], and [100] planes in zinc-blende structure. The shaded plane corresponds to the dashed line in the top panel

4.1.3 Lattice Vibration—Phonons

The vibration of crystal lattice at a finite temperature is represented, in quantum mechanics, by phonons. As discussed in Sect. 2.​5 and in later sections, phonon scattering due to lattice vibration plays a critical role in the carrier relaxation process, which determines many important basic properties of the semiconductor including effective mass, carrier recombination, and thermal conductivity, etc. The periodicity of the lattice means that phonons have band structure in much the same way as electrons, as discussed later in this section. The crystal lattice can be visualized as an array of point masses connected by springs. The masses are just the core ions. The length of the spring can be seen as the equilibrium bond length. It resists any attempt to change its length. The force between the neighboring atoms linked with the spring is governed by Hooke’s law, , where α is the Hooke’s law constant. To examine the lattice vibration property of III–V compound semiconductors, a 1D diatomic linear chain model shown in Fig. 4.2 will be used to represent the diatomic primitive unit cell structure. The lighter atoms with a mass m and the heavier atoms with a mass M are arranged alternately with an equal separation of a in between.

Fig. 4.2

One-dimensional row of springs and masses in a linear chain diatomic model of III–V compounds. These spring-linked masses are allowed to vibrate longitudinally through displacements u_i

Let us first discuss the 1D motion along the linear chain only, i.e., longitudinal vibration, without concerning the sideways motion. It is assumed that the only significant force interactions between atoms are direct nearest-neighbor interactions. Therefore, the net force acting upon the 2nth atom can be seen as interactions through two springs connecting to atoms (2n + 1) and (2n − 1). The force equation is given as

(4.3)

where u_i’s are the displacements of the ith atoms. In obtaining (4.3), we note that the force acting on the 2nth atom from the (2n–1)th atom is opposite to that from the (2n + 1)th atom. The force between nearest neighbors can be expressed by Hooke’s law and Newton’s second law as

(4.4)

The periodic solutions to these two equations are in the forms of

(4.5)

where ω is the oscillation frequency of atoms. The displacement of the (2n + 2)th and (2n − 2)th atoms can be expressed as

(4.6)

Replacing all the relevant u’s in the force equation for the (2n +1)th atom, we find

(4.7)

Solving for u_2n+1, we have

(4.8)

The force equation for the (2n)th atom has the form

(4.9)

Combining (4.8) and (4.9), it is found that

(4.10)

Rearranging the equation in terms of ω, we obtain

(4.11)

The solution of the ω-k dispersion relation has the form

(4.12)

The positive sign group of the ω-k dispersion curve is called the optical branch, and the negative sign group is called the acoustical branch because the energies associated with these two modes of vibrations are in the optical and acoustical range of the frequency spectrum, respectively. The dispersion curves of phonons vibrating longitudinally are shown in Fig. 4.3.

Fig. 4.3

Phonon dispersion curves for a 1D linear diatomic crystal with longitudinal vibrations

Near the zone center, at k = 0, ω₊ has a maximum value of

(4.13)

The oscillation frequency of the acoustical branch for small values of k, sinka ≈ ka, becomes

(4.14)

Here we assumed (1+ x)^1/2 = 1 + x/2 − x²/8 + … ≈1+ x/2. The frequency of the acoustical branch near the zone center is a linear function of k.

The smallest allowed wavelength in this diatomic system should equal twice the lattice constant (2a), which is λ = 4a. Therefore, the first zone boundary is located at k = ± 2π/λ = ±π/2a, and where the two roots become

(4.15)

For m ≠ M, a forbidden frequency band is formed.

The relative movement of the two atoms in the unit cell can be calculated by taking the ratio of u_2n+1 and u_2n.

(4.16)

or

(4.17)

For the acoustical branch, ω = ω_–(0) = 0 at k = 0, thus (4.17) leads to A = B. As shown in Fig. 4.4, the displacements in the longitudinal acoustic (LA) branch are like sound waves, where the two atoms in each unit cell move in the same direction by almost the same distance. It appears as if the whole crystal has been compressed or stretched over a small region. This creates alternating zones of compression and dilation. In a crystal, these waves can propagate in three directions. If we choose the direction in a cubic crystal, one acoustic mode is longitudinal and the other two are transverse. In the transverse acoustic (TA) modes, the atoms vibrate in the plane normal to the direction of wave propagation, similar to an electromagnetic wave in free space. The atoms and their center of mass move together as in long-wavelength acoustical vibrations.

Fig. 4.4

Motion of the atoms in longitudinal acoustic (LA) mode for a 1D chain of alternating light and heavy atoms. The two types of atoms move together in transverse acoustical (TA) mode of the same crystal system. Displaced positions of moving atoms are shown as gray circles

For the optical branch, using the values k = 0 and ω = ω₊(0) in (4.17), we find that B/A = – m/M. As shown in Fig. 4.5, the heavy and light atoms move in opposite directions and the displacement is inversely proportional to the mass of the atom. In the longitudinal optical (LO) mode, both the light and heavy atoms vibrate back and forth in the same direction in which the wave travels. Similar to the TA modes, in the transverse optical (TO) modes, the atoms vibrate in the plane normal to the direction of wave propagation, but the atoms move in opposite directions.

Fig. 4.5

Motion of the atoms in longitudinal optical (LO) mode for a 1D chain of alternating light and heavy atoms. The two types of atoms move in opposite directions in the transverse optical (TO) mode

Both Si and GaAs have two atoms in each primitive cell, allowing two different modes of vibrations close to that of Fig. 4.3. The experimental phonon dispersion curves of GaAs, shown in Fig. 4.6, are very similar to those predicted by the 1D model. The doubly degenerated transverse branches are also displayed. The LO phonon energy in the zone center of GaAs is 36 meV. We also notice that the oscillation frequency for TO phonons is smaller than that of LO phonons at the zone center. In compound semiconductors, the relative displacement of the two atoms, as in the optical mode, sets up an electric dipole. This leads to the formation of a polarization field and an electric field, which can interact with electromagnetic waves. Electrons can be scattered rapidly by the electric field associated with the polar LO phonons, raising their frequency. Purely transverse vibrations do not generate dipoles, so this effect is absent. This polar nature is not seen in elemental semiconductors because all atoms are identical and there is no transfer of charge.

Fig. 4.6

Left: Phonon dispersion curves along [100] from Γ to X for a GaAs crystal at room temperature. The transverse mode branches are doubly degenerated. (Reprinted with permission from [3], copyright American Physical Society) Right: Measured dispersion curves for phonons in Si along [100]. (Reprinted with permission from [4], copyright American Physical Society)

Since phonons are the quantum mechanical equivalents of lattice vibrations, there are abundant phonons available in semiconductor crystals at room temperature. For a wave of frequency ω, the crystal momentums of photons and phonons are both given by ħk and k = ω/υ, where υ is the velocity of the wave. For an electromagnetic wave, υ = c = 3 × 10⁸ m/s. The velocity of a lattice wave is similar to that of a sound wave, and its amplitude is smaller by about five orders of magnitude. Therefore, for the same frequency, the momentum of a phonon is much higher than that of a photon. This finite momentum associated with the phonon becomes very important in connecting electrons and photons in a semiconductor. In an indirect bandgap semiconductor, as discussed in Chap. 8, phonons can provide high enough momentum to satisfy momentum conservation required to excite an electron from the valence band to the conduction band by absorbing or emitting a photon with the proper energy. Phonons also play an equally important role in the light emission process in direct bandgap semiconductors. Electrons in higher states inside the conduction band need to relax to the band minimum before they can recombine with holes residing near the valence band maximum for photon generation. The relaxation process relies on phonons, with energy ħω, to dissipate extra energy into heat and to conserve momentum.

4.2 Electrical Properties

4.2.1 Effective Mass

The electron concentration in a non-degenerate semiconductor is calculated by integrating the electron distribution above the conduction band edge and formulated as

(4.18)

where D(E) is the density of states of a bulk semiconductor and has the form

(4.19)

where m_DOS stands for the DOS effective mass. In a non-degenerated semiconductor, where E_c − E_F > 3kT, the Fermi–Dirac distribution function f(E) can be approximated by the Boltzmann distribution function to simplify the calculation.

(4.20)

This leads to a non-degenerate electron concentration of

(4.21)

where N_c is the effective density of states in the conduction band and has a value of

(4.22)

For a parabolic conduction band with spherical energy surface, like GaAs, m_DOS simply equals m_e*. For non-spherical energy surface, the DOS effective mass used for carrier concentration calculation is modified according to the procedures outlined below. For example, in Si and AlAs, the conduction band is non-spherical (ellipsoidal) and has multiple valleys. For an ellipsoidal energy surface along k_x, it can be expressed as

(4.23)

This ellipsoidal surface can transform into a spherical surface through a coordinate transformation process and is expressed as

(4.24)

where , , and . The volume of space in the spherical shell bounded by radii and for a single ellipsoidal is and . Thus,

(4.25)

Since dV_p = dp_xdp_ydp_z and , dV_p and are related through

(4.26)

for one ellipsoidal energy surface. For g equivalent valleys, the total DOS effective mass becomes

(4.27)

In the valence band, there is a single peak containing multiple bands, i.e., light-hole and heavy-hole bands. Similarly, the DOS function is also modified to include the multiband effect.

(4.28)

where N_vl and N_vh are the effective density of states in the light-hole band and heavy-hole band, respectively. Therefore, the DOS effective mass of the valence band becomes

(4.29)

The calculated effective DOS and intrinsic carrier concentration (n_i) in Si and GaAs are listed in the Table 4.2.

