Exercises Chapter

1. Referring to Figure 1.2: if there are 10 million transistors uniformly distributed on a one centimeter square silicon chip, what is the linear size of each unit?

2. A contemporary computer chip dissipates 54 Watts on an area of one centimeter square. Assuming that transistor elements in succeeding computer generations require constant power independent of their size (a hypothetical assumption), estimate the power that will be needed for a one centimeter square silicon chip in 5 years. Base your estimate on the Moore's Law trend of doubling the transistor count every 1.5 years. (The industry is confident of continuing this trend.)

3. Extrapolate the line in Figure 1.2, to estimate in which year the size of the transistor cell will be 10 nm.

4. In Figure 1.1, the vibrational motions of the single-crystal-silicon bars are transverse (vertical as seen in the picture) and the lowest frequency vibration corresponds to an anti-node at the middle, with nodes at each end of the bar. The resonances occur when L = nk/2, n = 1,2,3... If the fundamental frequency of the 2 micrometer (uppermost) bar is 0.4 GHz, what frequency does that bar have when oscillating in its 2nd harmonic? How many nodes will occur across the bar in that motion?

5. Regarding Figure 1.1, the supporting article states that the vibrations of the bars are generated with electromagnetic radiation. If the radiation used to excite the shortest (2 micrometer) bar is tuned to the bar's fundamental frequency of 0.4 GHz, what is the vacuum wavelength of the radiation?

6. Continuing from Exercise 5, suggest possible means of detecting the motion of a bar in order to confirm a resonance. [Hints: look in a transmission electron microscope (TEM), or Scanning Electron Microscope (SEM) for blurring of the image: look for power absorption from the source..]

Nanophysics and Nanotechnology: An Introduction to Modern Concepts in Nanoscience. Second Edition. Edward L. Wolf

Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40651-4

1. It is stated that the linear vibration frequency of the CO molecule is 64.2 THz. Using the masses of C and O as 12 and 16, respectively (in units of u = 1.66 x 10-27kg), can you show that this vibration frequency is consistent with an effective spring constant K of 1860 N/m? [Hint: in a case like this the frequency becomes x =2pf = (K/i)1/2, with effective mass i= m1m2/(m1 + m2).] What is the wavelength of radiation matching this frequency?

2. Treating the CO molecule of Exercise 1 as a classical oscillator in equilibrium with a temperature 300 K, estimate the amplitude of its thermal vibration.

3. Show that the spring constant K' of one piece of a spring cut in half is K' = 2K.

4. A small particle of radius 10 micrometers and density 2000 Kg/m3 falls in air under the action of gravity. If the viscosity of air is 1.8 x10-5 Pa-s, show that the terminal velocity is about 23 mm/s.

5. For the particle considered in Exercise 4, find the diffusion length during a time 1 s, in air, at 300 K.

Exercises - Chapter 3

1. In the Millikan "oil drop experiment", find the electric field in V/m needed to arrest the fall of a 10 micrometer radius oil particle with net charge 4e. Take the density of the oil drop as 1000 Kg/m3; the acceleration of gravity, g, as 9.8 m/s2; and neglect the buoyant effect of the air in the chamber on the motion ofthe droplet.

2. An FM radio station transmits 1000 Watts at 96.3 MHz. How many photons per second does this correspond to? What is the wavelength? What is the significance of the wavelength, if any, from the photon point of view?

3. A photon's energy can be expressed as pc, where p is its momentum and c is the speed of light. Light carries energy and momentum but has zero mass! Calculate the force exerted by a 1000 Watt beam of light on a perfectly absorbing surface. Why is this force doubled if the surface, instead of absorbing, perfectly reflects all of the light?

4. It is found that short-wavelength light falling on a certain metal, causes emission (photoemission) of electrons of maximum kinetic energy, K, of 0.3 eV. If the work function, U, of the metal in question is 4eV, what is the wavelength of the light?

5. Explain, in the context of Exercise 4, how a value for Planck's constant h can be found as the slope of a plot of K vs. f = c/k, where the light frequency f is assumed to be varied. Such experiments give the same value of Planck's constant h as was derived by Planck by fitting the spectrum of glowing light from a heated body. Einstein was awarded a Nobel Prize in part for his (1905) analysis of this photoelectric effect.

