**1.1Crystal Structure of Nitrides**

Group III nitrides can be of crystalline structures: the wurtzite (Wz), zinc blende (ZB),

and rock salt. Under ambient conditions, the thermodynamically stable structure is

wurtzite for bulk AlN, GaN, and InN. The zinc blende structure for GaN and InN has

been stabilized by epitaxial growth of thin films on {0 1 1} crystal planes of cubic

substrates such as Si [1], SiC [2], MgO [3], and GaAs [4]. In these cases, the intrinsic

tendency to form the Wz structure is overcome by the topological compatibility.

However, Wz structure could very likely be present at the extended defect sites. The

rock salt form is possible only under high pressures and, therefore, is laboratory form

of exercise.

Let us now discuss the space groups for the various forms of nitrides. The rock salt,

or NaCl, structure (with space group Fm3m in theHermann–Mauguin notation and

O5

h in the Schoenflies notation) can be induced in AlN,GaN, and InN under very high

pressures. The reason for this is that the reduction of the lattice dimensions causes

the interionic Coulomb interaction to favor the ionicity over the covalent nature. The

structural phase transition to rock salt structure was experimentally observed at the

following estimated pressure values: 22.9GPa (17 GPa in other estimates) for AlN [5],

52.2GPa for GaN [6], and 12.1GPa for InN [7]. Rock salt III nitrides cannot be

produced by any epitaxial growth.

The space grouping for the zinc blende structure is F43m in the Hermann–

Mauguin notation and T2d

in the Schoenflies notation. The zinc blende structure has a

cubic unit cell, containing four group III elements and four nitrogen elements.

(Although the term zinc blende originated in compounds such as ZnS, which could

be in cubic or hexagonal phase, it has been used ubiquitously for compound

semiconductors with cubic symmetry. The correct term that should be used for the

cubic phase of GaN is actually sphalerite. However, to be consistent with the usage

throughout the literature, even at the expense of accuracy, the term zinc blende has

been used in this book). The position of the atoms within the unit cell is identical to

the diamond crystal structure. Both structures consist of two interpenetrating facecentered

cubic sublattices, offset by one quarter of the distance along a body diagonal.

Each atom in the structure may be viewed as positioned at the center of a tetrahedron,

with its four nearest neighbors defining the four corners of the tetrahedron. The

stacking sequence for the (1 1 1) close-packed planes in this structure is AaBbCc.

Lowercase and uppercase letters stand for the two different kinds of constituents.

The wurtzite structure has a hexagonal unit cell and thus two lattice constants, c

and a. It contains six atoms of each type. The space grouping for the wurtzite

structure is P63mc in the Hermann–Mauguin notation and C46

v in the Schoenflies

notation. The point group symmetry is 6mm in the Hermann–Mauguin notation

and C6v in the Schoenflies notation. The Wz structure consists of two interpenetrating

hexagonal close-packed (hcp) sublattices, each with one type of atom, offset along

the c-axis by 5/8 of the cell height (5c/8). The wurtzite and zinc blende structures are

somewhat similar and yet different. In both cases, each group III atom is coordinated

by four nitrogen atoms. Conversely, each nitrogen atom is coordinated by four group

III atoms. The main difference between these two structures lies in the stacking

sequence of closest packed diatomic planes. The Wz structure consists of alternating

biatomic close-packed (0 0 0 1) planes of Ga and N pairs, thus the stacking sequence of

the (0 0 0 1) plane is AaBbAa in the (0 0 0 1) direction.

Although the main interest is in Wz GaN as opposed to zinc blende GaN, a

description of stacking sequence of both GaN polytypes with the accepted Ramsdel

notation is warranted, so is the stacking order of SiC polytypes that are relevant to

GaN because they are used for substrates in GaN epitaxy. Therefore, a generic

description of stacking in Wz semiconductors is given below. A comprehensive

description of the tetrahedrally coordinated structures is imperative for a clear picture

of nitride semiconductors, particularly the extended defects that are discussed in

detail in Chapter 4. The bonds describe a tetrahedron denoted by T, which has one

atom species at each of the three corners and the other atom species in its center [8,9].

The basal plane of this structure is defined by one face of the tetrahedron and the

bond perpendicular to this plane defines the c-axis. A rotation of 180 around the caxis

produces a twin variant denoted by T0 as shown in Figure 1.1a (left).