Table 4.2

Effective density of states and intrinsic carrier concentration of Ge, Si, and GaAs

(cm−3)	Ge	Si	GaAs
Nc	1.04 × 1019	3.2 × 1019	4.7 × 1017
Nv	5 × 1018	1.8 × 1019	9.4 × 1018
ni	2.33 × 1013	1.02 × 1010	2.1 × 106

In addition, there is another type of effective mass associated with the conductivity. The conductivity is related to the effective mass in the form of

(4.30)

where n is the carrier concentration, q is the electron charge, τ is the carrier relaxation time, and is the conductivity effective mass. For semiconductors with a spherical energy surface, such as GaAs, the conductivity effective mass has a simple form of

(4.31)

For non-spherical energy surfaces of the conduction band, as for Si and AlAs, the conductivity has a tensor form of

(4.32)

for each ellipsoidal surface. Since for i ≠ j, so that all the off-diagonal elements of the tensor are zero. Using Si as an example, the major axes of the ellipsoidal energy surface lie along the coordinate axes of k-space. The conductivity tensor for, say, [001] and ellipsoids along k_z becomes

(4.33)

where m_x =  m_y = m_T and m_z = m_L. Since all six ellipsoids are equivalent energy minima (Fig. 3.​28), . By summing up conductivities of all six ellipsoids, one can write the total conductivity as

(4.34)

This leads to the conductivity and the conductivity effective mass of

(4.35)

(4.36)

In general, the carrier effective mass is a constant for non-degenerate semiconductors. However, the conduction band effective mass of a degenerated semiconductor becomes dependent on doping concentration. At high doping levels, in n-type semiconductor the Fermi level moves close to, or even into, the conduction band. Since the non-parabolicity of the band becomes significant, the effective mass increases with doping level for carrier density larger than 10¹⁸ cm⁻³.

4.2.2 Mobility

The mobility (µ) is defined as the drift velocity (v_d) per unit electric field intensity (F).

(4.37)

where τ is the relaxation time or the inverse of scattering probability (1/τ). The equilibrium drift velocity is determined from the balance between the electric field-induced acceleration and scattering generated deceleration. If the scattering mechanisms are assumed to act independently, the total probability of a scattering event occurring in the differential time dt is the sum of the individual events. Thus,

(4.38)

where τ_j is the relaxation time for each scattering mechanism. Since the mobility is proportional to τ, according to Matthiessen’s rule, the total carrier mobility is the inverse sum of each individual contribution.

(4.39)

The major scattering mechanisms included in compound semiconductors are impurity scattering (µ_i), lattice scattering (µ_l), polar scattering (µ_po), piezoelectric scattering (µ_pe), and alloy scattering (µ_al). The total mobility is thus limited by the dominant scattering mechanisms occurring in the semiconductor, as illustrated in Fig. 4.7.

Fig. 4.7

Temperature dependence of the mobility for high-purity n-type GaAs showing the separate and combined scattering processes.

Reprinted with permission from [5], copyright AIP Publishing

For holes, the mobility is low compared to electrons, mainly due to the heavy effective mass associated with the heavy-hole band. Due to the degeneracy of the heavy and light holes, the interaction between these two bands also contributes to the extra scattering.

In the following, we briefly discuss the mobility due to major scattering mechanisms.

1. (a)

   Impurity scattering (µ_i)

    

The calculation of the mobility for ionized impurity scattering is based upon the scattering of charged particles by Coulomb potential of nuclei, which was originally developed by E. Rutherford to explain the scattering of α-particles. The resultant mobility has a temperature dependence of T^3/2.

(4.40)

where N_i is the ionized impurity concentration. The impurity scattering is the major mobility limiting mechanism at low temperatures for both elemental and compound semiconductors.

1. (b)

   Lattice (deformation potential) scattering (µ_l)

    

The vibrations of lattice atoms at finite temperatures cause a local variation in the energies of the conduction and valence band edges. The abrupt changes in the band edges further cause increased electron deflections. The energy step height is related to the degree of lattice compression/dilation defined by the deformation potential. At low temperatures, the thermal energy available to excite optical-mode lattice vibrations is quite limited. So we shall consider only the scattering by LA phonons. The temperature dependence of the lattice scattering has a form of T^−3/2.

(4.41)

where Ξ is the deformation potential. This is the major scattering mechanism in the high-temperature region for elemental semiconductors, but not for III–V compounds.

However, experimental data show that, in both Si and Ge, the temperature dependences deviate quite considerably from the T^−3/2 law in the region dominated by lattice scattering. As discussed in the previous section, lattice vibrations generate both acoustic and optical branches of phonons. In (4.41), only the low-frequency acoustical mode is considered. In silicon or AlAs, there are six equivalent conduction band minima in the first Brillouin zone. Electrons scatter between these valleys, forming an inter-valley scattering process. The large momentum change involved requires the participation of high-energy phonons, both acoustical and optical. Consider the optical-mode phonon contribution, the inter-valley scattering mobility shows a different temperature dependence of the following form:

(4.42)

where is the optical phonon energy.

1. (c)

   Polar scattering (µ_po)

    

In compound semiconductors, when responding to lattice vibrations, the two neighboring atoms move in opposite directions in the optical phonon modes. Because of the nonzero ionicity, the relative movement of neighboring atoms constitutes an electric polarization, which in turn produces an electric field. Deflection of the motion of free carriers by the field limits the mobility of carriers. Thus, the electron mobility in III–V compounds is limited by polar scattering instead of acoustical mode scattering in the high-temperature region. Polar scattering becomes the most dominant scattering mechanism in compound semiconductors at high temperatures.

(4.43)

where is the equivalent temperature associated with longitudinal optical phonons. Since the polar scattering mobility is inversely proportional to (m*)^3/2, the large difference in electron mobility and hole mobility in III–V compounds is explainable by (4.43).

1. (d)

   Piezoelectric scattering (µ_pe)

    

A polarization electric field induced in an ionic crystal, such as II–VI compounds and III-N alloys, by the applied mechanical stress, or vice versa, is called the piezoelectric effect. The electric field accompanying the lattice vibrations can interact with electron motion through the lattice. Since zinc-blende III–V compounds are not pure covalent crystals, but mixtures of covalent and ionic bonding, there exist finite piezoelectric fields. The piezoelectric scattering mobility is

(4.44)

This is the main scattering mechanism which limits the mobility in ionic II–VI compounds, but it is less important for zinc-blende III–V compounds.

1. (e)

   Alloy scattering (µ_al)

    

When mixing binary compounds into ternary or quaternary alloys, the lattice constant, according to Vegard’s law, is proportional to the composition ratio of the binaries. A close examination indicates that the bond lengths associated with each individual binary do not change significantly, as shown in Fig. 4.8. Using an extended x-ray absorption fine structure (EXAFS) technique, in InGaAs alloys, the measured Ga-As and In-As near-neighbor distances change by only 0.04 Å over the whole alloy composition range. However, the virtual crystal model that follows Vegard’s law requires a change in average near-neighbor spacing of 0.17 Å. Therefore, the compound can be seen as a mixture of clusters of binary alloys instead of a completely random material. The carriers moving through regions containing different proportions of binaries may suffer scattering as

Fig. 4.8

Near-neighbor distance versus InAs mole fraction x in the In_xGa_1−xAs ternary alloy measured by EXAFS. The dashed line represents the average cation–anion spacing according to Vegard’s law.

Reprinted with permission from [6], copyright American Physical Society

(4.45)

where S represents the degree of randomness (S = 1 for complete random composition), and Δu is the alloy scattering potential. It turns out that this is the limiting scattering mechanism in the middle temperature range for ternary and quaternary compounds. Figure 4.9 shows the dominance of alloy scattering in electron Hall mobility near 100 K in an In_0.53Ga_0.47As ternary alloy lattice-matched to InP.

Fig. 4.9

Temperature dependence of the electron Hall mobility in In_0.53Ga_0.47As. The contribution due to alloy scattering dominates the total mobility near 100 K.

Reprinted with permission from [7], copyright Elsevier

1. (f)

   High-field phenomena

    

As shown in (4.37), the carrier mobility is determined by the total scattering events occurring inside the semiconductor. If scattering rates were fixed, mobility would remain constant and carrier velocity would be limited by the magnitude of the applied electric field. In fact, under the increasing applied electric field, new scattering mechanisms come into play as carrier energies increase. Under a moderately high-field (~10⁴ V/cm) condition, carriers may acquire sufficient energy to excite acoustic phonons within the semiconductor lattice. Increased phonon scattering is particularly pronounced between multiple conduction band minima in indirect bandgap materials. This leads to a sublinear rate of the mobility change with increasing applied field. For further increases of the field to above 10⁵ V/cm, the energy gained by carriers is transferred to lattice heating through optical phonon emission (~60 meV in Si and ~36 meV in GaAs), resulting in a saturated drift velocity (or mobility) with a value of about 10⁷ cm/s. The velocity–field characteristics for electrons in Si, GaAs, and InP are shown in Fig. 4.10.

Fig. 4.10

Velocity–field characteristics for GaAs, InP, and Si [8].

Reprint with permission from [9], copyright American Physical Society

Between the initial roll-off and final velocity saturation, the velocity–field behavior of electrons in the direct bandgap semiconductor is quite different from that in Si. As shown in Fig. 4.10, GaAs and InP display an anomalous peak in drift velocity corresponding to a negative differential resistance. This is the basis of microwave oscillation in compound semiconductors, known as the Gunn effect. To understand this, one must look into semiconductor band structures, e.g., Fig. 3.​25. For GaAs, the conduction band has a number of satellite valleys at L and X minima with 0.31 meV and 0.51 meV above the Γ valley minimum, respectively. At low field, electrons are in the Γ valley, characterized by a light effective mass of 0.063m₀ so that the low-field mobility is high. At intermediate field, electrons may gain enough energy to be transferred into the lowest satellite valleys. For GaAs, the lowest satellite energy band minima are located at L valleys with a heavy effective mass of 0.85m₀ and a high density of states (×50 than Γ valley) for scattering electrons, leading to lower mobility. Thus the electron transfer between Γ and L valleys results in a downturn in electron velocity as shown in Fig. 4.10. The transfer electron effect has been utilized in devices to generate negative resistance and microwave oscillations.

