**1.2.2 Mechanical Properties of GaN**

GaNhas a molecular weight of 83.7267 g mol1 in the hexagonalwurtzite structure.The

lattice constant of early samples of GaN showed a dependence on growth conditions,

impurity concentration, and film stoichiometry [151]. These observations were

attributed to a high concentration of interstitial and bulk extended defects. A case in

point is that the lattice constants ofGaNgrown with higher growth rates were found to

be larger. When doped heavily with Zn [152] and Mg [153], a lattice expansion occurs

because athighconcentrations thegroupIIelementbegins tooccupythe lattice sites of

the much smaller nitrogen atom. At room temperature, the lattice parameters ofGaN

platelets [18] prepared under high pressure at high temperatures with an electron

concentration of 5 · 1019 cm3 are a¼3.18900.0003Å and c¼5.18640.0001 Å.

The freestanding GaN with electron concentration of about 1016 cm3, originally

grown on sapphire (0 0 0 1) by hydride vapor phase epitaxy (HVPE) followed by liftoff,

has lattice constants of a¼3.20560.0002Å and c¼5.19490.0002Å. For GaN

powder, a and c values are in the range of 3.1893–3.190 and 5.1851–5.190 Å,

respectively. Experimentally observed c/a ratio for GaN is 1.627, which compares

well with 1.633 for the ideal case, and the u parameter calculated using Equation 1.1 is

0.367, which is very close to the ideal value of 0.375.

For more established semiconductors with the extended defect concentration from

low to very low, such as Si, GaAs, and so on, the effect of doping and free electrons on

the lattice parameter has been investigated rather thoroughly. In bulk GaN grown by

the high-pressure technique, the lattice expansion by donors with their associated free

electrons has been investigated [18]. However, large concentration of defects and

strain, which could be inhomogeneous, rendered the studies of this kind less reliable

in GaN layers. In spite of this, the effect ofMg doping on the lattice parameter in thin

films of GaN has been investigated. Lattice parameters as large as 3.220–5.200Å for a

and c values, respectively, albeit not in all samples with similar hole concentrations,

have been reported [154]. For GaN bulk crystals grown with high-pressure techniques

and heavily doped (a small percentage) with Mg, the a and c lattice parameters were

measured to be 3.2822–5.3602Å [155]. Suggestions have been made that the c

parameter of implanted GaN layers increases after implantation and languishes after

annealing [156]. However, the a parameter could not be precisely measured because

sharp off-normal diffraction peaks are needed to determine this parameter accurately.

For the zinc blende polytype, the calculated lattice constant, based on the measured

GaNbond distance inWzGaN, is a¼4.503 Å. The measured value for this polytype

varies between 4.49 and 4.55 Å, while that in Ref. [18] is 4.511 Å, indicating that the

calculated result lies within the acceptable limits [157]. A high-pressure phase

transition from the Wz to the rock salt structure has been predicted and observed

experimentally. The transition point is 50 GPa and the experimental lattice constant

in the rock salt phase is a0¼4.22 Å. This is slightly different from the theoretical

result of a0¼4.098 Å obtained from first-principles nonlocal pseudopotential

calculations [158].

Tables 1.6 and 1.10 compile some of the known properties of Wz GaN. Parameters

associated with electrical and optical properties ofWzGaNare tabulated in Table 1.11.

The same parameters associated with the zinc blende phase of GaN are tabulated in

Table 1.12.

The bulk modulus of Wz GaN, which is the inverse of compressibility, is an

important material parameter. Various forms of X-ray diffraction with the sample

being under pressure can be used to determine the lattice parameters. Once the

lattice parameters are determined as a function of pressure, the pressure dependence

of the unit cell volume can be obtained and fitted with an equation of state (EOS), such

as the Murnaghans EOS [159], and based on the assumption that the bulk modulus

has a linear dependence on the pressure:

where B0 and V0 represent the bulk modulus and the unit volume at ambient pressure,

respectively, and B0 the derivative of B0 versus pressure. X-ray diffraction leads to the

determination of the isothermal bulk modulus, whereas the Brillouin scattering

leads to the adiabatic one. Nevertheless, in solids other than molecular solids, there is

no measurable difference between the two thermodynamic quantities [160].

The bulk modulus (B) of Wz GaN has been calculated from first principles [161]

and the first-principle orthogonalized linear combination of atomic orbitals (LCAO)

method [158], leading to the values of 195 and 203 GPa, respectively. Another

estimate for B is 190 GPa [158]. These figures compare well with the value of

194.6 GPa estimated from the elastic stiffness coefficient [79] and a measured value

for 245 GPa [6].