The two variants (twins T and T0) are related to one another by mirror symmetry

about one of the {1 1 0 0} m-planes. A tetrahedron can occupy one of the three

possible positions in the basal plane. The representation of the tetrahedrally

coordinated materials in the Ramsdel notation is shown in Figure 1.1a for two

possible tetrahedra, one is the mirror image twin of the other with respect to the

(1 1 0 0) m-plane. The single bonds are on the (1 1 2 0) plane, called the a-plane. The

layers of the tetrahedra can then be denoted by T1, T2, T3, and by T01

; T02

; T03

for its

twin. An example of T1; T0; T1; T03

stacking order representing 2H ordering as in

wurtzitic GaN is shown in Figure 1.1b. The structure of nitride semiconductors and

most relevant polytypes of SiC can be completely described by a combinatorial

stacking of the aforementioned six tetrahedra layers. Naturally, not all the stacking

sequences must obey the following two rules to keep a corner sharing structure, as

such not all stacking orders are allowed:

(i) A tetrahedron T can be followed by another one of the same kind with the

following subscript: T1T2T3, and inversely for the twin variant: T03

T02

T01

.

(ii) A tetrahedron T1 must be followed by the twin variant of the preceding subscript:

T1T03

, and inversely for its twin variant: T01

T2.

In the Ramsdel notation, the stacking order for the wurtzite structure corresponding

to various polytypes can be denoted as

. T1T03

or T2T01

or T3T02

for the 2H polytype, which is also applicable to Wz nitride

semiconductors;

. T1T2T01

T03

for the 4H polytype;

. T1T2T3T02

T01

T03

for the 6H polytype;

. T1T2T3 or T03

T02

T01

for the 3C polytype.

The 3C, 4H, and 6H stacking sequences as well as 2H sequence on 6H sequence are

discussed in Chapter 3.

Recall that GaN crystallizes in the cubic structure (zinc blende or sphalerite, the

latter being the correct term and the former being the one used universally) or in the

more stable hexagonal structure (wurtzite). The anions (N3) form an hcp structure

in which the cations (Ga3þ) occupy half of the tetrahedral sites. The structure of a unit

cell of GaN projected along [0 0 0 1] is depicted schematically in Figure 1.2. The open

symbols represent g sites that are occupied by nitrogen atoms; theGa atoms are in the

tetrahedral sites, b. These latter sites can either be at heights (3/8)c above (b1) or below

(b2) N site, depending on the crystal polarity.

A stick-and-ball representation of Ga-polarity and N-polarity Wz structure is

depicted in Figure 1.3. The Wz and zinc blende structures differ only in the bond

angle of the second nearest neighbor (Figure 1.4). As clearly shown, the stacking

order of theWz along the [0 0 0 1] c-direction is AaBb, meaning a mirror image but no

in-plane rotation with the bond angles. In the zinc blende structure along the [1 1 1]

direction, there is a 60 rotation that causes a stacking order of AaBbCc. The point

with regard to rotation is illustrated in Figure 1.4b. The nomenclature for various

commonly used planes of hexagonal semiconductors in two- and three-dimensional

versions is presented in Figures 1.5 and 1.6. The Wz group III nitrides lack an

inversion plane perpendicular to the c-axis; thus, nitride surfaces have either a group

III element (Al, Ga, or In) polarity (referred to as Ga-polarity) with a designation of

(0 0 0 1) or (0 0 0 1)A plane or a N-polarity with a designation of (0 0 0 1) or (0 0 0 1)B

plane.We will use the former notations for each. The distinction between these two

directions is essential in nitrides because of their implications for the polarity of the

polarization charge. Three surfaces and directions are of special importance in

nitrides, which are (0 0 0 1) c-, (1 1 2 0) a-, and (1 1 0 0) m-planes and the directions

associated with them, h00 01i, h1120i, and h1100i as shown in Figure 1.7. The

(0 0 0 1), or the basal plane, is the most commonly used surface for growth. The other

two are important in that they represent the primary directions employed in

reflection high-energy electron diffraction (RHEED) observations in molecular beam

epitaxial growth, apart from being perpendicular to one another. They also represent

the direction of stripes employed in the epitaxial lateral overgrowth (ELO), details of

which are discussed in Section 3.5.5.2.