4.2.3 Intentional Impurity

In III–V compounds, intentional impurities are selected from columns adjacent to column III or V of the periodic table. Elements from group II predominantly occupy group III sites to form p-type impurities. Substitution of group V elements by group VI impurities would result n-type materials. Group IV impurities can occupy either cation or anion sites and are amphoteric dopants. The energy required to ionize a dopant is called the activation energy, which is mainly determined by the Coulombic interaction between the impurity atom and the attached charge carrier. The hydrogen atom model can be applied to understand the shallow donors in III–V compounds. We derived the ionization energy for a hydrogen atom as

(4.46)

where R_y is the Rydberg energy. To apply this model to shallow donors, we have to make two modifications. First, the effective mass of electrons in semiconductors is very different from the free electron mass. Second, the dielectric constant of semiconductors is much larger than the free space value. Then the donor ionization energy required for the n = 1 level is

(4.47)

The donor Bohr radius is given by

(4.48)

The calculated ionization energy and Bohr radius for a hydrogen-like donor in GaAs with _r = 12.85 and  = 0.067 m₀ are 5.5 meV and 102 Å, respectively. The bound state wave function of the impurity extends much farther than the unit cell of the lattice (~5 Å). This result provides the justification for using effective mass and the dielectric constant of the semiconductor in these calculations. The calculated activation energy is smaller than or comparable to the thermal energy kT at room temperature. Therefore, all shallow donor impurities can be considered as activated at room temperature.

The application of the hydrogen model to acceptors in III–V compounds is complicated by their degenerate valence band structure. In the limit of strong spin-orbit interaction—that is, for spin-orbit splitting (Δ_so) much larger than the acceptor energy—only the heavy-hole and light-hole bands are included for acceptor binding energy calculation. The calculated results of the effective Bohr radius and effective Rydberg energy are found to be functions of Luttinger parameters, γ₁, γ₂, and γ₃, which describe the valence band shapes near the center of the Brillouin zone. Table 4.3 lists the calculated acceptor ionization energy for the major III–V compounds, which agrees reasonably well with experimental results. The measured values for each compound are given below.

Table 4.3

Calculated acceptor ionization energy of selected III–V compounds [8]

Material	AlSb	GaP	GaAs	GaSb	InP	InAs	InSb
Ea	42.4	47.5	25.6	12.5	35.3	16.6	8.6

Following is a summary of the most commonly used dopants in III–V compound semiconductors.

1. (a)

   Group II impurities incorporate on the anion sites of III–V compounds to form shallow acceptors.

    

Beryllium (Be) is a major shallow acceptor in III–V compounds. It has an ionization energy of 28 and 41 meV in GaAs and InP, respectively. Except at the highest concentration ≥10¹⁹ cm⁻³, Be diffuses slowly in GaAs and the diffusion coefficient as a function of temperature has the form

(4.49)

with the activation energy E_a = 1.95 eV and D₀ = 2×10⁻⁵ cm²/s. However, at very high doping concentrations of ~10²⁰ cm⁻³, Be redistributes severely which can degrade the device performance. Be is widely used in molecular beam epitaxy (MBE)-growth, ion implantation, and as Au–Be alloy for p-type ohmic contact metallization.

Zinc (Zn) is another major shallow acceptor in III–V compounds besides Be. The ionization energy is 31 meV in GaAs and 47 meV in InP. Due to its high vapor pressure, Zn is not suitable for MBE. In chemical vapor deposition (CVD), Zn is a popular dopant. Zn diffuses rapidly at high concentration and its diffusion rate is concentration dependent. The Au-Zn alloy is commonly used as a p-type ohmic contact metallization.

Magnesium (Mg) is a shallow acceptor impurity in III–V compounds with an ionization energy of 28 meV in GaAs and 41 meV in InP. Mg has a strong affinity for oxygen. Elemental Mg can form MgO easily during the doping process, which leads to a low doping incorporation coefficient at high growth temperature (>500 °C). Recently, it was found that Mg is an efficient p-type dopant in III-nitride compounds grown by both MBE and metalorganic chemical vapor deposition (MOCVD). Nevertheless, the doping efficiency is low with deep ionization energy of ≥150 meV.

Cadmium (Cd) is a shallow acceptor in GaAs and InP with ionization energies of 35 and 56 meV, respectively. It has a very high vapor pressure, approximately 1–2 orders of magnitude higher than Zn. It is a useful dopant in liquid phase epitaxy (LPE) growth of InP for temperature ≤550 °C with a maximum doping concentration of ~10¹⁸ cm⁻³.

1. (b)

   Group VI impurities incorporate on the cation sites of III–V compounds to form shallow donors.

    

Sulfur (S) is a shallow donor in III–V compounds with an ionization energy of 6 meV in GaAs. It has a high vapor pressure, and the incorporation depends strongly on the epitaxial growth temperature. This makes the accurate control of doping concentration from a solid sulfur source difficult. However, hydrogen sulfide (H₂S) is the most common doping precursor used in CVD for n-type doping of III–V compounds.

Selenium (Se) is also a high vapor pressure shallow donor in III–V compounds similar to sulfur. It has an ionization energy of 6 meV in GaAs. Hydrogen selenide (H₂Se) is a commonly used gaseous doping source of Se. However, the doping memory effect of using H₂Se is the major drawback. The memory effect manifests itself by a lack of abruptness of doping profiles.

Tellurium (Te) is an efficient shallow donor in III–V compounds. It has an ionization energy of 30 meV in GaAs. This is much larger than the calculated 5 meV due to a correction associated with the big Te atom. Elemental Te has been used as an n-type doping source in LPE growth of GaAs. Diethyltellurium [(C₂H₅)₂Te, DETe] has been used as a Te doping precursor for MOCVD and metalorganic MBE (MOMBE) growth of GaAs and InP.

1. (c)

   Group IV dopants

    

Group IV impurities incorporated into III–V compounds share some interesting characteristics. First, they all are amphoteric, i.e., they can occupy either the group III or the group V sites to become donors or acceptors, respectively. Second, they autocompensate. A group IV impurity that predominantly occupies the cation sites as a donor can also occupy the anion sites. This causes compensation and lowers the doping efficiency. Third, the electrically activated impurities saturate at high doping concentration. The autocompensation increases at high doping level and leads to the saturation of the free carrier concentration. Fourth, the free carrier concentration depends on growth temperature and V/III flux ratio during growth. For example, as shown in Fig. 4.11, when growing silicon-doped GaAs by liquid phase epitaxy, the type of conductivity depends on the growth temperature and the Si concentration in the liquid. At a fixed Si concentration above 4 × 10⁻⁴ atomic fraction in the growth solution, the GaAs shows n- and p-type conductivities when grown at high and low temperatures, respectively.

Fig. 4.11

Effect of growth temperature and Si concentration in the liquid on the doping behavior of Si in LPE grown GaAs

Carbon (C) is a shallow acceptor in GaAs (26 meV) but a donor in InP (43 meV) and InAs. Carbon is a stable impurity and diffuses very slowly. Its temperature-dependent diffusion coefficient in GaAs follows (4.49) with an activation energy E_a= 1.75 eV and D₀ = 5 × 10⁻⁸ cm²/s. Compared to Be, at 800 °C, the diffusion coefficient of C is almost two orders of magnitude smaller. It is an unintentional dopant in MOCVD when using metalorganic Ga and Al precursors. CCl₄ and CBr₄ are commonly used gaseous carbon doping sources. A very high acceptor level (~10²⁰ cm⁻³) is easily achieved in many III–V compounds. These behaviors make C the most ideal p-type dopant for the base region of III–V heterojunction bipolar transistors (HBT).

Silicon (Si) is a stable shallow donor (4–6 meV) in most III–V compounds at low concentration. The temperature-dependent diffusion coefficient in GaAs has an activation energy E_a = 2.45 eV and D₀ = 4 × 10⁻⁴ cm²/s. It strongly compensates in the high concentration regime (~5×10¹⁸ cm⁻³). It has been widely used as an n-type dopant for epitaxial growth and ion implantation. Gaseous silicon dopant precursors including silane (SiH₄) and disilane (Si₂H₆) have been widely used in MOCVD and MOMBE growth.

Germanium (Ge) is a strongly amphoteric impurity in III–V compounds. It is a p-type dopant for LPE growth of GaAs, but not popular for MBE and MOCVD growth. Au-Ge alloy is used for n-type ohmic contact metallization.

Tin (Sn) is a weakly compensating shallow donor in III–V compounds with 4–6 meV ionization energy in GaAs and InP. The major problem of Sn impurity during growth is the tendency to strongly redistribute due to the Sn diffusion and surface segregation. Nevertheless, high electron mobilities in GaAs:Sn are expected due to its low compensation.

1. (d)

   Hydrogen passivation

    

In III–V semiconductors, hydrogen is not an electrically active impurity. However, during the doping process, hydrogen generated from the carrier gas and/or dopant flux can passivate or neutralize shallow donor or acceptor in semiconductors. Using Si in (Al)GaAs as an example, under normal conditions, the substitutional silicon atom replaces a Ga and bonds with four nearest neighboring As atoms. When hydrogen atoms are incorporated into the lattice, a hydrogen can bond to the Si donor by breaking one of the Si-As bonds and sit in an interstitial site, as shown in Fig. 4.12a. The broken Si-As bonds of the As atom opposite the hydrogen atom are replaced with a lone pair of electrons—one provided by the donor atom and the other from the dangling bond of the As atom. The Si-H complex yields a neutral electron count, and the hydrogen has passivated the donor atom. The active donor concentration becomes very low compared to the impurity concentration. Hydrogen passivation of donor atoms has been found for a large number of donors including Si, Sn, Se, and Te. The dissociation of donor hydrogen bond by thermal annealing at temperatures above 400 °C allows hydrogen to diffuse out of the hydrogenated semiconductor and recover the doping concentration. Hydrogen also passivates shallow acceptors in III–V compounds as shown in Fig. 4.12b. The acceptor As bonds are broken, and the hydrogen atom is bonded to the arsenic atom and occupies a site between the acceptor impurity and the neighboring arsenic. The entire hydrogen acceptor complex is neutral, and the hydrogen passivates the acceptor.

Fig. 4.12

a The group IV donor (Si)-hydrogen complex with hydrogen in an interstitial site and bound to the donor atom. b The group II acceptor (Be)-hydrogen complex with hydrogen bound to an As atom.