The bulk modulus is related to the elastic constants through

and the range of bulk modulus values so determined is from about 173 to

245 GPa [160].

Using the room-temperature elastic constants of single-crystal GaN calculated by

Polian et al. [38] yields an adiabatic bulk modulus, both Voigt and Reuss averages, of

210 GPa [91].

However, the results obtained later point to a Poissons ratio of more near 0.2 as

tabulated in Table 1.6 and depend on crystalline direction. The Poissons ratio for the

ZB case can be calculated from the elastic coefficients for that polytype as n or

s0¼(C12/C11þC12) leading to values of about 0.352 as tabulated in Table 1.8. The

Poissons ratio varies along different crystalline directions as tabulated in Table 1.13

for AlN. It should be noted that there is still some spread in the reported values of

elastic stiffness coefficients, as discussed in detail in the polarization sections of

Section 2.12. More importantly, Kisielowski et al. [39] pointed out that expression

Chetverikova et al. [163] measured the Youngs modulus and Poissons ratio of their

GaN films. From the elastic stiffness coefficients, Youngs modulus Eh0001i is

estimated to be 150 GPa [157,162]. Sherwin and Drummond [164] predicted the

elastic properties of ZB GaN on grounds of values for those Wz GaN samples

reported by Savastenko and Sheleg [162]. The elastic stiffness coefficients and the

bulk modulus are compiled in Table 1.24. Considering the wide spread in the

reported data more commonly used figures are also shown.

Wagner and Bechstedt [178] calculated the elastic coefficients of Wz GaN using a

pseudopotential plane wave method and pointed out the discrepancies among the

results from different calculations and measurements tabulated in Table 1.24. It is

argued that reliable values produce 2C13/C33=0.50–0.56 and n=0.20–0.21 [178].

The agreement between ab initio calculations [42,178] and some measure

Table 1.24 Experimental and calculated elastic coefficients (Cii),

bulk modulus (B) and its pressure derivative (dB/dP), and

Youngs modulus (E or Y0) and (in GPa) of Wz GaN and ZB GaN

(in part from Ref. [160]).

[28,169–171] suffer fromdeviations in one or more of the values of elastic

constants. The results fromSavastenko and Sheleg [162] show excessive deviation for

all the elastic constants and, therefore, should be avoided completely. The results

from surface acoustic wave measurements of Deger et al. [165] on epitaxial epilayers

have been corrected for piezoelectric stiffening and, therefore, are among the most

reliable.

The vibrational properties of nitrides can best be described within the realm of

mechanical properties. These vibrations actually serve to polarize the unit cell [172].

Phonons can be discussed under mechanical and optical properties.Here an arbitrary

decision has been made to lump them with the mechanical properties of the crystal.

Using GaN as the default, a succinct discussion of vibrational modes, some of which

are activeRamanmodes,someare active in infrared (IR)measurements,andsomeare

optically inactive called the silent modes, is provided [173]. Vibrational modes, which

go to the heart of the mechanical properties, are very sensitive to crystalline defects,

strain, and dopant in that the phonon mode frequencies and their frequency broadening

can be used to glean very crucial information about the semiconductor. The

method can also be applied to heterostructures and strained systems. Electronic

Raman measurements can be performed to study processes such as electron–phonon

interaction in the CW or time-resolved schemes. Time-resolved Raman measurements

as applied to hot electron and phonon processes under high electric fields have

important implication regarding carrier velocities. A case in point regarding GaN is

treated in this context in Volume 3, Chapter 3.

The wurtzite crystal structure has the C46

v symmetry and the group theory

predicts the existence of the zone center optical modes A1, 2B1, E1, and 2E2. In

a more simplified manner, one can consider that the stacking order of the Wz

polytype is AaBb while that for the ZB variety is AaBbCc. In addition, the unit cell

length of the cubic structure along [1 1 1] is equal to the width of one unit bilayer,

whereas that for the hexagonal structure along [0 0 0 1] is twice that amount.

Consequently, the phonon dispersion of the hexagonal structure along [0 0 0 1]

(G ! A in the Brillouin zone) is approximated by folding the phonon dispersion for

the ZB structure along the [1 1 1] (G ! L) direction [174], as shown in Figure 1.12.