The cohesive energy per bond in the wurtzite form is 2.88 eV (63.5 kcal mol1),

2.2 eV (48.5 kcal mol1), and 1.93 eV (42.5 kcal mol1) for AlN, GaN, and InN,

respectively [10]. The calculated energy difference DEW-ZB between wurtzite

and zinc blende lattice is small [11]: DEW-ZB¼18.41 meV/atom for AlN,

DEW-ZB¼11.44 meV/atom for InN, and DEW-ZB¼9.88 meV/atom for GaN.

Wurtzite form is energetically preferable for all three nitrides compared to zinc

blende, although the energy difference is small.

TheWzstructure can be represented by lattice parameters a in the basal plane and c

in the perpendicular direction, and the internal parameter u, as shown in Figure 1.8.

The u parameter is defined as the anion–cation bond length (also the nearest

neighbor distance) divided by the c lattice parameter. The c parameter depicts the

unit cell height. The wurtzite structure is a hexagonal close-packed lattice, comprising

vertically oriented X–N units at the lattice sites. The basal plane lattice

parameter (the edge length of the basal plane hexagon) is universally depicted by a

and the axial lattice parameter, perpendicular to the basal plane, is universally

described by c. The interatomic distance in the basic unit is described by the

internal parameter u. In an ideal wurtzite structure represented by four touching

hard spheres, the values of the axial ratio and the internal parameter are

approximation (GGA) and local density approximation (LDA). The experimental data

are from Leszczynski et al. [15].

In all Wz III nitrides, experimentally observed c/a ratios are smaller than ideal and

it has been postulated that not being so would lead to the zinc blende phase [16]. There

are two avenues that can lead to a deviation from ideal: changing the c/a ratio or

changing the u value. It should be pointed out that a strong correlation exists between

the c/a ratio and the u parameter so that when c/a decreases, the u parameter

increases in a manner to keep the four tetrahedral distances nearly constant through a

distortion of tetrahedral angles. For equal bond length to prevail, the following

relation must hold:

Table 1.1 Structural parameters for GaN reported by Bechstedt,

Großner, and Furthm€uller (BGF) [13] and by Wei and Zunger

(WZ) [14] using the local density approximation (LDA).

The nearest neighbor bond length along the c-direction (expressed as b in

Figure 1.8) and off c-axis (expressed as b1 in Figure 1.8) can be calculated as

The bond angles, a and b, are given by [17]

Table 1.2 tabulates the calculated as well as experimentally observed structural

parameters discussed above, inclusive of the lattice parameters, the nearest and

second nearest neighbor distances, and the bond angles for three end binaries, GaN,

AlN, and InN. The distances are in terms of Å.

The lattice parameters are commonly measured at room temperature by X-ray

diffraction (XRD), which happens to be the most accurate one, using the Bragg law. In

ternary compounds, the technique is also used for determining the composition;

however, strain and relevant issues must be accounted for as the samples are in the

Table 1.2 Calculated (for ideal crystal) and experimentally

observed structural parameters for GaN, AlN, and InN [17].

form of epitaxial layers on foreign substrates. The accuracy of X-ray diffraction and

less than accurate knowledge of the elastic parameters together allow determination of

the composition to only within about 1% molar fraction. In addition to composition,

the lattice parameter can be affected by free charge, impurities, stress (strain), and

temperature [18]. Because the c/a ratio correlates with the difference of the electronegativities

ofthetwoconstituents,componentswiththe greatestdifferencesshowthe

largest departure from the ideal c/a ratio [19]. ForGaN, the c/a ratio and the value of u

are measured as 1.627 (1.634 in Ref. [17]) and 0.377, respectively, which are close to the

ideal values [20]. AlN deviates significantly from the ideal parameters: c/a¼1.601

(1.606 in Ref. [17])andu¼0.382.Althoughthe data forInNare notas reliable, values of

u¼0.379 and c/a¼1.601 have been reported [17]. Inhomogeneities, strain, partial

relaxation of strain, and high concentration of structural defects may distort the lattice

constants from their intrinsic values and cause a wide dispersion among the reported

values. Table 1.3 lists a comparison of measured and calculated lattice parameters

reported for AlN,GaN, and InN crystallized in the wurtzite structure in more detail in

terms of the specifics of the sample used for measurements and complements. The

dispersion is even a greater concern in ternary and quaternaries, as compositional

inhomogeneities, in addition to the aforementioned issues, cause an additional

dispersion. The particulars of the ternaries are discussed in Section 1.5.