Reprinted with permission from [11], copyright American Physical Society

4.2.4 Deep Levels

During the growth or doping of semiconductors, chemical impurities, native defects, or the combination of both, can form deep energy level(s) in the forbidden gap. These deep levels act like traps to capture and annihilate electrons and holes, making the material electrically inactive. Therefore, the deep traps may have many negative effects on the properties of semiconductors, including compensation of intentional impurities, minority carrier lifetime reduction, and reduced mobility and luminescence efficiency. In most cases, we should avoid deep levels in the fabrication of semiconductor materials and devices. Under certain circumstances, we utilize deep levels to control the conductivity of semiconductors. In this section, two particularly important deep levels in III–V compound semiconductors are discussed.

1. (a)

   DX Centers

    

During the 1970s, the rapid development of compound semiconductors was mostly focused on the Al_xGa_1−xAs system due to the availability of quality GaAs substrates, among other reasons. It was found that the n-type Al_xGa_1−xAs with 0.2 ≤ x ≤ 0.4 has some unusual properties as compared to GaAs. Most noticeable are the large donor activation energy and a strong sensitivity of the conductivity to illumination. These properties have profoundly negative effects on the device performance of Al_xGa_1−xAs/GaAs high-electron-mobility transistors (HEMT), as will be discussed in Chap. 9. The origin of the deep center observed in n-type Al_xGa_1−xAs involves a donor atom and another constituent to form a complex called the donor-complex (DX) center. The DX center formation mechanism and its properties are discussed below.

In Al_xGa_1−xAs, donors such as Si, Te, or Sn can exist in two different configurations—substitutional and interstitial—as shown in Fig. 4.13. When the incorporated Si atom takes the substitutional site, as shown in Fig. 4.13a, it replaces one of the anions of Al_xGa_1−xAs, e.g., Al or Ga. The Si atom is ~0.8 Å above the tetrahedron base formed by three adjacent As atoms. Due to the covalent bond length difference between Si (1.16 Å) and Al or Ga (1.26 Å), the substitutional configuration is the unrelaxed state for the Si atom. On the other hand, if the donor atom occupies an interstitial site, one of the four arsenic donor bonds is broken (Fig. 4.13b) and the interstitial configuration is the relaxed state. In this configuration, the Si atom is 0.2 Å below the tetrahedral plane defined by three As atoms. When the Si atom relaxes from the substitutional to interstitial configuration, the total displacement of the Si atom between the two configurations is approximately 1 Å. This displacement is large compared to the equilibrium bond length of the host crystal (~2.44 Å). This structural configuration change leads to a change in the electronic configuration of the defect.

Fig. 4.13

a Substitutional and b DX configuration of a Si donor in (Al)GaAs. In DX configuration, the Si occupies an interstitial site and has three cations in its immediate vicinity.

Reprinted with permission from [11], copyright American Physical Society

To illustrate the interplay between structural and electronic configurations, consider the hydrogen molecule as an example. The total energy and nuclei distance of bonding and anti-bonding states of two hydrogen atoms are shown in Fig. 4.14. The equilibrium distance between the two protons corresponds to a minimum total energy which depends on the electronic state of the molecule. The bonding state has a shorter core distance and lower total energy minima than the anti-bonding state. For small lattice relaxation, e.g., between bonding and anti-bonding states of a hydrogen molecule, the direct transitions between the ground (bonding) state and the excited (anti-bonding) state through absorption and emission are readily allowed. The electron of the bonding state can be excited to the anti-bonding state vertically and then relaxed to a new core position (configuration) before returning to the bonding state. This process leads to different absorption energy ħω_a and emission energy ħω_e for electron transitions between bonding and anti-bonding states, known as the Franck-Condon shift. However, due to the heavier mass of the nuclear as compared to the electronic mass, the two nuclei do not move during the short time of the electronic transition but the molecule slowly moves to the new configuration, i.e., new core distance. This situation is changed dramatically for large lattice relaxation. Figure 4.15 shows two configurations of such defects. Photon energy of ħω_a is required to excite the defect from the ground (relaxed) state to the excited (unrelaxed) state. Once the defect has minimized its energy in the excited state, the defect becomes metastable and cannot return to the ground state by means of an optical (vertical) transition. In order for the defect to return from the metastable state to the ground state, it has to overcome the capture energy barrier E_c through other allowed paths such as thermally excited transitions. Thermal emissions are also allowed to excite the defect from the ground state to the excited state. The thermal emission energy E_E = E_c + E_t is measured from the minimum of the ground state to the top of the thermal barrier.

Fig. 4.14

Bonding and anti-bonding state energy of a hydrogen molecule as a function of distance between two core atoms. Note, ħω_a > ħω_e

Fig. 4.15

Schematic configuration coordinate diagram for large lattice relaxation

We can use the same analysis to understand the physical mechanism of the DX center. The schematic representations of the relevant energies of the DX center and their representation in the configuration coordinate diagram are shown in Fig. 4.16. Experimentally, these relevant energies have been determined for Si-doped Al_xGa_1−xAs: E₀ ≥ 1 eV, E_t ≈ 150 meV, E_c ≈ 150 meV, and E_E ≈ 300 meV. Apparently, this is a very deep level, as expected. Since the DX interstitial configuration in Si-doped Al_xGa_1−xAs is a stable (relaxed) configuration, the substitutional configuration is the excited (unrelaxed) configuration, and a neutral substitutional donor can capture an electron and transform into a DX center, i.e.,

Fig. 4.16

a Energy diagram model of a DX center in n-type AlGaAs; E_c—capture energy barrier, E_E—transit emission energy, E_t—activation energy, and E₀—absorption energy of the DX center. b Configuration coordinate diagram of the DX center.

Reprinted with permission from [12], copyright American Physical Society

(4.50)

where D⁰ and DX^– represent a neutral substitutional donor and a negatively charged DX center, respectively. An ionized shallow donor can also capture two electrons by assuming the DX configuration, i.e.,

(4.51)

As a consequence, both deep DX^– donors and shallow D⁰ donors coexist in n-type Al_xGa_1−xAs (0.2 ≤  x  ≤ 0.4) at low temperatures. As an example, Fig. 4.17 shows the dependence of the Hall carrier concentration of Si-doped n-type Al_0.32Ga_−.68As on temperature. When measured in dark, the electron concentration remains constant for temperature below ~150 K, with a value of 3.5 × 10¹⁷ cm⁻³, only a fraction of the Si impurity concentration of 1.5 × 10¹⁸ cm⁻³. The electron concentration increases at temperatures higher than 150 K indicating the thermal ionization of the deep donor center. However, the saturation of the free carrier concentration does not occur even at 300 K confirming that the DX center has large thermal activation energy. Upon illumination with an infrared radiation having a wavelength of 820 nm, which is below the bandgap of Al_xGa_1−xAs at low temperature, the carrier concentration increases. The increase in carrier concentration persists even after the illumination has been turned off hours or days. This phenomenon is known as the persistent photoconductivity (PPC) effect, a key characteristic of DX center in n-type Al_xGa_1−xAs. The model of two atomic configurations can explain the phenomenon of PPC. The negatively charged DX center resumes a neutral state, DX⁰, under photoionization. This state is unstable, and the Si donor assumes the lower-energy substitutional D⁰ configuration. Accompanying the neutral Si donors are the extra free carriers released from the DX^– state. At sufficiently low temperatures, the Si cannot assume the DX^– configuration due to the large capture barrier (E_c). Thus, the extra carriers remain free persistently. At temperatures higher than ~150 K, the neutral donor can capture an electron and return to the state of a DX center. Therefore, the high PPC concentration can be quenched by heating the sample to a temperature >150 K.

Fig. 4.17

Temperature dependence of the Hall electron concentration in n-type Al_0.32Ga_0.68As with a Si doping concentration of 1.5 × 10¹⁸ cm⁻³. Solid and open circles indicate experimental data measured in the dark and after illumination at low temperatures, respectively.

Reprinted with permission from [13], copyright American Physical Society

1. (b)

   EL2 Level

    

When growing bulk III–V semiconductors, it is unavoidable that some unintentional impurities (Si, C, B, Fe, S, and Mn) are incorporated into crystals. The dominant residual impurity in bulk GaAs crystal grown by liquid encapsulated Czochralski (LEC) technique is the shallow acceptor C with a concentration in the low 10¹⁵ cm⁻³ range. This impurity comes from the graphite heating elements and/or the wall of the stainless steel growth chamber. The residual conductivity would be p-type with a resistivity of 0.1–10 Ω cm. For high-speed device applications, it is necessary to minimize the parasitic capacitance and inductance by using semi-insulating (SI) substrates with a resistivity >10⁷ Ω cm. Many deep impurities have been intentionally introduced in order to achieve high resistivity in compound semiconductors. To achieve SI GaAs, the most noticeable dopant used was Cr. The major drawback is the Cr diffusion and redistribution during high-temperature processing and epitaxial growth. Later, the EL2 defect was identified as the key to achieve undoped SI GaAs grown by the LEC technique. The incorporation of a sufficiently high concentration of EL2 defects allows one to obtain semi-insulating GaAs materials with a resistivity higher than 10⁷ Ω cm.

In GaAs there are many deep levels whose chemical origins are unknown. The EL2 defect is one of these deep levels (forming an electron trap) in undoped GaAs. The EL2 level has been identified to be the native As anti-site defect, As_Ga. The arsenic atom has two excess electrons if occupying a Ga site, i.e., a double donor. The main EL2 donor level, located approximately 0.75 eV below the conduction band edge, compensates for the shallow C acceptor impurities. For high EL2 concentrations, due to their large ionization energy, only a fraction of the EL2 centers will be ionized, while the remaining EL2 centers will be neutral. As a result, the Fermi level is pinned at the EL2 level which is approximately in the mid-bandgap of the GaAs. Thus, GaAs with sufficiently high concentration of EL2 centers has near-intrinsic characteristics. The EL2 concentration can be controlled during bulk crystal growth via the As/Ga composition in the growth melt (Fig. 4.18). For high As/Ga ratios (As atom fraction ≥0.475) in the growth melt, the concentration of EL2 defects increases, which compensates the shallow C acceptors. At low As/Ga ratios in the growth melt, the shallow C acceptor impurities are not fully compensated, leading to residual p-type conductivity of the bulk GaAs.

Fig. 4.18

Resistivity of unintentional doped GaAs grown by LEC technique as a function of As atom fraction in the growth melt. An As-rich melt produces semi-insulating GaAs with resistivity >10⁷ Ω cm and a Ga-rich melt gives p-type conductivity.

