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NEOCLASSICAL THEORY OF INTERACTION 103

2.9.6 Graphene-Based Electromechanical Switch

Figure 2.9.12 demonstrates schematically two
Graphene
cross-sectional views (not in scale) of Micro- Membrane
Electro-Mechanical Switch (MEMS) in two
stages, off (top image) and on (bottom image). Microstrip Trace
The switch comprises the graphene membrane Substrate
suspended above the substrate and the bottom 0 a)
microstrip trace printed on the substrate Before pull-in
located above the ground screen. The typically
silver electrodes are plated on the substrate and Silver Electrodes
support the graphene membrane. The DC or After pull-in
AC bias voltage from an external source is
applied between electrodes and microstrip
trace.
Substrate
At the initial stage, namely, “before pull-in,” V b)
an open circuit is expected due to no electrical
contact between the membrane and bottom Figure 2.9.12 Schematic image of
trace. Then the electrostatic force exerted by graphene Micro ElectroMechanical
the bias voltage (5V approximately) pulls the Switch (MEMS)
membrane down into electrical contact with
the trace, as shown in Figure 2.9.12b, due to graphene elasticity. Thereby, the microstrip line
becomes shorted. As soon as the bias is removed the contact is broken again by mechanical
restoring forces. These switches exhibit the sharp switching, reversible operation cycles and
high on/off ratio from about 10 to below 10 depending on bias force. Furthermore, such
5
flexible membrane can be transferred into a mechanical resonator applying the AC voltage of
proper frequency and avoiding electrical contact with the trace. If so, such device becomes a
variable and voltage-dependable capacitor. Summarizing note that the question, ultimately, isn't
if graphene will change everything from computing to manufacturing, but how and when.

2.10 SOME ADDITIONAL PROPERTIES OF MATERIALS

2.10.1 Kramers-Kronig (K-K) Relations
31
Now, let us establish so-called the Kramers-Kronig (K-K) relations linking the real and
imaginary part of material dielectric or magnetic constant. It means, that () and () in
′
′′

the same way as () and () are not the pair of independent functions. For example, the
′
′′

measured frequency-dependence of dielectric loss through () strictly and unambiguously
′′

determines the frequency behavior of the dielectric permittivity () and vice versa. In
′

practice, the K-K relations allows to halve the time of the broadband test by evaluating the real
or imaginary part only, or to cross-check the consistency of the test results measuring both of
them, or to verify the validity of numerical simulation data. In particular, the K-K relations
enable us to find the frequency characteristic named spectral signature of remote objects, even
from non-perfect experimental data. Let follow in four steps:
31 The material in this section is far from trivial and required some knowledge of improper integral. Please
refresh it from the calculus course.