The wurtzite polytypes of GaN, AlN, and InN form a continuous alloy system

whose direct bandgaps range, according to data that adorned the literature for years,

from 1.9 eV for InN, to 3.42 eV for GaN, and to 6.2 eV for AlN. A revisit of the InN

bandgap indicates it to be about 0.78 eV [30] and the same for AlN is about 6 eV in

which case the energy range covered would be about 0.7–6 eV. Thus, the III–V

nitrides could potentially be fabricated into optical devices, which are active at

wavelengths ranging from the red well to the ultraviolet. The bandgaps of nitrides,

Table 1.3 Measured and calculated lattice constants of wurtzite AlN, GaN, and InN.

substrates commonly used for nitrides, and other conventional semiconductors are

shown in Figure 1.9 with respect to their lattice constants.

All III nitrides have partially covalent and partially ionic bonds. The concept of

fractional ionic character (FIC) is useful in interpreting many physical phenomena in

the crystals [31,32]. The FIC may be defined for a binary compound AB as

FIC ¼ jQA QBj=jQA þQBj, where QA and QB are effective charges on atoms A

and B. The FIC values range from zero for a covalent compound (each atom has four

electrons) to 1 for an ionic compound (all eight electrons belong to the anion).

Figure 1.10 displays the charge distribution along the AB bond for all three compounds.

The arrow along the bond charge indicates the atomic boundaries in the

crystals that are not always at the minimumof the line charge along the bond AB. This

should be expected taking into account the partial covalent bond of the compounds,

because only in the ionic crystals, the atomic boundary is clearly defined.Table 1.4 lists

the calculated effective radii, rIII and rN, the effective charges, and FIC for AlN, GaN,

and InN.Theionicity ofAlNis high. Thismayexplain the difficulties withAlNdoping.

It is well known that only covalent semiconductors or semiconductors with a large

covalent component can form hydrogen-like shallow levels in the bandgap by

substitution of a host atom with a neighbor with one more or one less electron. GaN

and InN have a smaller than AlN but nearly equal ionicity.GaN was doped both p- and

n-type. Thus, one can expect that InN can also be doped n- and p-type. To date, only

n-type InN has been obtained because of high volatility of nitrogen and easiness of

nitrogen vacancy formation that acts as a donor in this compound.

The III-nitrides are commonly grown on mismatched substrates because of the lack

of suitable native substrates. Thus, the epitaxial layers are strained during cool down, if

they are sufficiently thick for themto relax at the growth temperature. The mechanical

forces related to strain dramatically change the band structure of the epitaxial layers.

The pressure dependence of the bandgap energy Eg can be expressed as Eg¼Eg(0)

gPþdP2, where Eg(0) is the bandgap of stress-free semiconductor, g and d are the

Table 1.4 Calculated ionic radii (Å), effective charges (electrons),

and the fractional ionic character (FIC) for III nitrides [32].

and1.8 · 105, respectively [33,34].The bandgap is in termsof eVand the pressure is

in terms of kbar. The pressure dependence has, in general, a sublinear character. The

variation of the GaN energy gap with pressure, both theoretical [35] and experimental

[6], is shown in Figure 1.11. The calculated pressure coefficients for III nitrides are

given in Table 1.5.

Parameters associated with mechanical properties of GaN in wurtzitic phase are

tabulated in Tables 1.6 and 1.7, the latter dealing with the sound wave velocity. The

same parameters for the zinc blende phase ofGaN are tabulated in Tables 1.8 and 1.9.

Table 1.5 Calculated pressure coefficients for III nitrides including

wurtzitic, zinc blende, and rock salt phases (g in units of

meV GPa1 and d in units of meV GPa2) [35].

Table 1.6 Parameters related to mechanical properties of wurtzitic GaN (in part after Ref. [36]).

Table 1.7 Wave propagation properties in wurtzitic GaN [36].

Table 1.9 Wave propagation properties in zinc blende GaN (after Ref. [36]).

Table 1.11 Parameters related to electrical and optical properties

of Wz GaN (in part after Refs [36,44]).

Table 1.11 (Continued)

Table 1.11 (Continued)

The phonon energies are discussed later on in Table 1.27 in detail and effective

masses are discussed in Chapter 2. More details of effective masses can be found in

Table 2.9.