1. Using the Uncertainty Principle, estimate the minimum velocity of a bacterium (modeled as a cube of side 1 micrometer, and having the density of water), known to be located with uncertainty 0.1nm at x =0, in vacuum and at T = 0.

2. Explain why the preceding question makes no sense if the bacterium is floating in water at 300 K. Explain, briefly, what is meant by Brownian motion.

3. Find the minimum speed of a C60 molecule in vacuum, if it is known to be located precisely at x,y,z = 0 plus or minus 0.01 nm in each direction.

4. A Buckyball C60 molecule of mass 1.195 x 10-24 Kg is confined to a one-dimensional box of length L = 100 nm. What is the energy of the n = 1 state? What quantum number n would be needed if the kinetic energy is 0.025 eV (appropriate to room temperature)?

5. A particle is in the n = 7 state of a one-dimensional infinite square well potential along the x-axis: V = 0 for 0 < x< L =0.1 nm; V = infinity, elsewhere. What is the exact probability at T = 0 of finding the particle in the range 0.0143 nm < x <0.0286 nm? (Answer is 1 /7, by inspection of the plot of P(x).)

6. Consider a hypothetical semiconductor with bandgap 1 eV, with relative electron mass 0.05, and relative hole mass 0.5. In a cube-shaped quantum dot of this material with L = 3 nm, find the energy of transition of an electron from the (211) electron state to the (111) hole state. (This transition energy must include the bandgap energy, and the additional energy is referred to as the "blue shift" of the fluorescent emission by the "quantum size effect".) Note that the "blue shift" can be tuned by adjusting the particle size, L.

Exercises - Chapter 5

1. Make an estimate of the vibrational frequency of the H2 molecule, from the lower curve in Figure 5.1. (You can do this by approximating the minimum in the curve as 1/2 K(r-ro)2, where r is the interatomic spacing, and K is an effective spring constant, in eV/(Angstrom)2. Estimate K from Figure 5.1; then adapt the usual formula for the frequency ofa mass on a spring).

2. Use the result from Exercise 5.1 to find the "zero point energy" of di-hydro-gen treated as a linear oscillator. Estimate the "zero-point motion" of this oscillator, in nm, and as a fraction of the equilibrium spacing. Compare this answer to one obtained using the uncertainty principle.

3. At T = 300 K, what is the approximate value of the oscillator quantum number, n, for the di-hydrogen vibrations, expected from kT/2 = (n + 1/2)hm? (At 300 K, an approximate value for feT is 0.025 eV.)

4. Explain, in a paragraph or two, the connections between the covalent bond energy of symmetric molecules, such as di-hydrogen; the notion of anti-symmetry under exchange for fermi particles, such as electrons; and the phenomenon of ferromagnetism (a basis for hard disk data storage).

5. Explain (a) why mean free paths of electrons and holes in semiconductors can be much larger than the spacing between the atoms in these materials. (b) Explain, in light of the nanophysical Kronig-Penney model, why the mean free paths of electrons and holes in exquisitely ordered solids (such as the semiconductor industry's 6 inch diameter single crystal "boules" of silicon) are not infinitely large (i.e., limited by the size, L, of the samples). Consider the effects of temperature, T, the impurity content N and possible isotopic mass differences among the silicon atoms.

6. Why cannot a filled band provide electrical conductivity?

7. Why can the effective mass in a solid differ from the mass of the electron in vacuum?

8. Why does a hole have a charge of +e, and a mass equal to the mass of the valence band electron that moves into the vacant site?

9. How does Bragg reflection (nk = dsinh) influence the motion of carriers in a (one-dimensional) semiconductor with lattice constant a? Why does this lead to the answer that the "zone-boundary" is at k = p/a?

10. Explain why the group velocity of carriers is zero at "the zone boundary", k = p/a.

11. Explain the difference between the two (standing wave) wavefunctions valid at k = p/a. Explain, qualitatively, the difference in their electrostatic energies, and why this energy difference is precisely the "band gap" into the next higher band.

12. Estimate the lifetime s for the H (or D) atom to lose its electron by field ionization at (a) E =25 V/nm and (b) E = 2.5 V/nm, using a one-dimensional tunneling model.

