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7.5 THE PIEZOELECTRIC EFFECT Certain types of crystal materials exhibit a behaviour such that under stress, an extra electric charge gathers on their surfaces. The effect is called 'piezoelectricity' and is a consequence of a non-compensated electric polarisation generated in the volume of the crystal. Piezoelectric behaviour is exhibited in quartz, Rochelle Salt and Tourmaline, as well as in cubic and hexagonal semiconductors such as InGaAs and GaN. The changes in the crystal lattice geometry due to stress act to separate the charges and break the local electrostatic neutrality in the crystal, a schematic diagram illustrating this effect for wurtzite GaN is given in Fig. 7.8. The uncompensated positive and negative charges induce a global polarisation vector, the properties of which are defined by the material and the nature of the applied stress.

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arithmetic } (15.6) with P,r,d := P,P,0 }

Figure 7.8 Schematic illustration of the charge separation in GaN grown along the [0001] direction which causes the appearance of an uncompensated polarisation. The left hand diagram shows the unstrained crystal lattice, and on the right the atomic positions when under strain.

(15.9) }

where ta,n-i is a constant obtained from the t-distribution with n 1 degrees of freedom (n is the number of simulations) and

If an arbitrary stress is acting on the crystal then the total induced polarisation (in matrix notation similar to that in Sec. 7.1) can be written as:

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O ^ P < Q A P = QxO + P P = PmodQ A 0 = P-=-Q P = PmodQ . Combining with (15.9), ( O ^ P < Q ) = (P = PmodQ) = (0 = P + Q) . The case that P = 0 is important. From (15.10), we get 0 = OmodQ . (15.11) (15.10)

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where dik is the piezoelectric modulus. The piezoelectric polarisation is a vector which is described by three components, i.e. P1 and P2 in the growth plane and P3 along the growth direction, which in turn define a 3 x 6 matrix of the piezoelectric moduli (d). As for the case of the elastic constants, the symmetry in the crystals reduces the number of independent elements of the d matrix. For cubic and hexagonal crystal geometries the number of independent moduli relies on the crystal class e.g. one modulus for InGaAs and three moduli for wurtzite GaN. So it is stress induced symmetry breaking in a crystal which leads to a piezoelectric polarisation, hence the underlying symmetry of the crystal is important. A good example of this is InGaAs. When grown on a GaAs substrate in the [111] direction it exhibits a piezoelectric behaviour. However, layers grown along the [001] direction are polarisation free, while those grown along the [110] direction exhibit a piezoelectric polarisation in the plane of the layers. In the following, the piezoelectric properties of epitaxial layers will be considered in more details for the example of the increasingly more widespread wurtzite GaN. In order to determine the piezoelectric polarisation for GaN grown in the [0001]

Now, we calculate (Qxm + n) mod Q and (Qxm + n) + Q for arbitrary m, 0 ^ m:

This section discusses various aspects that have an influence on the performance of supervised NNs. These aspects include data manipulation, learning parameters, architecture selection, and optimization methods.

direction consider equation (7.49) which assuming a biaxial stress can be written as:

{ { { = { (15.6) with P := Qxm + n } arithmetic } (15.6) with P := n } arithmetic }

Note that the factor is a consequence of adopting the matrix notation as discussed in Sec 7.1. The matrix equation can be decoupled into three linear equations defining the polarisation vector components, the solution of which gives a single non-zero component of the piezoelectric polarisation in the growth direction as:

0 ^ r < Q A n = Qx(d-m) +r r = nmodQ A d-m = n + Q r = nmodQ A d = m + (n + Q) . We conclude that (Qxm + n) modQ = nmodQ A (Qxm + n) + Q = m + (n + Q) . (15.12) In particular, combining (15.11) and (15.12) by instantiating n to 0, (Qxm) modQ = 0 A (Qxm) + Q = m . (15.13)

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