b) To determine the linearly independent and non-zero elements of the stress tensor we shall start by examining the stress tensor for the zincblende crystal and ask what is the difference in the symmetry between zincblende and wurtzite. We notice that the wurtzite properties along the c-axis (equivalent to say the z-axis in zincblende) is different from those along the x- and y-axes. This suggest that while all the x, y and z directions are equivalent in the zinblende crystal this is only true for the x and y direction in the wurtzite structure. This leads us to conclude that the coefficients C11=C22 but both are not equal to C33. Similarly C13=C23 but they are not equal to C12. Similarly C1313 =C2323=C44 but not necessarily equal to C1212=C66. To obtain the relation between C1212 and C1111-C2222 we have to consider some symmetry operation that transforms x into y such as a 3-fold rotation. If we rotate the coordinate axis by 120 degrees in the counterclockwise direction about the z-axis we will transform the axis (x,y,z) into (x',y',z') with transformation matrix given by:
or x'i=aijxj.
The effect of this operation on C1212 is given (via the definition
of a tensor):
C1212'=a1k a 2l a 1m a
2n Cklmn with summation over repeated indices implied.
In principle the summation has to be performed over all values of k,l,m
and n. This operation is greatly simplified by the observation that Cklmn
is zero for many combination of k,l,m and n. The non-zero summands are:
C1212, C2222, C2121, C1111,
C1122, C2211, C1221, C2112,
and C1212. In terms of the contracted notation we obtain:
C66'= C66(a112 a 222+ a 122 a 212+2 a 11 a 22 a 12 a 21)+ C11(a 122 a 222+ a 112 a 212)+ C12(2 a 11 a 22 a 12 a 21)
Substituting in the transformation coefficients a ij etc:
C66'= C66[(1/4)x(1/4)+(3/4)x(3/4)+2(1/4)(-3/4)]+
C11[(3/4)x(1/4)+(1/4)x(3/4)]+ C12[(-1/4)x(3/4)+(1/4)(-3/4)]=
C66(4/16)+ C11 (6/16)+C12(-6/16).
Since the crystal is invariant under this operation we expect C66'=C66. Finally by substituting back into the equation C66' we obtain: C66=(1/2)( C11- C12).
(c) We shall adopt an coordinate system in which the z-axis is parallel to the c-axis of the wurtzite crystal. The subscripts 1,2,3 will correspond to x,y and z.
The Newton's Equation of motion for a small volume xyz of the wurtzite crystal along the x-axis can be written as:
or
where is the density, u=(u1,u2,u3) is the displacement vector, t is the time and Xij is the stress tensor. There are two other similar equations for u2 and u3. Using the results of part (b) the stress tensor can be expressed in terms of the strain tensor eij (or ei in the contracted notation) and the stiffness tensor Cijkl (or as Ckl in the contracted notation) as:
Substituting this stress tensor elements into the equation of motion we obtain the following equation:
with two similar diffferential equations for u2 and u3.
Copyrighted by Peter Y. Yu, 1997. You are allowed to download this material for your personal use. Reproduction for commercial sale prohibited.