Chapter 2 Theoretical background
2.1 Crystal structures and Lattice dynamics
2.1.1 Crystal structures [1]
Most of the group II-VI binary compound semiconductors crystallize in either cubic
zinc-blende or hexagonal wurtzite structure in which each anion is surrounded by four cations at the corners of a tetrahedron, and vice versa. This tetrahedral coordination is
typical of sp3 covalent bonding, but these materials also have a substantial ionic character.
ZnO is a II-VI compound semiconductor whose ionicity resides at the borderline between
covalent and ionic semiconductor. The crystal structures shared by ZnO are wurtzite (B4), zinc blende (B3), and rocksalt (B1), as schematically shown in Fig. 2-1. At
ambient conditions, the thermodynamically stable phase is wurtzite. The zinc-blende ZnO structure can be stabilized only by growing on cubic substrates, and the rocksalt
structure may be obtained at relatively high pressures.
Fig. 2-1 Stick and ball representation of ZnO crystal structures: (a) cubic rocksalt (B1), (b) cubic zinc blende (B3), and (c) hexagonal wurtzite (B4). The shaded gray and black spheres denote Zn and O atoms,
respectively.
The wurtzite structure has a hexagonal unit cell with two lattice parameters, a and c,
in the ratio of c/a= 83 =1.633
and belongs to the space group of C or P66v4 3mc. A schematic representation of the wurtzitic ZnO structure is shown in Fig. 2-2. The structure is composed of two interpenetrating hexagonal-close-packed (hcp) sublattices,
each of which consists of one type of atom displaced with respect to each other along the threefold c-axis by the amount of u = 3/8 = 0.375 in an ideal wurtzite structure. The
fractional coordinate, the u parameter, is defined as the length of the bond parallel to the c axis in unit of c. Each sublattice includes four atoms per unit cell and every atom of one
kind (group-II atom) is surrounded by four atoms of the other kind (group VI), or vice versa, which are coordinated at the edges of a tetrahedron.
Fig. 2-2 Schematic representation of a wurtzitic ZnO structure having lattice constants a in the basal plane and c in the basal direction; u parameter is expressed as the bond length or the nearest-neighbor distance b
divided by c, and α and β are the bond angles.
In a real ZnO crystal, the wurtzite structure deviates from the ideal arrangement, by changing the c/a ratio or the u value. It should be pointed out that a strong correlation
exists between the c/a ratio and the u parameter when the c/a ratio decreases. The u parameter increases in such a way that those four tetrahedral distances remain nearly
constant through a distortion of tetrahedral angles due to long-range polar interactions.
The lattice parameters of a semiconductor usually depend on the following factors: (i)
free-electron concentration acting via deformation potential of a conduction-band minimum occupied by these electrons; (ii) concentration of foreign atoms and defects and
their difference of ionic radii with respect to the substituted matrix ion; (iii) external strains (for example, those induced by substrate); and (iv) temperature. The lattice
parameters of any crystalline material are commonly and most accurately measured by high resolution x-ray diffraction (HRXRD). For the wurtzite ZnO, the lattice constants
at room temperature determined by various experimental measurements and theoretical calculations are in good agreement. The lattice constants mostly range from 3.2475 to
3.2501 Å for the a parameter and from 5.2042 to 5.2075 Å for the c parameter. The c/a ratio and the u parameter vary in a slightly wider range, from 1.593 to 1.6035 and from
0.383 to 0.3856, respectively. The deviation from that of the ideal wurtzite crystal is probably due to lattice stability and ionicity. It has been reported that free charge is the
dominant factor responsible for expanding the lattice proportional to the deformation potential of the conduction-band minimum and inversely proportional to the carrier
density and bulk modulus. The point defects such as zinc antisites, oxygen vacancies, and extended defects, such as threading dislocations, also increase the lattice constant,
albeit to a lesser extent in the heteroepitaxial layers.
2.1.2 Symmetry properties and phonon modes of inelastic cross sections [2, 3]
The symmetry properties of the scattering cross sections are determined by the
symmetry properties of second-order susceptibility for the excitation concerned. The spatial symmetry properties of the scattering medium lead to further connections between
the cross section measured in different experiments on the same sample. The Stokes cross section for different polarization of the incident and scattered light are often related
by the spatial symmetry, and the cross section is sometimes required to vanish for certain polarizations that depend on the nature of the excitation.
The spatial symmetry of the scattering medium is formally specified by its symmetry group, the group of all spatial transformations that leave the medium invariant.
Individual atoms and molecules have spatial symmetries characterized by a point group consisting of rotations and reflections that leave the atom or molecule invariant. The
atomic arrangements in a regular crystal lattice are characterized by a space group that contains translations in addition to rotations and reflections. The effects of the
translational invariance of a crystal are largely accounted for in the momentum conservation conditions, and the residual effects of the spatial symmetry derive from the
crystal point group that remains on removal of translations from the space group. There are 32 different crystal point groups. The effects of spatial symmetry are particularly
important for scattering by crystal samples, and the anisotropy of the cross section is generally different for the different crystal symmetries. We consider in the present
section the restrictions imposed on the cross section by the point symmetry of a crystal.
