10. Appendix

614 Appendix B
⎛ ux
⎞
⎜ x ⎟
⎜ uy
⎟
⎜ ⎟
⎜ y ⎟
⎜ ⎟
⎜ uz ⎟
⎜ ⎟
⎜ z ⎟
e ⎜ ⎟
⎜ uy uz ⎟
⎜ ⎟
⎜
z x ⎟
⎜ ux uz ⎟
⎜
⎟
⎜ z x ⎟
⎝ ux uy ⎠
y x
Once the strain tensor and the electromechanical tensor are known the polarization
P induced by the strain can be calculated from the definition:
Pi (e)ijej while the electric field E is related to P by: E ( 4appleP
 ),  being
the dielectric constant.
To simply the notation we can assume that the z-axis is parallel to the
c-axis of the wurtzite structure and the y-axis lies in the z-q plane so that
q (0, q sin ı, q cos ı). With this notation we obtain the strain tensor as:
⎛
⎞
0
⎜ iuyq sin ı ⎟
⎜
⎟
⎜ iuzq cos ı ⎟
e ⎜
⎟
⎜ iq(uz sin ı uy cos ı) ⎟
⎝
iuxq cos ı
⎠
iuxq sin ı
From this strain tensor we obtain the polarization:
P e15iqux cos ı, e15iq(uz sin ı uy cos ı), iq(e31uy sin ı e33uz cos ı)
and hence the electric field.
In particular we are interested in the longitudinal component of this piezoelectric
field since this will couple most strongly to an electron. The longitudinal
electric field El is given by the projection of E along q:
E · q
El
q
4appleiq
e15 sin ı(uz sin ı uy cos ı) e31uy cos ı sin ı e33uz cos
Â
2 ı
From this longitudinal field we can define a scalar potential V such that
Hpiezoelectric |e|V
4apple|e|
e15 sin ı(uz sin ı uy cos ı) e31uy cos ı sin ı e33uz cos
Â
2 ı
Note that a factor of 4apple may be missing in some earlier editions of the book.