Hints: The ground state with E = 0 is spherically symmetric, depending only on r. With an electric field Ex = E, the situation approximately depends only on x:

electron potential energy U(x) = -kCe2/x - eEx, where kC = 9x 109, with E in V/m. Assume that the ground-state energy is unchanged by E, so

The electron barrier potential energy is U(x) - E0. For E = 0, the electron is assumed to have energy U = -E0 at x = ao = 0.053 nm, the Bohr radius. When E >0, there is a second location, near x2 = E0/eE, where U is again E0. The problem is to find the time s to tunnel from x = ao to x = x2 = E0/eE through the potential barrier U(x) - E0. For small E, the maximum height of the barrier seen by the electron is » E0 = 13.6 eV, and the width t of the tunnel barrier is essentially Ax = t = x2 = E0/eE. If the barrier potential U(x) were VB (independent of x), the tunneling transmission probability T2 would be given by (4.57) and (4.64), retaining only the exponential part of (4.64). Thus

T2 = exp[-2(2mVB)1/2Ax/"], with barrier width Dx and " = h/2p, with h Planck's constant. [A more accurate conventional approach would replace (VB)1/2Dx by

To simplify, consider the average value of the barrier. Approximate the barrier function U(x) - E0 by half its maximum value, v2 VBmax, over the width Dx. The escape rate, _/escape = 1 /s, where s is the desired ionization lifetime, is T*fapproach. The orbital frequency of the electron will be taken from Bohrs's rule for the angular momentum L = mvr = h/2p, equation (4.2), so fipproach = h/[(2p)2mao2] = 6.5 x 1015 s-1.

The working formula _/escape = 1/s is then

So as remaining steps for the reader:

(A) Solve the quadratic -E0 = -fee2/x - eEx, show that Dx = [(E0/e)2 - 4keE]/E. (This limits the applicability of the approximation to E < 32.1 V/nm, where Dx = 0. This alone is a rough criterion for the onset of rapid field ionization.)

(B) Find dU/dx and set it to zero, to show that the position x' of the maximum in U(x) occurs at x' = (ke/E)1/2. Insert this value x' into U(x) to show that its maximum value is U(x') = -2e(keE)1/2. Then show that VBmax= E0 - 2e(keE)1/2.

(C) Evaluate _/escape from the working formula, taking E =25 V/nm. (The exponential should be about exp(-2.35). The lifetime is about 1.7 x 10-15 s.

(D) Evaluatef,scape from the working formula for E =2.5 V/nm. (The exponential should be exp(-118.8). The lifetime is about 6x1033 s or about 1.9 x 10 y. (The famous result of Oppenheimer is a lifetime of(1010)10 s for E = 1000 V/m.)

13. Estimate the ion current in the neutron generation apparatus using a positively charged tip as described in relation to Figure 5.23. Assume that the radius r around the tip center leading to 100% ionization of the deuterium is 600 nm. The assumption is that all D2 molecules which fall on the front half of this surface (of area 2pr2) contribute 2e to the ion current. The rate of molecules crossing a surface in a dilute gas is nv/4 (molecules per unit area per unit time), where n is the number per unit volume of molecules whose rms speed is Iv.

(A) In a gas of deuterium molecules at temperature T = 270 K, show that the rms speed is 1294 m/s. The deuterium molecule has a mass of 4 amu.

(B) The gas is said to be at a pressure of 0.7 Pa at T = 270 K. Using the ideal gas law pV = RT (for one mole, corresponding to Avogadro's number of molecules), show that the number density n of molecules is 1.9 x1020 m-3. On this basis, show that the ion current would be about 44.5 nA (the observed value is 4 nA).

(C) Verify that the mean free path of the D ion exceeds the dimension of the container, so that straight-line trajectories can be assumed. A formula for the mean free path is k =1 /ns, where the cross-section can be taken as s = pp2, and take the radius p of the D2 molecule as 0.037 nm (look at Figure 5.1).

14. Under adiabatic compression of an ideal gas bubble of initial radius 0.1 mm at 1 bar (one atmosphere = 101.3 kPa) and 300 K to final radii (A) rmin= 14.1 mm and (B) rmin= 10 nm, find the resulting final pressures, expressed in atmospheres (bar). See text related to Figure 5.25.

15. For adiabatic compression of an ideal gas, find expressions for the pressure as a function of the temperature, volume and radius r of the bubble. Follow the method used in the text in deriving the relation T (1/V)y-1.

16. The classical equation of Laplace relates the pressure difference across the surface of a spherical bubble of radius r in a liquid as

Dp = 2y/r, where y is the surface tension, which is 0.0728 N/m for water at 25 °C. Evaluate this equilibrium pressure difference in atmospheres for water bubbles of radii 0.1 mm, 1 mm and 10 nm.

