10. Appendix

650 Appendix B
To determine the symmetry of the Raman-active phonons in wurtzite crystals,
we need to determine the irreducible representations of a vector in the
wurtzite structure. From Problem 3.7 we find that the character table for the
wurtzite structure becomes identical to that of point group C6v in the limit
when the phonon wave vector approaches zero. Assuming that one chooses
the z-axis to be parallel to the C6 axis of this point group or the c-axis of the
wurtzite structure (this axis will also be labeled as 3 in subsequent tensor subscripts,
while the x and y axes will be labeled as 1 and 2, respectively). Since
the largest dimension of irreducible representations in C6v is 2, the representation
of a vector in 3D (with components x, y and z) has to be reducible. One
way this representation can be reduced is to separate {z} from {x, y}. {z}
must belong to the A1 irreducible representation since it is invariant under all
the symmetry operations of C6v. The remaining components {x} and {y} form
a 2D irreducible representation. Whether this irreducible representation is of
symmetry E1 or E2 can be decided by applying to {x, y} the C2 operation (a
rotation by 180 ◦ about the z-axis): xy → xy. The character for this operation
is 2. Hence {x, y} belongs to the E1 irreducible representation. The above
result can be summarized as: the irreducible representations to which a vector
in the group C6v belongs are A1 and E1.
Based on this result we can predict that the zone-center phonons of symmetry
°1 and °5 in the wurtzite crystal (corresponding to A1 and E1 representations,
respectively, in C6v) are infrared-active. To obtain the symmetry of the
corresponding Raman active phonons we have to calculate the direct product:
(A1 ⊕ E1) ⊗ (A1 ⊕ E1). By inspection of the character table of C6v in Problem
3.7 it can be shown that: E1 ⊗ E1 A1 ⊕ A2 ⊕ E2. Thus one obtains the result:
(A1 ⊕ E1) ⊗ (A1 ⊕ E1) 2A1 ⊕ A2 ⊕ 2E1 ⊕ E2.
In summary, in systems with C6v point group symmetry the Raman-active
phonon modes must belong to the A1, A2, E1 or E2 irreducible representations.
Similarly, the Raman-active zone-center optical phonons in the wurtzite
crystal structure must belong to the °1, °2, °5, or°6 irreducible representations.
Thus, of all the 9 zone-center optical phonons in the wurtzite structure the
ones with °1 and °5 symmetries are both infrared and Raman active; the two
phonons with °6 symmetry are only Raman-active while the two phonons with
°3 symmetry are neither infrared nor Raman active (such modes are said to
be silent).
To obtain the Raman tensor for the Raman-active phonons we have to
derive, in principle, the form of the phonon displacement vector Qk and the
electric susceptibility tensor ¯ and then apply Eq. (7.37) to obtain the Raman
tensor Rij. In reality, we can choose any basis function in lieu of the phonon
displacement vectors, provided they belong to the same irreducible representation.
As, an example, let us consider the °1 optical phonon. As shown in Problem
3.15 the third order electromechanical tensor (em) in the wurtzite crystal
has only three linearly-independent and non-zero elements: (em)15, (em)31,
and (em)33. Since the third rank tensor (¯/Q) has the same symmetry prop-