Doing so reduces the TO phonon mode at the L point of the Brillouin zone in the

zinc blende structure to the E2 mode at the G point of the Brillouin zone in the

hexagonal structure. This vibrational mode is denoted as EH2

with superscript H

depicting the higher frequency branch of the E2 phonon mode. As indicated in the

figure there is another E2 mode at a lower frequency labeled as EL2

. This has its

genesis in zone folding of the transverse acoustic (TA) mode in the zinc blende

structure. It should be noted that in the hexagonal structure there is anisotropy in

the macroscopic electric field induced by polar phonons. As a result, both the TO

and LO modes split into the axial (or A1) and planar (or E1) modes where atomic

displacement occurs along the c-axis or perpendicular to the c-axis, respectively.

This splitting is not shown in Figure 1.12 as it is very small, just a few meV, near

zone center; phonon dispersion curves forGaNincluding the splitting of the A1 and

E1 modes can be found in Volume 3, Figure 3.84. As discussed below, in the context of hexagonal structures, group theory predicts

eight sets of phonon normal modes at the G point, namely 2A1þ2E1þ2B1þ2E2.

Among them, one set of A1 and E1 modes are acoustic, while the remaining six

modes, namely A1þE1þ2B1þ2E2, are optical modes. As shown in Figure 1.12, one

A1 and one B1 mode (BH1

) derive from a singly degenerate LO phonon branch of the

zinc blende system by zone folding, whereas one E1 and one E2 mode (EH2

) derive

from a doubly degenerate TO mode in the cubic system.

The first-order phonon Raman scattering is due to phonons near the G point zone

center, that is, with wave vector k

0, because of the momentum conservation rule in

the light scattering process. Raman measurements typically are employed to probe

the vibrational properties of semiconductors. When performed along the direction

perpendicular to the c-axis or the (0 0 0 1) plane, the nomenclature used to describe

this configuration is depicted as ZðXY;XYÞZ.Here, following Portos notation [175]

A(B, C)D is used to describe the Raman geometry and polarization, where A and D

represent the wave vector direction of the incoming and scattered light, respectively,

whereas B and C represent the polarization of the incoming and scattered light. In

Raman scattering, all the above-mentioned modes, with the exception of B1 modes,

are optically active. Because of their polar nature, the A1 and E1 modes split into

longitudinal optical (A1-LO and E1-LO) meaning beating along the c-axis, and

transverse optical (A1-TO and E1-TO), meaning beating along the basal plane. To

reiterate, the A1 and B1 modes give atomic displacements along the c-axis, while the

others, E1 and E2, give atomic displacements perpendicular to the c-axis, meaning on

the basal plane.Here, the A1 and E1 modes are both Raman and IR active whereas the

two E2 modes are only Raman active and the two B1 modes are neither Raman nor IR

mode–Raman configuration relationship are tabulated in Table 1.25. Shown in

Figure 1.13 are the modes in the Raman backscattered geometries in relation to

hexagonal crystalline orientation that can be used to sense the various phonon modes

indicated.

The acoustic modes, which are simple translational modes, and the optical modes

for wurtzite symmetry are shown in Figure 1.14. The calculated phonon dispersion

curves [57] for GaN are shown in Figure 1.15. There is another way to describe the

number of vibrational modes in zinc blende and wurtzitic structures, which is again

based on symmetry arguments. In the wurtzite case [66], the number of atoms per

unit cell s¼4, and there are total of 12 modes, the details of which are tabulated in

Table 1.26. This table also holds for the zinc blende polytypes with s¼2. This implies

a total of six modes in zinc blende as opposed to 12 in wurtzite, three of which are

acoustical (1 LA and 2 TA) and the other three are optical (1 LO and 2 TO) branches.

These phonon modes for a wurtzite symmetry, specifically the values for wurtzite

GaN, are listed in Table 1.27 obtained from Refs [56,157,176,177] along with those

obtained from first-principles pseudopotential calculations [161,178]. Also listed are

TO and LO optical phonon wave numbers of ZB GaN [25,179].

Table 1.25Raman measurement configuration needed to observe the phonon modes in hexagonal nitrides.

Table 1.26 Acoustic and optical phonon modes in a crystal withwurtzite symmetry such as GaN, AlN, and InN, where s represents

the number of atoms in the basis.

between the luminescence or Raman shifts and the corresponding biaxial stress are

seldom directly measured data. They are either obtained using elastic constants or are

constructed from deformation potentials, which have been obtained by means of

additional hydrostatic pressure coefficients. Owing to these varying procedures and

different sets of parameters used to extract the conversion coefficients from the raw

experimental data, discrepancies in the experimental reports of deformation potentials

are present.