More should be said about the dielectric constants. Electromagnetic theory indicates

that for any longitudinal electromagnetic wave to propagate, the dielectric function

e(o) must vanish. Doing so leads to [66]

Likewise, Equation 1.5 can be expanded in the direction perpendicular to the c-axis or

in the basal plane or the (x, y) plane, the z-direction (in k-space between theGpoint and

M(1/2, 0, 0) or K (1/3, 1/3, 0) points), the low- and high-frequency dielectric functions

are related each with the help of A1(TO) and E1(LO) phonons through

where and // indicate in the basal plane and along the c-direction, respectively.

Table 1.12 Parameters related to electrical and optical properties

of zinc blende GaN (in part after Refs [36,44]).

which will be used to determine the optical frequency dielectric constant from the

knowledge of A1(LO) and A1(TO) phonon frequencies along the c-direction and

E1(LO) and E1(TO) in the basal plane. This relationship is used very often.

Parameters related to the energy bandgap, carrier mass, and mechanical properties

of AlN have been determined [71–76]. Extensive data on all the binary and ternary

band structure parameters can be found in Chapter 2. For example, additional

parameters on the critical point energies for Wz GaN, AlN, and InN are given in

Tables 2.1–2.3, respectively. Tables 2.4–2.6 list the critical point energies for ZB GaN,

AlN, and InN, respectively. Effective masses and other band parameters for Wz GaN

are listed in Table 2.9. Table 2.10 tabulates the Luttinger band parameters for ZBGaN.

Table 2.14 lists the effective band parameters for Wz AlN, whereas the effective

masses and band parameters for Wz AlN are tabulated in Table 2.15. Luttinger

parameters for ZB AlN are listed in Table 2.16. The band parameters and effective

masses for Wz InN are tabulated in Tables 2.19 and 1.20, respectively.

Returning to the content of this chapter, parameters associated with the mechanical

properties of AlN in wurtzitic and zinc blende phases are tabulated in Tables 1.13

and 1.14, respectively. The parameters related to the sound wave velocity in wurtzitic

AlN are listed in Table 1.15.

Table 1.13 Parameters related to mechanical properties of wurtzitic AlN (in part after Ref. [36]).

Parameters associated with thermal properties of wurtzitic AlN are tabulated in

Table 1.16.

Parameters associated with electrical and optical properties of wurtzitic AlN are

tabulated in Table 1.17. The same range of parameters associated with the zinc blende

Table 1.14 Parameters related to mechanical properties of zinc blende AlN.

Table 1.15 Acoustic wave propagation properties in wurtzite AlN [36].

Table 1.16 Parameters related to thermal properties of wurtzitic AlN (in part after Ref. [36]).the surface of c-Si [95]. It is worth noting that, owing to the extremely weak Raman

signal usually presented by AlN films, it is not uncommon to ascribe some of the

features erroneously to AlN [96].

Conduction band first- and second-order pressure derivatives [36]:

where P is pressure in kbar.

Table 1.17 Parameters related to optical and electrical properties

of wurtzitic AlN [97–102] (in part from Ref. [36]).

Table 1.17 (Continued)

Phase transition from the wurtzite phase to the rock salt structure (space

group O5

h; lattice parameter 4.04 Å) takes place at the pressure of 17 GPa (173 kbar)

[109,110].

Parameters associated with the electrical and optical properties of zinc blende AlN

are listed in Table 1.18. For details regarding the Luttinger parameters for the valence

band in zinc blende AlN, refer to Table 2.16. Parameters associated with the

mechanical properties of wurtzitic InN are tabulated in Table 1.19.

For wurtzite crystal structure, the surfaces of equal energy in G valley should be

ellipsoids, but effective masses in the z-direction and perpendicular directions are

estimated to be approximately the same.

Table 1.18 Parameters related to optical and electrical properties of zinc blende AlN.

The parameters associated with thermal properties of wurtzitic InN are tabulated

in Table 1.20.

The specific heat Cp of InN at constant pressure for 298 K<T<1273 K [51] is

Cp¼38.1þ1.21 · 102T (J mol1 K1). Refer to Table 1.31 for a detailed treatment of

mechanical properties of InN.

The parameters associated with electrical and optical properties of wurtzitic InN

are tabulated in Table 1.21.

Available parameters associated with the mechanical properties of zinc blende

InN, primarily calculated, are tabulated in Table 1.22. Other parameters dealing with

electrical and optical properties of zinc blende InN, primarily calculated, are listed in

Table 1.23.

Table 1.19 Parameters related to mechanical properties of wurtzitic InN (in part after Ref. [36]).