Exercises - Chapter 6

1. The basic measure of thermal agitation energy is 1/2 kT per degree of freedom, where k is Boltzmann's constant. How does thermal energy (at 400 K) compare with the energy differences among the computer-modeled simulations of octane presented in Figure 6.1?

2. In Figure 6.1, 1011 Hz is a characteristic rate at which the octane molecule switches from one shape (conformation) to another. For comparison, find the lowest longitudinal vibration frequency, using the methods of Chapter 2, modeling octane as a chain of eight carbon atoms (mass 12) connected by springs of strength K = 440 N/m and length 0.15 nm.

3. The variety of different conformations of octane, exhibited in Figure 6.1, seem incompatible with using octane to carry a single (small-diameter) tip, which is a required function of the proposed "molecular assembler". With regard to stability, give reasons why acetylene C2H2 (a) and a small-diameter carbon nanotube (b), should be superior.

4. Consider the FET device shown in Figure 6.4. Why is the forward current in this FET so large? How could such FET devices be assembled at a density of 107/cm2on a silicon chip? (If you have a good answer, have your statement notarized and hire a patent lawyer!)

5. How fast can the FET device shown in Figure 6.4 respond to a gate voltage turned on at time zero? It is said that the electron transport along a nanotube is "ballistic", which means that, if the force is eE, then the velocity is eEt/m, and the distance covered in time t is 1/2(eE/m)t2. For the device geometry shown, where E = V/L, set V = 1 Volt and find: (a) the corresponding value of t. Secondly, (b) for the device geometry shown, convert this result into an effective mobility i, such that v = lE = iV/L, and t = L2/wV. Note that the mo bility i is conventionally given in units of cm2/Vs. Compare this mobility to values listed in Table 5.1.

6. Consider the magnetotactic bacterium shown in Figure 6.7. (a) Calculate the torque exerted on the bacterium in an earth's magnetic field of 1 Gauss (10-4T) if the bacterium is oriented at 90°. (b) Explain why a strain of magne-totactic bacteria found in the northern hemisphere, if transported to an ocean in the southern hemisphere, would surely die out.

7. In the STM study of ordered arrays of C60 shown in Figure 6.9 it was observed that the molecules were locked into identical orientations at 5 K, but were apparently rotating at 77 K. (a) From this information, make an estimate ofthe interaction energy between nearest-neighbor C60 molecules. Is this energy in the right range to be a van der Waals interaction? (b) Compare this temperature information to the melting point of a "C60 molecular solid" which you can estimate from Figure 5.2.

Exercises - Chapter 7

1. Compare the conventional, state-of-the-art, spatial resolution for modern photolithography, presented as 180 nm, to the resolution, in principle, that is available at wavelength 248 nm, see Figure 7.3.

2. Explain why e-beam writing is not competitive as a production process for writing wiring on Pentium chips.

3. What features of Figure 7.3 are inconsistent with the stated exposure wavelength of 248 nm?

4. Compare e-beam writing to conventional lithography with regard to resolution and throughput.

5. It is reported in Consumer Reports (October 2003, p. 8) that the "Segway Human Transporter" has five independent electronic gyroscope sensors, with, presumably, a corresponding set of servo systems. Write an essay on how such rotation sensors might be fabricated (clue, one method might be based on Figure 7.1). In principle, how would the performance of such rotation sensors be expected to change with a uniform reduction in scale, xyz ofx 10?

6. What are the chief advantages of the superconductive RSFQ computing technology, and what are the chiefdisadvantages ofthe same technology?

7. Why is the spatial resolution of the STM better than that of the AFM?

8. Why can't a single tip in the STM/AFM technology possibly perform operations at a rate greater then 1 GHz?

9. Why cannot a single tip in the STM/AFM technology perform operations beyond observing, nudging, and exciting an atom (or molecule) in its view? (Explain what is meant by "nudging", and why it is easier for atoms, like Xe, or molecules, like benzene, with a large number of electrons.) Other desirable, but unavailable, functions for a single tip would be picking up, orienting, and depositing, atoms or molecules. Why are these functions unavailable?

1. With reference to the initial discussion of qubits in Chapter 8, explain in words why several independent measurements of identically prepared quantum states are needed to determine the coefficients a and b. Can a and b ever be precisely known in such a case?