The spatial properties of the excitations of the scatter are described by irreducible representations of its symmetry group. Let ΓX be the irreducible representation appropriate to the excitation; we callΓX the excitation symmetry. In the microscopic theory with an initial state i of symmetry Γi and a final state f of symmetryΓf , the excitation symmetry is that of the operator f i , which projects the initial state onto
the final state. Thus
ΓX =Γf ×Γi*, (2-1)
where the asterisk denotes complex conjugation. The transformation properties of the incident and scattered light are described by the three-dimensional polar-vector
representation ΓPV of the point group considered, since the quantities incident field (EI),
Stokes field (ES), and Stokes polarization (PS), which characterize the light, are all polar
vectors.
The relation between Stokes polarization, excitation amplitude, and incident field
must be invariant under all the spatial transformations of the symmetry group of the scattering. This invariance condition (for detailed discussion, see Nye 1957 [4]) is
common to all equations that relate properties of a system with given spatial symmetry.
It has two main consequences for light scattering.
The first consequence is the existence of selection rules. In group theoretical language, only thoseΓX are allowed for which the direct product ΓX ×ΓPV includes the
polar vector representationΓPV (or an irreducible part of it) in its decomposition. An equivalent statement is that those ΓX that occur in the decomposition of ΓPV* ×ΓPV are
the allowed excitation symmetries. The scattering of light by all excitations whose symmetries do not satisfy this condition is a forbidden process.
The second consequence of the invariance condition is the imposition of restrictions on the components of the second-order susceptibility for those excitation symmetries that allowed in light scattering. For each allowed ΓX, some of the Cartesian components, i
and j, are required to have related values. Nye (1957) gives details of similar
determinations of the symmetry properties of a wide range of tensor quantities in various crystal symmetries.
Poulet and Mathieu [5] give the fullest account of the calculation of selection rules and symmetry properties of second-order susceptibilities for inelastic light scattering.
These calculations are not repeated here, but the main results for the 32 crystal point groups are set out in Ref. 5, which were divided into three parts, for crystals with biaxial,
uniaxial, and isotropic dielectric properties. GaN-, AlN- and InN-based materials are highly stable in the hexagonal wurtzite structure (uniaxial) although they can be grown in
the zinc blende phase and unintentional phase separation and coexistence may occur.
The wurtzite crystal structure belongs to the space group C and group theory predicts 6v4 zone-center optical modes are A1, 2B1, E1 and 2E2. The A1 and E1 modes and the two E2
modes are Raman active while the B modes are silent. The A and E modes are polar,
resulting in a splitting of the LO and TO modes (Hayes and Loudon, 1978 [6]). The Raman tensors for the wurtzite structure are as follows:
Here x, y or z in brackets after an irreducible representation indicates that the vibration is also infra-red active and has the direction of polarization indicated. Such vibrations
occur only in piezo-electric crystals (i.e. crystals with no center of inversion symmetry).
In crystals which do have a center of inversion symmetry, only even-parity vibrations,
whose representations have a subscript g, can be Raman active and only odd-parity (subscript u) vibrations can be infrared active. This fact leads to the important
complementary nature of infra-red absorption and Raman effects measurements.
Directly above each irreducible representation is a matrix, which gives the non-vanishing components of the Raman tensor, i.e., of αρσ,μorR . The different elements of the ρσμ
matrices are the nine components of the tensor obtained by allowing both ρ and σ to take
on the values x, y and z. Here x, y, and z are the crystal principal axes chosen to be identical with the principal axes x1, x 2 and x3 defined for all the crystal classes by Nye.
The component μ of the phonon polarization for the case of infrared-active vibrations is the quantity given in brackets after the irreducible representation symbol.
Fig. 2-3 Phonon dispersion curves for ZnO crystal of wurtzite structure. (after Calleja et al. [7])
The vibrational modes in ZnO wurtzite structures are given in Figure 2-3. At the Γ point of the Brillouin zone, it can be seen that the existence of the following optic phonon
modes: A1+2B1+E1+2E2; A1 and E1 modes are both Raman and infrared active; and B1
(low) and B1 (high) modes are silent. For the lattice vibrations with A1 and E1
symmetries, the atoms move parallel and perpendicular to the c axis, respectively. The low-frequency E2 mode is associated with the vibration of the heavy Zn sublattice, while
the high-frequency E2 mode involves only the oxygen atoms. The displacement vectors of the phonon normal modes are illustrated in Fig. 2-4. In the case of highly oriented
ZnO films, if the incident light is exactly normal to the surface, only A1 (LO) and E2
modes are observed, and the other modes are forbidden according to the Raman
selection rules. Table 2-1 gives a list of observed zonecenter optical-phonon wave numbers along with those calculated for wurtzite ZnO.
Fig. 2-4 Displacement vectors of the phonon modes in ZnO wurtzite structure. (after Jephcoat et al. [8])
Table 2-1 Phonon mode frequencies of wurtzite ZnO at the center of the Brillouin zone obtained from
infrared spectroscopic ellipsometry and Raman scattering measurements in comparison with theoretical
predictions [1].