Reprinted with permission from [14], copyright AIP Publishing

4.3 Free Carrier Concentration and the Fermi Integral

4.3.1 Free Carrier Concentrations in 3D Semiconductors

The electron and hole concentrations in 3D semiconductor structures are calculated by integrating carrier distributions over appropriate energy ranges.

(4.52)

where D(E) and f(E) are the DOS and the Fermi–Dirac distribution function, respectively. The DOS of the conduction band and the valence band are

(4.53)

In a semiconductor, the carriers are indistinguishable and have to obey the Pauli exclusion principle. The occupation behavior in the energy space is governed by the Fermi–Dirac distribution function

(4.54)

where E_F is the Fermi level. For non-degenerated semiconductors where E − E_F > 3kT, the Fermi–Dirac distribution function is simplified to the Boltzmann distribution function and (4.52) is solved analytically. For degenerated semiconductors where |E – E_F| is less than a few kT, the Fermi–Dirac distribution function cannot be simplified. Using the DOS and f(E) expressed above, one obtains

(4.55)

By replacing and , the electron concentrations has the form of

(4.56)

We can reduce (4.56) to a more convenient form by using the Fermi–Dirac integrals defined as

(4.57)

where Γ(j + 1) is the Gamma function. With j = 1/2 and Γ(3/2) = /2 in (4.56) and (4.57), then

(4.58)

where N_c is the effective DOS and F_1/2(η) is the Fermi–Dirac integral of the order j = 1/2. Their expressions are

and

Table 4.4 lists the Fermi–Dirac integral of orders 1/2 as a function of the reduced Fermi energy η.

Table 4.4

Fermi–Dirac integral of order +1/2, F_1/2(η), as a function of reduced Fermi energy η

η	F1/2	η	F1/2	η	F1/2	η	F1/2	η	F1/2
–4.0	0.01820	–2.0	0.12930	1.0	1.5756	4.0	6.5115	7.0	14.290
–3.9	0.02010	–1.8	0.15642	1.2	1.7900	4.2	6.9548	7.2	14.886
–3.8	0.02220	–1.6	0.18889	1.4	2.0221	4.4	7.4100	7.4	15.491
–3.7	0.02451	–1.4	0.22759	1.6	2.2720	4.6	7.8769	7.6	16.104
–3.6	0.02706	–1.2	0.27353	1.8	2.5393	4.8	8.3550	7.8	16.725
–3.5	0.02988	–1.0	0.32780	2.0	2.8237	5.0	8.8442	8.0	17.355
–3.4	0.03299	–0.8	0.39154	2.2	3.1249	5.2	9.3441	8.2	17.993
–3.3	0.03641	–0.6	0.46595	2.4	3.4423	5.4	9.8546	8.4	18.639
–3.2	0.04019	–0.4	0.55224	2.6	3.7755	5.6	10.375	8.6	19.293
–3.1	0.04435	–0.2	0.65161	2.8	4.1241	5.8	10.906	8.8	19.954
–3.0	0.04893	0.0	0.76515	3.0	4.4876	6.0	11.447	9.0	20.624
–2.8	0.05955	0.2	0.89388	3.2	4.8653	6.2	11.997	9.2	21.301
–2.6	0.07240	0.4	1.0387	3.4	5.2571	6.4	12.556	9.4	21.986
–2.4	0.08794	0.6	1.2003	3.6	5.6623	6.6	13.125	9.6	22.678
–2.2	0.10671	0.8	1.3791	3.8	6.0806	6.8	13.703	9.8	23.378
–2.0	0.12930	1.0	1.5756	4.0	6.5115	7.0	14.290	10	24.085

For the hole concentration in the valence band, it is convenient to set a new parameter . The integral becomes

(4.59)

or

(4.60)

We assume E_v = 0. For non-degenerate n-type semiconductors, where E_c − E_F kT or F_1/2(η)  1 or n N_c, the Fermi–Dirac distribution can be approximated by Boltzmann distribution function.

(4.21)

For p-type semiconductors, carrier concentration equation analogous to this is expressed as

(4.61)

For example, the calculated carrier concentration of a non-degenerate n-type semiconductor with E_F − E_c = –2kT, using Boltzmann approximation, is

The Fermi–Dirac integral approach using η = –2 leads to n = 0.12390N_c. The calculated electron concentrations using both approaches are quite comparable with a small error of ~4.7%. Now, if the Fermi level is moving closer to E_c, say E_F − E_c = –kT, the calculated carrier concentration error using Boltzmann approximation increases to 12%.

4.3.2 Free Carrier Concentrations in 2D Semiconductor Structures

For semiconductors with only 2D spatial freedom, e.g., in a quantum well, the carrier density is

(4.62)

By replacing

(4.63)

the electron concentrations has the form of

(4.64)

The integral is the Fermi–Dirac integral of zero order (j = 0)

(4.65)

where Γ(1) = 1 is the Gamma function. The Fermi–Dirac integral of zero order can be solved analytically and has a solution

(4.66)

Thus the 2D carrier density has a form of

(4.67)

where  = m* kT/πħ². For a non-degenerate 2D system, the Fermi–Dirac distribution can be approximated by Boltzmann distribution function. Thus,

(4.68)

In the highly degenerate 2D case, E_F> E_c, exp() ≫ 1 and

(4.69)

4.3.3 Carrier Concentration in the Multiple Valley Limit

When the energy differences between conduction band minimums along different crystallography directions are small, electron distribution is not limited to the lowest band. Depending on the energy separation between E_F and the conduction band minima in Γ, L, and X-directions, the electron distribution in each valley can be calculated. Let

(4.70)

In a non-degenerate semiconductor the free carrier concentration including contributions from all valleys is

(4.71)

where

In the case of degenerate semiconductors, one obtains .

4.4 Surface States in Compound Semiconductors

In bulk compound semiconductor, the atomic bonds take the sp³ configurations to form the zinc-blende crystal structure. A side view of the (110) surface along the [001] direction is shown in Fig. 4.19a where both group III and group V atoms are presented. When cleaving, the surface atoms take a relaxed configuration. In GaAs, due to their bond length difference and to minimize energy, on the (110) surface, the As bonds (~1.18 Å) move outward and the Ga bonds (~1.26 Å) move inward after the bonds are broken from the surface (Fig. 4.19b). To minimize their bond energies, the surface bonds of As and Ga atoms are taking s²p³ and sp² configurations, respectively, rather than the sp³ configuration of the bulk bonds, and form ‘dangling’ surface bonds. These bonds are available for strong interactions with atoms and molecules of the ambient. The surface states are formed near the center of the forbidden gap as a result of interactions between the 'dangling' surface bonds and contamination and surface imperfections. The surface imperfections are generated by deposited metal, which induces anti-site defects or vacancies near the interface. In silicon, the exposed surface atoms are the same and impose less severe surface problems. On polar (100) and (111) surfaces of III–V compounds, the surface atoms do not retain their ideal bulk structure, but undergo surface reconstruction. Depending on the surface condition, whether it is group III- or group V-rich, the surface can have different atomic arrangements that are more complex than on a (110) surface. For example, under As-rich conditions, the (100) GaAs surface atoms have periodicities twice and four times larger than the bulk atomic arrangement in the and [110] directions, respectively.

Fig. 4.19

a Bulk structure of the (110) surface, and b the relaxed configuration of (110) surface of a zinc-blende semiconductor

As a result of the surface states, the energy bands are modified near the surface. For an ideal metal–semiconductor interface, the Schottky barrier height (ϕ_b) is determined by the difference between the work function of the metal (ϕ_m) and the electron affinity of the semiconductor (χ_s) as shown in Fig. 4.20a.

Fig. 4.20

Energy band diagram of a ideal Schottky barrier, and b metal–semiconductor barrier with high density of surface states. The Fermi level at the semiconductor surface is pinned near the mid-gap and represented by ϕ₀

(4.72)

In 1947, John Bardeen first recognized the importance of the semiconductor surface while studying metal–semiconductor contacts. He found that a surface space charge layer (surface states) is formed by the presence of surface defects and contaminations (Fig. 4.20b) and characterized by a mid-gap energy level ϕ₀. If the density of surface states is sufficiently high (Bardeen limit: >10¹² cm⁻²), it can pin the surface Fermi level near the middle of the bandgap independent of the doping density. He further showed that if the density of the surface states is sufficiently high (≥10¹³ cm⁻²), the Schottky barrier height becomes a fixed value for each semiconductor independent of the metal work function.

(4.73)

Thus, the surface states play an important role in the physical properties of carrier transport near the semiconductor surface. Due to the nature of high surface state density, this observation has been verified in III–V compounds where the pinning of Schottky barrier heights is commonly observed. Because of this high density of surface states and the lack of robust native oxides to unpin the surface Fermi level, a high performance inversion-mode metal–oxide–semiconductor field-effect transistor (MOSFET) based on III–V compounds was not demonstrated until late 1990s. Using in situ electron beam evaporated Ga₂O₃(Gd₂O₃) dielectric film on a clean as-grown GaAs in an ultra-high vacuum connected multiple chamber MBE growth system, MOS structures have been successfully fabricated on GaAs with a low interface trap density for the first time. The unpinning of the GaAs Fermi level results from the Gd₂O₃ restoring the surface As and Ga atoms to near-bulk charge. Later, high performance MOSFETs were demonstrated using ex situ atomic layer deposited (ALD) high-κ Al₂O₃ and HfO₂ films as the gate dielectrics on GaAs and other III–V materials. The metal alkyls used in ALD dielectric process, in particular, trimethyl aluminum for Al₂O₃ deposition, enable unpinning the Fermi levels on III–V semiconductors. A more detailed discussion of the development of III–V MOSFETs can be found in Chap. 9.

