Conclusion

A silicon cantilever beam 500 nm thick, 10 μm wide, and 100 μm long has a quality fac-tor Q = 60 in air (see Figure 5.9). A driving force causes the cantilever’s tip to resonate at its damped natural frequency (f_d = 70 kHz) with an amplitude Ad = 10 nm. What is the approxi-mate bandwidth Δf of this system? What frequencies are at either end of the bandwidth?

What is the amplitude of the beam’s oscillation at these frequencies?

The bandwidth, Δf, is given by Δf f

≈ Q^d = Hz = 70 000 Hz

60 1170

,

Thus, the frequencies on either side of the band are 70 ± 1.17/2 kHz, or 70.585 and 69.415 kHz. At these frequencies, the amplitude is

Ad nm 2 nm

10

2 7

= =

70.585 69.415

f_d = 70.0

l = 100 μm

w = 10 μm t = 500 nm Q = 60

A_d = 10

√2 = 7 A_d A [nm]

[kHz]

f_drive Δf

FIGURE 5.9 The quality (Q) factor determines the sharpness of the resonance peak. Q can be esti-mated for the cantilever beam shown (see Back-of-the-Envelope 5.3) by the ratio of the resonance frequency and the bandwidth: f_n/Δf.

140    ◾   Nanotechnology

5.3.2 Atoms

Only at absolute zero do atoms cease to move. Or at least that is what we think: the lowest temperatures ever achieved on Earth are around 1 billionth of a degree above absolute zero, or 1 nanokelvin, and the atoms were still moving around a little so we do not know what happens for sure at 0 K.

Temperature, in a quantum mechanical sense, is a measure of how fast atoms are mov-ing. Atoms in a solid vibrate back and forth; and as the temperature rises, the atoms vibrate with bigger and bigger amplitudes. Sometimes they will vibrate right off the solid lattice into a fluid state, where they are free to bounce around like balls in a bingo spinner. Later on in Chapter 7, the discussion of nanoscale heat transfer will draw on what is presented in this chapter about the motions of atoms.

However, this now-familiar classic mass–spring system does not lose its validity just because we are discussing atoms. The concept of a simple harmonic oscillator can be approached from both classical and quantum viewpoints, and works wonderfully in understanding the vibrations of atoms. We will delve into both viewpoints during our discussion here.

5.3.2.1 The Lennard–Jones Interaction: How an Atomic Bond Is Like a Spring

For now, let us focus our discussion on a pair of atoms bonded together as a molecule. We restrict the motion of these two atoms to a single dimension, along the x-axis. They can move toward each other or away from one another. We will view them as an undamped simple harmonic oscillator. We looked in detail at this situation when discussing the Lennard–Jones potential in Chapter 4, “Nanomaterials.” This is also an excellent model for our purposes here—providing us with a helpful description of the interactions between atoms. We use the Lennard–Jones potential to model the potential energy, PE, of a pair of atoms separated by a distance, x. These atoms will be attracted or repelled by each other, depending on x. For the case of two similar atoms, the attractive interaction varies with the inverse-sixth power of the separation distance, while the repulsive interaction tends to vary by the inverse twelfth power of the distance:

PE x C

x C

( )

6 (5.24)

The constants C₁ and C₂ represent the strengths of the attractive and repulsive interactions, respectively, and vary by atom. This relationship is shown graphically in Figure 5.10. The Lennard–Jones model applies specifically to van der Waals interactions (and is very accu-rate for noble gas atoms such as argon, krypton, and xenon), but the shapes of the curves in Figure 5.10 are common to almost every kind of bond type, not just van der Waals bonds.

As we learned in Section 5.3.1.2, energy is a force applied over a distance (E = Fx). Thus, if we wish to know the interaction force, we can take the derivative of Equation 5.24. The force F between the atoms is therefore

F x C

x C

( )

7 (5.25)

This relationship is also plotted in Figure 5.10.

Nanomechanics    ◾   141

At very large separation distances, the atoms do not even “feel” the presence of one another at all. Therefore, both the potential energy and the force between them are zero. In our model, this equates to there being no spring between the balls. As the atoms near one another, they are at first subject to attractive forces (as if there is an elongated spring between the atoms pulling them together). The closer they get, lesser energy is added (the spring is less and less elongated). Eventually, the potential energy curve reaches a mini-mum at the equilibrium separation distance, x = x_e. It is here that the curve of the interaction force also goes to zero. Unless additional energy is added to the system, or the  system is in motion exchanging kinetic and potential energy, the atoms remain at this  separation distance. In our model, this is just like the equilibrium position of the spring (x = 0), where it is neither compressed nor elongated.

Potential Energy, PEForce, F

x

x x_e

x k

dx

dF k = dF dx

PositiveNegativeRepulsiveAttractive

FIGURE 5.10 Potential energy, PE, and force, F, as a function of separation distance, x, for a pair of atoms. The force curve is the derivative of the energy curve. The Lennard–Jones interaction model tells us that at large separations, the atoms do not “feel” one another. In terms of the ball–spring model, this equates to no spring at all between the balls. As the atoms approach one another, attractive forces tug them to the point of least energy at x = x_e. The spring constant, k, of the bond between the atoms is determined by the slope of the force curve at the equilibrium point.

142    ◾   Nanotechnology

The force of a spring is proportional to the spring constant, k, measured in units of force per unit displacement (typically N/m). How then do we determine the spring constant of the bond shared by two atoms? The answer is on the force curve in Figure 5.10. If we zoom in on the region near the equilibrium position, x_e, we see that the force is relatively linear.

For values of x less than x_e, the atoms have been forced closer together than they would be naturally—like a compressed spring. Values greater than x_e are the equivalent of an elongated spring. Because we know that the spring constant is directly proportional to the force and inversely proportional to the distance, the spring constant is then the slope of the curve (or dF/dx) in this region.

The spring force, F_spring, holding together the atoms in the ball–spring model is

Fspring = −k x x

(

)

Here, x is the separation distance, x_e is the equilibrium separation, and k is the spring constant.

We are very close to a full description of the two-atom model from the classical stand-point of a ball–spring system. A lingering unknown is the mass of the system. We cannot simply add up the masses of the two atoms in this case. When two objects are acted upon by a central force (e.g., the gravitational force between the Earth and the moon), we treat