Electric Clock for NanoMagnet Logic Circuits - Field-Coupled Nanocomputing: Paradigms, Progress, and Perspectives - page 78

the strain matrix and D is the electric charge matrix, s is the compliance (the
inverse of the stiffness), T is the stress matrix, d is the matrix for the direct
piezoelectric effect while d t is the matrix for the converse piezoelectric effect,
finally E is the electric field and ʳ is the permittivity. The matrix of d coecients
therefore links the various components of the applied electric field to the resulting
strain. The two most important d coecients for this kind of applications are
the d 31 and the d 33 . When the applied electric field is perpendicular to the strain
the d 31 coecient is used, while, when the electric field and the strain lie in the
same plane and in the same direction, the d 33 coecient is instead used. d 31
coecient is normally smaller than the d 33 coecient.
The second problem is related to the electrodes placement. One of the elec-
trodes is placed on top of the piezoelectric layer, between the piezoelectric mater-
ial and the magnets, reducing the mechanical coupling between them. Moreover,
considering the fabrication of a complex NML circuit, these electrodes must be
contacted with wires to distribute the clock voltage, but this is dicult consider-
ing that electrodes are covered by magnets. In the second solution (Figure 6 (B))
electrodes are placed at both sides of the piezoelectric layer, so the electric field
and the strain lie along the same plane and are coupled through the d 33 coef-
ficient, which is higher than the d 31 as can be seen from Table 1 . The coupling
between piezoelectric layer and magnets is stronger and electrodes can be con-
tacted freely because there is no physical interference caused by magnets. We
therefore choose this second structure to build our magnetoelastic clock system
for NML technology.

Table 1. Comparison between piezoelectric thin films. d 31 and d 33 are the two main
piezoelectric coecients. Ef MAX is dielectric strength and ʳ r is the dielectric constant.

d 31 (pm/V) d 33 (pm/V) Ef MAX (MV/m) ʳ r

PZT − 30 ÷− 80 50 ÷ 150 > 50 300 ÷ 1300

PVDF 23 − 33 5 12

ZnO 0.26 5.9 25 ÷ 40 10.9

BT − 33 82 2 1250 ÷ 10000

Table 1 shows a comparison between some of the piezoelectric material most
commonly used. d 31 and d 33 coecients bond the mechanical strain with the
applied voltage, and as can be clearly seen, d 33 is normally higher than d 31 .
Ef MAX is the maximum electric field that the material can tolerate while ʳ r
is the relative dielectric constant. PZT (Lead-Zirconate-Titanate) is one of the
most commonly employed material and, as can be seen from Table 1 , it has also
the best performance. The other materials shown in Table 1 are PVDF, which is
a polymer, Zinc Oxide (ZnO) and Barium-Titanate (BT), but they show worst
performance than PZT, so in this work we use PZT as our reference choice for
the piezoelectric substrate.