In semiconductors, the dangling-bond energy is typically located in the middle of the bandgap, and the interface trap density (D_it) increases exponentially in the energy ranges close to the band edges. Over the years, a variety of models of semiconductor surface pinning energy were developed. Among others, the charge neutrality level (CNL)-based model is more related to the band structures of III–V channel materials and offers a realistic explanation of all experimental results on III–V MOSFETs. The CNL energy level represents a weighted average value over the density of states (DOS). CNL is pushed away by the large DOS of the conduction and valence bands, as shown in Fig. 4.21. Therefore, the semiconductor surface pinning energy CNL is located inside the forbidden gap. If the Fermi level E_F is above CNL, the states are of acceptor type and negatively charged if the states are occupied. If the Fermi level E_F is below CNL, the states are of donor type and positively charged if the states are occupied. The CNL values above the valence band edges are determined by averaging values derived from various theoretical models as shown in Table 4.5. In general, the calculated CNL is in close agreement with Schottky barrier heights (ϕ_bv) on p-type III-V compounds.

Fig. 4.21

CNL is a weighted average of DOS. A high DOS in the valence band tends to push the CNL toward the conduction band and vice versa.

Reprinted with permission from [15], copyright AIP Publishing

Table 4.5

Schottky barrier height (p-type semiconductor) and charge neutrality level (CNL) data of selected III–V compounds [15]

Material	AlP	AlAs	AlSb	GaP	GaAs	GaSb	InP	InAs	InSb
ϕbv (eV)	1.27	1.002	0.47	0.797	0.562	0.07	0.857	0.58	0.04
CNL (eV)	1.3	0.92	0.4	0.8	0.55	0.06	0.6	0.50	0.15

In general, the surface pinning energies (ϕ₀) calculated from different models are in close agreement with Schottky barrier heights (ϕ_bv) on p-type III–V compounds (Table 4.5). Figure 4.22 shows the measured Fermi levels on cleaved (110) GaAs, GaSb, and InP surfaces deposited with different metals and overlayers in vacuum. Indeed, the Fermi level is pinned on these compounds and the surface pinning energy matches well with ϕ_bv. Note that in GaSb and InP, both n- and p-type materials, the pinning energies are closer to the valence band and conduction band, respectively. This indicates that the nature and density of surface states are different for each material.

Fig. 4.22

Fermi pinning level positions for a range of overlayers on GaAs, GaSb, and InP (110) surfaces. Circles represent n-type and triangles represent p-type materials.

Reprinted with permission from [16], copyright AIP Publishing

4.5 III–V Compound Semiconductors

4.5.1 Lattice Constant

For binary compounds, all physical parameters of the material are fixed. Thus, they have zero degrees of freedom. By mixing two binaries with a common element, one can form a ternary compound. Two group III or group V elements can share the same sublattice to form III–III′–V or III–V–V′ ternary alloy, respectively. Ternary compounds have one degree of freedom in selecting lattice constant or energy bandgap. The lattice constant, a₀, varies linearly with the alloy composition, following Vegard’s law. Thus, the lattice constant of a ternary compound A_xB_1−xC can be expressed as a linear combination of lattice constants of binaries AC and BC.

(4.74)

where x is the composition fraction of binary AC and a_MN is the lattice constant of binary MN. The lattice constant and bandgap energy relationship of a number of III–V binary compounds are shown in Fig. 4.23.

Fig. 4.23

Lattice constant as a function of bandgap energy of III–V compound semiconductors. The bandgap energy of ternaries follows the line connecting the constituent binaries. Solid and dashed lines indicate direct and indirect bandgap, respectively

Furthermore, the quaternary compounds have two degrees of freedom in selecting the lattice constant and energy bandgap independently. The quaternary region in Fig. 4.23 is bounded by either three or four ternaries. In the former case, a type-II quaternary A_xB_yC_1−x−yD such as Ga_xAl_yIn_1−x−yAs is formed by three ternaries ABD, BCD, and ACD. Elements A, B, and C share the same sublattice sites as group III atoms, and D is a group V element. The lattice constant of this III–III′–III″–V quaternary follows Vegard’s law as

(4.75)

The other quaternary compounds AB_xC_yD_1−x−y formed by one group III element and three group V elements also fall into this category. It has a III–V–V′–V″ form and follows Vegard’s law as

(4.76)

In the latter case, a type-I quaternary A_xB_1−xC_yD_1−y is formed by four ternaries ABC, ABD, ACD, and BCD. The group III and group V sublattices are shared by A, B, and C, D, respectively, to form the III–III′–V–V′ quaternary. Its lattice constant is

(4.77)

For example, the lattice constant of the Ga_xIn_1−xAs_yP_1−y alloy is given by

Figure 4.24 shows the linear variation of lattice constant with composition in Ga_xIn_1−xAs_yP_1−y alloy following Vegard’s law.

Fig. 4.24

Lattice constant of the Ga_xIn_1−xAs_yP_1−y quaternary alloy. The vertical axes at four corners indicate the lattice constants of binaries. The shaded rectangles indicate the composition range lattice-matched to InP and GaAs, respectively

In theory, one can mix three III–III′–V ternaries with the same group III elements but three different group V elements to form a penternary or quinternary alloy III–III′–V–V′–V″. The lattice constant and bandgap coverage do not extend beyond what one can achieve with quaternaries. Therefore, there have been few reported results on quinternary. In addition, there are physical limits that prevent the formation of compounds in certain composition ranges, under equilibrium growth conditions, where a miscibility gap exists. As shown in Fig. 4.8, the nearest-neighbor distance of alloys is not changed significantly. In ternary and quaternary compounds, the distance between group III and group V atoms only changes by <2% as determined by EXAFS technique. For two binaries having very different bond lengths, they will not form homogeneous ternary compounds over a certain composition range called the miscibility gap. Instead, the mixture contains multiple solid phases (or phase separation). However, homogeneous thin films can be achieved grown by non-equilibrium methods such as MBE and MOCVD.

4.5.2 Bandgap Energy

In ternary and quaternary compounds, the crystal structure is seldom completely ordered. The periodic lattice potential is broken due to the randomness in lattice atom distribution. Additionally, the bond length variation induces a local strain. Vegard’s law, therefore, does not extend to some physical parameters including E_g, m*, and µ in ternaries and quaternaries. First consider a ternary semiconductor A_xB_1–xC which is an alloy of two semiconductor binaries i (BC) and j (AC). A ternary semiconductor parameter can be expressed in a general form as

(4.78)

where T_ij(x) is the value of the physical property of the ternary alloy composed of binaries i and j, the B_i and B_j are the material property of the given binaries, and C_ij is the ternary bowing parameter which relates to the deviation from a linear interpolation relationship between two binaries. Thus, the relationship between the bandgap and composition becomes

(4.79)

where C is the bowing parameter. The bowing parameter varies between 0.2 and 0.8 in III–V compounds.

The other feature in ternary alloys is that, when mixing a direct bandgap binary with an indirect bandgap binary, the lowest conduction bandgap crosses over from direct to indirect at a composition between the two binaries. The energy corresponding to the crossover composition is the highest energy that can be efficiently used for this compound as a light emitter. The bandgap energy as a function of composition in Al_xGa_1−xAs is shown in Fig. 4.25. This ternary alloy has a direct energy gap from GaAs (x = 0) up to x ≈ 0.4 (40% of Al mole fraction). Beyond the crossover composition (x > 0.4), the alloy has a minimum energy gap in the X valley and the bandgap becomes indirect.

Fig. 4.25

Composition dependence of the direct energy gap Γ and the indirect energy gap X and L for Al_xGa_1−xAs

For determining the band parameters of quaternary alloy from those of the constituent ternary alloys, a number of approaches with different degree of uncertainties have been developed. A widely adopted method derived using interpolation procedure is shown first. For the type-I quaternary A_xB_1–xC_yD_1−y, the band parameter is expressed as a weighted sum of the related ternary values as

(4.80)

where T_ij’s represent the ternary semiconductor parameters following (4.78). For type-II quaternaries such as AB_xC_yD_1−x−y, the band parameter has the form of

(4.81)

where B_ij’s represent the binary semiconductor parameters, and C’s are ternary bowing parameters. Nevertheless, fitting calculated quaternary alloy parameters to experiments is necessary to reach a suitable description of a particular quaternary alloy system. For example, the bandgap of the most important quaternary Ga_xIn_1−xAs_yP_1−y as a function of alloy composition is listed as [17]

(4.82)

The composition dependence of energy gap for a quaternary alloy is constructed from energy bandgap versus composition (E_g–x, y) relations of the four constituent ternaries. As an example, Fig. 4.26 shows a 3D composition–bandgap energy plot for the quaternary alloy Ga_xIn_1−xAs_yP_1−y. On each sidewall is the E_g–x, y relation of one of the four ternary alloys. The four ternaries form the boundary of the quaternary energy gap surface. The base of the 3D plot gives the composition in terms of x and y. The intersection of the direct and indirect energy gap surfaces indicates that most of this system is in the direct bandgap region covering 0.36 to ~2 eV. For practical applications, the compositions lattice-matched to InP are of considerable interest because Ga_xIn_1−xAs_yP_1−y has an energy gap that covers from 0.74 to 1.35 eV. This allows the construction of a wide range of Ga_xIn_1−xAs_yP_1−y/InP heterostructures for photonic as well as high-speed device applications.

Fig. 4.26

Composition dependence of the bandgap energy surface of the quaternary Ga_xIn_1−xAs_yP_1−y alloy. Each sidewall represents the bandgap energy–composition relationship of a constituent ternary alloy. The square base shows the composition in terms of x and y

From Fig. 4.26 we can construct the energy gap–lattice constant–composition diagram using the projection of the energy gap surface onto the x-y plane. This diagram is given in Fig. 4.27. The constant bandgap energy lines are plotted for a function of composition (x, y). The two straight lines originating from GaAs and InP corners represent the lattice-match compositions to GaAs and InP, respectively.

Fig. 4.27

x-y compositional plane for Ga_xIn_1−xAs_yP_1−y. The x-y coordinate of any point in the plane gives the composition. The curved lines are constant direct bandgap energy values that were obtained by projection from the direct energy surface in Fig. 4.26. The composition lattices matched to InP and GaAs are shown as straight lines connected to InP and GaAs corners, respectively

The interpolations used above by requiring only the boundary binary and ternary values satisfy a necessary but not sufficient condition for describing the quaternary alloy property inside those boundaries. Thus, it is necessary to determine the quaternary bowing parameter for a better description of the quaternary function Q(x, y). A multivariable quadratic interpolation algorithm has been developed to identity the intrinsic quaternary bowing parameter for modeling the compositional dependence of quaternary alloy bandgaps [18].