2. Verify the size of effective magnetic field B* needed to produce an energy difference between (electron) spin up and spin down of about twice 4.74 eV, as suggested in Figure 5.1.

3. Make an argument based on symmetry to explain why the average magnetic dipole-dipole interaction for the proton spin and the ground-state electron spin in hydrogen is zero. If this is so, what is the origin of the 21.1 cm line emission observed from hydrogen in outer space?

4. With regard to Figure 8.2, explain, for ferromagnetic hcp-Co, the statement that the "density of spin-down states at EF exceeds the spin-up density", while the majority spins are "up".

5. Explain why the DNA strands seen in Figure 8.17, deposited initially in a drop of solvent directly over the trench in the Si, eventually, after evaporation of the solvent, cross the trench in straight lines which are accurately perpendicular to the trench.

Exercises - Chapter 9

1. Evaluate the capacitance of a nanotube of radius 0.5 nm and length 1 m, assuming a coaxial counter-electrode of radius 250 nm (see text near Figure 9.7). [Answer: 36 pF/m]

2. Using the result from above, and the stated capacitance, 10 nF, for the chemical sensing device of Figure 9.7, estimate the total length of the nanotube network forming the chemically sensitive electrode. [Answer: 279 m]

3. Published data have shown roughly 0.1 mA as an observed source-drain current in carbon nanotube-based FET, at limits of the gate voltage range. Estimate the current density, in A/cm2, assuming the nanotube of diameter 1.4 nm is (A) a solid cylinder and (B) a cylindrical shell of diameter 1.4 nm and shell thickness 0.1 nm.

4. In connection with Figures 9.9 and 9.10, estimate the electrostatic attractive force in newtons between two crossing (10,10) single-wall nanotubes, assuming the voltages are +5 V and -5 V, and the tubes are 20 nm long. Take the initial wall-to-wall spacing s to be 1.2 nm. The diameter of the (n,m) nanotube is given as dtube = ap-1(nn + mm + nm)1/2 where a = 0.249 nm.

5. Assume that a (10,10) single-wall carbon nanotube as in Exercise 9.4 (see Figure 9.9) of length L =20 nm is clamped at each end. Estimate the transverse vibrational frequency of one such section.

Hints: Use the formulas in Section 2.1, applicable to an elastic beam (assume Young's modulus is 1 TPa) clamped at each end. Estimate the mass per unit length of the (10,10) single-wall nanotube from the mass per carbon atom, 12x 1.66x 10-27 kg, the C-C bond length of 0.142 nm, and the tube's surface area pdtubeL for length L. (A published estimate for this frequency is about 100 GHz.)

6. Continuing from Exercise 9.5, the formulas in Section 2.1 give an approach to estimate the spring constant K for the (10,10) nanotube. Using this estimate, find the force needed to displace the tube by 1 nm. Compare this force to the electrostatic force estimated in Exercise 9.4.

Exercises - Chapter 10

1. Estimate the diameters of the proposed molecules in Figure 10.1.

2. Provide a critique of the suggested size of the "molecular assembler" as one billion atoms (it may be on the small side?) What would be the expected radius for a spherical assembly of one billion atoms, in nm? Can you estimate the number ofatoms in a typical enzyme, as a comparison?

3. It has been suggested that the time for a typical bacterium to grow is about an hour. Suppose a rectangular (cube) bacterium has dimension L = 1 micrometer, the density of water, and is made of carbon 12. Roughly how many atoms are there in this bacterium? Calculate the rate of growth in atoms per second if this forms in one hour. Compare your answer to the performance of the IBM Millipede AFM, discussed in Chapter 7.

4. The example of the magnetotactic bacterium (see Chapter 6) proves that some robust inorganic solids, eg. Fe3O4, can be grown at room temperature from (ionic) solution. Compare this case with the case of diamond (and the proposed diamondoid inorganic molecules), which require high temperatures/pressures to form (and apparently cannot be grown from solution). What is the crucial difference?

5. What is the most significant difference between a virus and a living organism? (Can a virus live forever?)

6. Poll your colleagues to ask if they think that "computers" (or artificially intelligent robots) will ever be able to establish separate identities, consciousness, and other human qualities. If so, how threatening does this prospect seem?

Nanophysics and Nanotechnology

Edward L. Wolf

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