PROCESO LECTOR

A convenient quantity to describe the effect the AFM tip exerts on the nanomechanical resonator is the mechanical impedance. Therefore this section introduces the concept of mechanical impedance of basic mechanical elements such as a spring or a mass and then treats combinations of such basic elements in lumped element models. An elaborate treatment of mechanical impedance theory as well as a table of the impedances of various combined elements can be found in [Har09].

v₁ v₂

F₁ F₂

P

1 2

c

Figure 3.4: Schematic illustration of a mechanical resistance, symbolized by a dashpot with damping constant c.

In vibrating systems, the mechanical impedance Z at a given position is defined as the ratio of the applied forceF to the velocityv at this position.

Z= F

v (3.2)

In the following the forceF is assumed to vary periodically in time with frequencyωand magnitudeF0, such that

F =F0eiωt. (3.3)

Applying this force to a linear mechanical system leads to a velocity

v=v0ei(ωt+φ). (3.4)

Herev0is the magnitude of the velocity andφis a phase betweenF andv. In the follow-

ing, the mechanical impedance of basic elements that can be used to build lumped element models of mechanical systems is introduced. These elements are mechanical resistances, springs and masses.

Mechanical Resistance

In a mechanical resistance or damper, the relative velocity between two points is pro- portional to the applied force. Because this relation is fulfilled by viscous friction, a mechanical damper can be symbolized by a dashpot as depicted in Fig. 3.4. The compo- nents comprising this ideal resistance are assumed to be massless and infinitely rigid. The velocityv1 of point 1 with respect to the velocityv2 at point 2 (measured with respect to

the stationary point P) is

v = (v1−v2) =

F1

c (3.5)

where c is the damping constant or mechanical resistance. If the force given by equa- tion 3.3 is applied at point 1 and point 2 is fixed, the velocityv1becomes

v1=

F0eiωt

c =v0e iωt

v₁ v₂

F₁ F₂

P

1 2

k

Figure 3.5:Schematic illustration of an ideal spring with a spring constant k.

Because the damping coefficient is real, the force and velocity are in phase. The mechan- ical impedance can then be identified to be equal to the damping coefficient:

Zres=

F

v =c (3.7)

Spring

The displacement between the end points of a linear spring is proportional to the applied force (c.f. Fig. 3.5). Withx1, x2 being the displacements at points 1 and 2 relative to the

reference point P and a spring constantkthis can be written as

x1−x2=

F1

k (3.8)

The applied force is transmitted by the spring, so that F1 = F2. If the force given by

equation 3.3 is applied at point 1 and point 2 is fixed, the displacement of point 1 can be expressed as

x1=

F0eiωt

k =x0e

iωt _(3.9)

The derivative of this equation with respect to time gives the velocity. Using equation 3.2 one finds the mechanical impedance of a spring to be

Zspring= −

ik

ω (3.10)

From this expression one can see that the velocity is 90° ahead of the force in a spring, just as the current in a capacitor is 90° ahead of the applied voltage. In fact, by identifying the spring constant k in the mechanical case with the capacitance C in the electrical case, where the impedance is given by the ratio of voltage to current, one finds that the mechanical impedance of a spring is identical to the electrical impedance of a capacitor. Mass

For a massm, the accelerationx¨is given by

¨

x= F m =

F0eiωt

By integrating equation 3.11 and using equation 3.2, the mechanical impedance is found to be

Zmass=iωm (3.12)

From this expression one can see that the velocity lags 90° behind the force for a mass, just as the current in an inductor lags 90° behind the applied voltage. By identifying the mass min the mechanical case with the inductanceLin the electrical case one sees that the mechanical impedance of a mass is identical to the electrical impedance of an inductance.

Parallel and series elements

Just as in the electrical case, the combined impedances of assemblies of basic elements can be easily calculated. Such assemblies can be divided into two basic groups, parallel elements and series elements. In contrast to the electrical case however, the total me- chanical impedance for parallel elements is the sum of the impedances of the individual elements [Har09]

Zpar = ∑

l

Zl (3.13)

and the total impedance for series elements is given by

1

Zser = ∑l

1

Zl

. (3.14)

This inverted behavior with respect to the electrical case comes from the fact that for parallel mechanical elements, the (connected) end points have to have the same velocity. In the electrical case, the voltage is equal for every individual component in a parallel circuit. From the respective definitions of the impedance (mechanical/electrical) one can directly see that this leads to the inversion between the parallel and series element case. Reflection at interfaces

As in the electrical realm, an impedance discontinuity causes a reflection of an incident wave. The same applies for the mechanical case, and a reflection coefficientRfor waves can be written as [Har09]

R= ∣Z1−Z2 Z1+Z2

∣2, (3.15)

whereZ1andZ2 are the different impedance values at the discontinuity.

As an exemplification of such an impedance discontinuity in an elastic system, consider the geometry depicted in Fig. 3.6. A region of widthbis interfaced with a region of width

a. If one assumes a mechanical impedancez per unit length and - as parallel mechanical impedances add - integrates over the width, one obtains the impedanceZb =z ⋅b at the

impedance per length z

a

b

interface

Figure 3.6:Schematic illustration of an impedance discontinuity. A region with width b is interfaced with a region of width a.

Za=z⋅a. The reflection coefficient is thus only given by the geometric parametersaand b: R= ∣Za−Zb Za+Zb∣ 2 = ∣a−b a+b∣ 2 (3.16) In the limit a → b, the reflection coefficient becomes zero, and in the limit a → 0, it approaches unity. Identifying a high reflection coefficient with low radiation loss for a resonator comprised of two such junctions, one can see that the resonator width should be small, whereas the clamp should be very wide in order to obtain high reflection. Note however that this is quite an oversimplification of the problem, and much more elab- orate calculations are necessary to obtain realistic values for specific resonator geome- tries [Col11]. Nevertheless, the presented example descriptively shows that such support losses strongly depend on the geometry of the support.