For a quaternary alloy A_xB_1−xC_yD_1−y, the material parameter function Q(x, y) can be plotted as a surface domain enclosed by four boundaries similar to the composition dependence of the bandgap energy surface shown in Fig. 4.26. The four corners of the surface are defined by four binary alloys, and their material parameters are B_AC, B_AD, B_BC, and B_BD. Following (4.78), the ternary parameter expressions are

(4.83a)

(4.83b)

where C_III−V is the alloy bowing parameter. The quaternary parameter can be expressed in two paths with either x mixing or y mixing:

(4.84a)

(4.84b)

Requiring the quaternary function Q(x, y) to be unique inside the quaternary surface domain gives Q_AB = Q_CD. D is the quaternary surface bowing parameter and can be estimated as

(4.85a)

(4.85b)

Therefore, the quaternary material parameter function Q(x, y) can be expressed as

(4.86)

Letting B_BC = B₁, B_AC = B₂, B_AD = B₃, B_BD = B₄, and C_ABC = C₁₂, C_BCD = C₁₄, C_ACD = C₂₃, C_ABD = C₃₄, the above equation can be compacted into the following form:

(4.87)

By matching experimental bandgap values of a quaternary alloy with calculations, the quaternary bowing parameter D is identified and listed in Table 4.6. This equation is valid for both type-I and type-II quaternary alloys. For example, the composition of Ga_xIn_1–xAs_yP_1−y type-I alloy (III–III′–V–V′) can be expressed as [B_1yB_4(1–y)]_(1–x)[B_2yB_3(1–y)]_x with B₁ = InAs, B₂ = GaAs, B₃ = GaP, and B₄ = InP. For type-II quaternary alloys with the form of III–III′–III″–V, one binary can be assigned twice in the same row or column of the alloy matrix with a zero ternary bowing parameter (i.e., B₂ = B₃, C₂₃ = 0). The composition of Al_x(Ga_yIn_1–y)_1−xP type-II alloy is then given by [B_1yB_4(1–y)]_(1–x)B_2x. The necessary bandgap and bowing parameters of binary and ternary alloys as well as the quaternary bowing parameters needed in (4.87) for calculating III–V quaternary semiconductor alloy bandgaps are listed in Table 4.6. As examples, the composition dependence of bandgap energy diagrams of both type-I and type-II quaternary alloys are calculated and shown in Fig. 4.28.

Table 4.6

Bowing parameters of ternary and quaternary III–V semiconductor alloys

The numbers between two binary alloys are: First row—the bowing parameter of the direct bandgap ternary alloy (C_ij), second row—the numbers in the parentheses are the ternary bowing parameters of the indirect bandgap, and third row—the ternary bowing parameter of spin-orbit splitting Δ_so. Numbers in the shaded boxes are bowing parameters of quaternaries (D) associated with four neighboring binary alloys that form type-I quaternary alloys. Bowing parameters of type-II quaternaries associated with three constituent binary alloys in the same row are shown in the last column [18, 19]

Fig. 4.28

Examples of calculated composition dependence of bandgap energy diagrams of type-I (Ga_xIn_1−xAs_ySb_1−y, top) and type-II (Al_xGa_yIn_1−x−yP, bottom) III–V quaternary alloys. Available lattice-matched substrates are shown in thick straight lines, and the shaded area indicates the region with indirect bandgap.

Reprinted with permission from [18], copyright AIP Publishing

4.6 III–N and Dilute III–V–N Compound Semiconductors

4.6.1 III–N Compounds

AlN, GaN, InN, and their alloys can crystallize in both wurtzite and zinc-blende lattice structures. Most of the current substrate materials used to grow these compounds have wurtzite structure, which enhances the formation of the same lattice structure. Therefore, all III–N compounds considered here have a wurtzite crystal structure and can be described by a- and c-axis lattice constants. However, under certain conditions, it is possible to grow zinc-blende structure GaN crystals. Due to the lack of suitable substrates, wurtzite III–N epitaxial layers have been grown on sapphires (Al₂O₃) and SiC along the [0001] direction. Usually, the resulting epilayers contain high density of threading dislocations (≥10⁹ cm⁻²) extending from the hetero-interface toward the surface. Therefore, there has been a continuous effort to develop large area GaN substrate technology. The wurtzite structure GaN consists of two closely spaced hexagonal lattices as described in Sect. 2.​4, one formed by gallium atoms and the other by nitrogen atoms. It has ionicity about 0.5 and shows more ionic properties than covalent properties. Thus, one can simply say that an electron is transferred from an N atom to a Ga atom in the wurtzite GaN crystal to form a Ga^– anion and N⁺ cation. Looking along the c-axis, as shown in Fig. 4.29, reveals that the (0001) planes are alternating anion layers and cation layers with the Ga-plane and N-plane terminating (0001) and surfaces, respectively. The ionic bonds linking neighboring (0001) planes with opposing charges generate a spontaneous polarization (P_SP) in the direction of . In addition, we also notice a large lattice-mismatch between GaN and AlN (Δa/a = 2.4%, Δc/c = 3.9%). When coupled with the large piezoelectric constants (~1 versus 0.01 in III–V’s), a strong strain-induced polarization or piezoelectric polarization (P_PE) forms in III-N heterostructures. The direction of the polarization follows dilation or contraction of the c-axis. The existence of significant spontaneous and piezoelectric polarizations in III-N heterostructures leads to unique electronic and photonic properties.

Fig. 4.29

Crystal structure of wurtzite GaN along [0001] direction.

Reprinted with permission from [20], copyright AIP Publishing

III-N layer structures are currently grown using either MBE or MOCVD. These epitaxy techniques will be described in more detail later in Chap. 5. In MBE, molecular beams of Ga and Al are generated from effusion cells using elemental Ga and Al as source materials. Due to the large bond strength of N₂, it is impractical to thermally dissociate N₂ into atomic N. Instead, in plasma-assisted MBE (PAMBE), an RF plasma source is generally used to generate atomic nitrogen and N⁺ for III-N growth. Ammonia (NH₃) has also been used as a nitrogen source by direct thermal dissociation on the heated substrate surface at a temperature of ~800 °C. MBE layers are usually grown at ~800 °C directly on (0001) sapphire or c-Al₂O₃ under Ga-rich surface conditions to achieve a high layer quality. This particular MBE growth condition yields N-terminated B-face or N-face layers, i.e., the GaN growth direction is along capped with a top nitrogen plane. Polarity reversal into Ga-terminated A-face or Ga-face layer is achievable by inserting a thin AlN layer into the structure.

For III-N layers grown by MOCVD, metalorganics such as trimethylaluminum (TMA) and triethylgallium (TEG) are used to deliver Ga and Al, respectively, and ammonia is used as the nitrogen source. In MOCVD process, III-N materials are grown at much higher temperatures (~1050 °C) than in MBE. To maintain material quality, very high nitrogen flow inside the MOCVD growth chamber is required. In addition, a low-temperature grown Al(Ga)N nucleation layer on (0001) sapphire is needed before the growth of GaN. The nitrogen-rich growth condition for the nucleation layer yields a Ga-terminated A-face or Ga-face layer and a growth direction along [0001] capped with a top gallium plane. This growth direction is opposite to the MBE grown nitrides and causes a sign change in spontaneous polarization.

The composition dependences of the energy gaps for the ternary alloys AlGaN, GaInN, and AlInN follow the usual quadratic form, similar to other III–V alloys, of (4.79).

(4.88)

where C is the bowing parameter. The bowing parameter is always negative for these alloys, which reflects a reduction of the alloy energy gaps. The lattice constants, both in (0001) plane (a) and along c-axis (c), and energy gaps of the III-N binaries are listed in Table 4.7. Using bowing parameters of −0.7, −1.4, and −2.5 in AlGaN, GaInN, and AlInN, respectively, along with bandgap energies of binaries, the composition dependence of the direct energy gap on the lattice constant in (0001) plane (a₀) for these wurtzite structure III-N ternary alloys is plotted in Fig. 4.30.

Table 4.7

Band structure parameters for wurtzite III-N binaries [21, 22]

Parameters	GaN	AlN	InN
a (Å)	3.189	3.112	3.54
c (Å)	5.185	4.982	5.70
(eV) @ 300 K	3.438	6.138	0.64

Fig. 4.30

Composition, in terms of lattice constant in the (0001) plane (a), dependence of the direct energy gap Γ for AlGaN, GaInN and AlInN alloys

One interesting property about GaN is the doping behavior. GaN can be doped easily with shallow donors of Si and Ge above 10¹⁹ cm⁻³. The Si donor ionization energy is about 12–17 meV. However, doping GaN with acceptors to obtain a high concentration of holes was a difficult problem until the late 1980s. It was discovered that p-type doping had been limited by hydrogen passivation of acceptors. To activate hydrogen passivated Mg acceptors, the Mg-doped GaN requires a low-temperature (~300 °C) heat treatment in the form of low-energy electron beam irradiation treatment or thermal annealing in vacuum or in nitrogen atmosphere to dissociate hydrogen atoms which form complexes with Mg atoms. One drawback of Mg doping in GaN is its high acceptor ionization energy of ~150 meV which leads to a low doping efficiency. Currently, hole concentrations of mid 10¹⁷ cm⁻³ and mid 10¹⁸ cm⁻³ are achievable in MOCVD and MBE grown layers, respectively.

4.6.2 Dilute III-V-N Compounds

The composition dependence of the energy gap for the III–V ternary alloys usually does not follow Vegard’s law due to the randomness in lattice atom distribution and bond length variation. The deviation from a linear interpolation between the two binaries is accounted for by the bowing parameter. When mixing III–V compounds with a small fraction (usually ≤ 2–3%) of nitrogen in dilute nitride compounds such as GaAs_xN_1−x and GaP_xN_1−x, a very large bowing in bandgap energy exists (Fig. 4.31). This unusual behavior is probably due to the extremely large bond length difference between N and other group V elements (Table 4.1) and/or the unique isoelectronic property of N in III–V alloys. Isoelectronic centers are formed by substituting one atom of the crystal with another atom of the same valence but with large differences in electronegativity and bond length. The isoelectronic center was first observed in GaP:N where a localized potential well is formed around the nitrogen atom. This potential well allows a nitrogen atom to capture an electron, which in turn can bind a hole by Coulomb attraction, thus forming a bound exciton. A more detailed discussion of isoelectronic traps can be found in Chap. 8.

Fig. 4.31

a Energy bandgap of GaAs_1−xN_x as a function of nitrogen fraction x. b BAC model calculated indirect bandgap energy of GaP_1−xN_x along with experimental data.

Reprinted with permission from [23], copyright IOP

It is the interaction between the spatially localized N isoelectronic level and the conduction band of the underlying non-nitride semiconductor that leads to a splitting of the conduction band into two subbands and a large reduction of the fundamental bandgap. According to the band anti-crossing (BAC) model, if the effect on the valence bands is completely neglected, the dispersion relations for the two coupled subbands are expressed as

(4.89)

where E^C(k) is the conduction band dispersion of the unperturbed non-nitride semiconductor, E^N is the position of the nitrogen isoelectronic impurity level in that semiconductor, V is the interaction potential between the two bands, and x is the fraction of nitrogen in the alloy. The values of E^N and V for the dilute III–V–N quaternary alloy Ga_1−xIn_xAsN are [1.65(1–x) + 1.44x − 0.38x(1–x)] and [2.7(1–x) + 2.0x − 3.5x(1–x)], respectively. The values of E^N and V are reduced to 1.65 and 2.7, respectively, for GaAsN. As an example, the anti-crossing characteristics of the dispersion relations for the two coupled conduction bands in GaAs_0.99N_0.01 are shown in Fig. 4.32. The fundamental energy gap is now governed by the transition from E_– to the top of the valence band. One precaution is that the BAC model considers only the interaction between a single, spatially localized nitrogen level and the conduction band of the underlying non-nitride semiconductor. It neglects not only mixing with the L and X valleys, but also nitrogen pairs and clusters. Therefore, the predictive power of the BAC model for an indirect bandgap material, e.g., GaP_xN_1−x, is more limited.

Fig. 4.32

Conduction band dispersion relations for GaAs_0.99N_0.01 at room temperature from the BAC model (solid curves). The unperturbed GaAs conduction band (dashed curve) and the position of the N level (thin dashed line) are also shown.

Reprinted with permission from [21], copyright AIP Publishing

Problems

1. 1.

   The diatomic chain model considered for phonon (lattice vibration) characteristics have identical springs but different masses. A model with alternating spring constants, α and β, but the same mass, m, is appropriate for Si, Ge or diamond crystal. Calculate the ω-k dispersion relation for crystals with a diamond structure.

    
2. 2.

   Determine what fraction of holes in Si and InP are heavy holes. Let (Si) = 0.537, (Si) = 0.153m₀, (InP) = 0.56m₀, (InP) = 0.12m₀.

    
3. 3.
   For Ge:  = 1.59m₀,  = 0.0823m₀,  = 0.28m₀,  = 0.043m₀; and for GaAs:  = 0.067m₀,  = 0.50m₀,  = 0.076m₀.

   1. (a)

      Determine the density of states effective mass m_DOS of Ge.

       
   2. (b)

      Calculate N_c and N_v of Ge at 300 K.

       
   3. (c)

      Determine what fraction of holes in Ge and GaAs are heavy holes.

       
   4. (d)

      Comment on the room-temperature mobility difference between Ge (µ_e ~ 3900, µ_h ~ 1800 cm²/Vs) and GaAs (µ_e ~ 9000, µ_h ~ 400 cm²/Vs).

       
    
4. 4.
   The energy differences E_c − E_F≥ 3kT and E_F − E_v≥ 3kT are defined as thresholds for non-degenerate n-type and p-type semiconductors, respectively.

   1. (a)

      Determine the n- and p-type carrier densities for Si and GaAs at the threshold of non-degeneracy. Commenting the differences between n- and p-type materials as well as between Si and GaAs.

       
   2. (b)

      Calculate n- and p-type carrier densities of Si and GaAs at 300 K for E_c – E_F = 0.2kT and E_F – E_v = 0.2kT, respectively.

       
    
5. 5.
   For a linear extrapolation of the density of states effective mass between GaAs and AlAs, the electron effective mass for the indicated conduction band in Al_xGa_1−xAs is taken as

   

   The energy bandgap variations in the X-, L-, and Γ-conduction bands are also provided as follows:

   

   Determine the fraction of the electrons is in the X-, L-, and Γ-conduction band valleys of non-degenerately doped n-type (a). Al_0.3Ga_0.7As, and (b). Al_0.5Ga_0.5As.
    
6. 6.

   Electron–hole pairs (EHPs) are generated by photoexcitation of an undoped Ga_0.47In_0.53As layer. The increasing EHP concentration causes the quasi-Fermi level of electrons and holes to move away from the equilibrium Fermi level toward the conduction band and valence band, respectively. One special condition, under a strong photoexcitation, which is important for lasers, occurs when the separation in quasi-Fermi levels (F_c − F_v) = E_g, a condition known as ‘transparency’. At what electron concentration (n = p) does the material reach transparency? You have to use Fermi–Dirac integral for both electrons and holes.

   For Ga_0.47In_0.53As:  = 0.041m₀,  = 0.465m₀, and  = 0.0503m₀. Note, at transparency, η – ζ = 0, where

   
    
7. 7.
   For Ge:  = 1.59m₀ and  = 0.0823m₀; for Si:  = 0.9163m₀ and  = 0.1905m₀; and for GaAs:  = 0.067m₀. The measured room-temperature electron mobility of n-type Ge, Si, and GaAs samples doped to 2×10¹⁷ cm⁻³ are 2750, 600, and 4000 cm²/V-s, respectively.

   1. (a)

      Calculate the average relaxation time, τ (scattering probability = 1/τ), for Ge, Si, and GaAs.

       
   2. (b)
      Assume, at room temperature, the dominant scattering mechanisms in elemental semiconductors are due to lattice scatterings. The scattering due to optical phonons for Ge can be approximated as

      

      The LO phonon frequency, ω_op, for Ge is 6 THz. Calculate the lattice scattering mobility, µ_l
       
   3. (c)

      Although Ge and GaAs have a similar average relaxation time, the mobility difference is quite large. Comment on the room-temperature mobility difference between Ge and GaAs.

       
    
8. 8.
   The ternary GaAs_xSb_1−x is direct bandgap over the full composition range. It has a bandgap energy of 0.775 eV at x = 50% where it is lattice-matched to InP. The room-temperature bandgap energy and lattice constant of GaAs and GaSb are 1.424 eV and 5.6533 Å, and 0.727 eV and 6.0959 Å, respectively.

   1. (a)

      Find the coefficients a, b, and c of the bandgap energy quadratic equation.

       
   2. (b)

      Sketch E_g versus x.

       
    
9. 9.
   The ternary InAs_xSb_1−x is direct bandgap over the full composition range. It has the smallest bandgap among all III–V compounds at x = 35%. The corresponding emission wavelength at that composition is 11.5 µm. The room-temperature bandgap energy and lattice constant of InAs and InSb are 0.35 eV and 6.0584 Å, and 0.17 eV and 6.4794 Å, respectively.

   1. (a)

      Find the coefficients a, b, and c of the bandgap energy quadratic equation.

       
   2. (b)

      Sketch E_g versus x and indicate on the sketch the bandgaps when the ternary is lattice-matched to AlSb (6.1355Å) and GaSb (6.09593Å).

       
    
10. 10.

    Derive the quaternary direct bandgap energy as a function of compositions x and y of the quaternary alloy Ga_xIn_1−xSb_yAs_1−y using (4.87). Make a sketch of the E_g versus compositions x and y. For the iso-energy curves, show only the curve with E_g = 0.6 eV, which corresponds to an eye-safe wavelength of ~2.1 µm. Estimate the compositions lattice-matched to InAs and GaSb on this curve.

     
11. 11.
    From the E_g versus composition sketch of the Ga_xIn_1−xAs_yP_1−y quaternary alloy (Fig. 4.27):

    1. (a)

       Find the range of bandgap energies covered by the lattice-matched alloys on InP and GaAs. (The lattice constant of GaP is 5.45117 Å.)

        
    2. (b)

       Find the x and y values for the alloy lattice-matched to Ga_0.75In_0.25As and has bandgap energy of 1.4 eV.

        
    3. (c)

       Derive the quaternary direct bandgap energy as a function of compositions x and y using (4.87). Compare the results at x = y = 0.5 with that derived from (4.82).

        
     
12. 12.
    The alloy Al_xGa_yIn_1−x−yAs is a type-II quaternary alloy.

    1. (a)

       Derive the quaternary direct bandgap energy as a function of compositions x and y using (4.87).

        
    2. (b)

       Make a sketch of the E_g versus composition x and y. Mark clearly the boundary of the direct–indirect crossover and show the compositions lattice-matched to InP, which has a lattice constant of 5.86875 Å. The lattice constants of AlAs, GaAs, and InAs are 5.6605 Å, 5.65325 Å, and 6.0584 Å, respectively.

        
    3. (c)

       What is the range of the bandgap energy covered by this quaternary alloy lattice-matched to InP? At which compositions does it have the bandgaps corresponding to 1.3 and 1.55 µm?

        
     
13. 13.
    In an effort to achieve efficient luminescence from group IV semiconductors, the Ge/α-Sn alloy has been investigated. The lattice constants of α-Sn and Ge are 6.48 Å and 5.6575 Å, respectively. The calculated transition energies along Γ, L, X valleys of the Ge_xSn_1−x compound are shown below. Deduce the bandgap energy as a function of composition x for the Γ valley and determine the composition range where this alloy is a semiconductor with direct bandgap energy. Note the material becomes semimetal at E_g ≤ 0 eV.

    

     

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Further Reading

1. 1.
   S. Wang, Fundamentals of Semiconductor Theory and Device Physics(Prentice-Hall, 1989)
2. 2.
   S. Adachi, Physical Properties of III–V Semiconductor Compounds (Wiley, 1992)
3. 3.
   E. F. Schubert, Doping in III-V Semiconductors (Cambridge, 1993)