Chapter 2



Chapter 2

Problems

1. Calculate the wavelength of an electron which has a kinetic energy of 4 eV.

2. What should be the energy of an electron so that the associated

electron waves have a wavelength of 600 nm?

3. Since the visible region spans between approximately 400 nm and

700 nm, why can the electron wave mentioned in Problem 2 not

be seen by the human eye? What kind of device is necessary to

detect electron waves?

4. What is the energy of a light quantum (photon) which has a wavelength

of 600 nm? Compare the energy with the electron wave energy

calculated in Problem 2 and discuss the difference.

5. A tennis ball, having a mass of 50 g, travels with a velocity of 200

km/h. What is the equivalent wavelength of this “particle" ? Compare

your result with that obtained in Problem 1 above and discuss the

difference.

6. Derive (2.9) by adding (2.7) and (2.8).

7.“Derive” (2.3) by combining (1.3), (1.5), (1.8), and (2.1).

*8: Computer problem.

(a) Insert numerical values of your choice into (2.9) and plot the result.

For example, set a constant time (e.g. t = 0) and vary Δk.

(b) Add more than two equations of the type of (2.7) and (2.8) by using

Different values of Δω and plot the result. Does this indeed reduce the

number of wave packets, as stated in the text? Compare to Fig. 2.3.

Chapter 3

Problems

1. Write a mathematical expression for a vibration (vibrating string, for

example) and for a wave. (See Appendix 1.) Familiarize yourself with

the way these differential equations are solved. What is a “trial

solution?” What is a boundary condition?

2. Define the terms “vibration” and “wave.”

3. What is the difference between a damped and an undamped vibration?

Write the appropriate equations.[pic]

4. What is the complex conjugate function of:

(a) [pic]=a+ bi; and

(b) Ψ= 2Ai sinαx

Chapter 4

Problems

1. Describe the energy for:

(a) a free electron;

(b) a strongly bound electron; and

(c) an electron in a periodic potential

Why do we get these different band schemes?

2. Computer problem. PlotΨΨ* for an electron in a potential well. Vary n from 1 to ~100. What conclusions can be drawn from these graphs? (Hint: If for large values for n you see strange periodic structures, then you need to choose more data points!)

3. State the two Schrödinger equations for electrons in a periodic

potential field (Kronig-Penney model). Use for their solutions, instead

of the Bloch function, the trial solution

Ψ(x) = Aeikx

Discuss the result. (Hint: For free electrons V0 = 0.)

4. When treating the Kronig-Penney model, we arrived at four equations

for the constants A, B, C, and D. Confirm (4.61)

5. The differential equation for an undamped vibration is

[pic] (1)

whose solution is

[pic] (2)

where

[pic] (3)

Prove that (2) is indeed a solution of (1).

6. Calculate the “ionization energy" for atomic hydrogen.

7. Derive (4.18a) in a semiclassical way by assuming that the centripetal

force of an electron, mv2/r, is counterbalanced by the Coulombic

attraction force, -e2/4πε0r2, between the nucleus and the orbiting

electron. Use Bohr's postulate which states that the angular momentum

L = mvr (v = linear electron velocity and r = radius of the orbiting

electron) is a multiple integer of Planck's constant (i.e., nћ). (Hint: The kinetic energy of the electron is E = [pic]mv2.)

8. Computer prοblem. Plot equation (4.67) and vary values for P.

9. Computer problem. Plot equation (4.39) for various values for D and γ.

10. The width~ of the potential well (Fig. 4.2) of an electron can be

assumed to be about 2 Ǻ. Calculate the energy of an electron (in

Joules and in eV) from this information for various values of n. Give

the zero-point energy.

Chapter 5

Problems

1. What is the energy difference between the points [pic] and [pic](upper) in the band diagram for copper?

2. How large is the “gap energy” for silicon? (Hint: Consult the band

diagram for silicon. )

3. Calculate how much the kinetic energy of a free electron at the corner

of the first Brillouin zone of a simple cubic lattice (three dimensions!)

is larger than that of an electron at the midpoint of the face.

4. Construct the first four Brillouin zones for a simple cubic lattice in two dimensions.

5. Calculate the shape of the free electron bands for the cubic primitive

crystal structure for n = 1 and n = -2 (see Fig. 5.6).

6. Calculate the free energy bands for a bcc structure in the kx-direction

having the following values for h1/h2/h3的: (a)[pic]; (b) 001; and

(c) 010. Plot the bands in k- space. Compare with Fig. 5.18.

7. Calculate the main lattice vectors in reciprocal space of a fcc crystal.

8. Calculate the bands for the bcc structure in the 110 [Γ- N] direction

for: (a) (000); (b) ([pic]); and (c)([pic]).

9. If b1(tl = 1 is given (see equation (5.14)), does this mean that b1 is parallel to tl?

Chapter 6

Problems

1. What velocity has an electron near the Fermi surface of silver? (EF =

5.5 eV).

2. Are there more electrons on the bottom or in the middle of the valence

band of a metal? Explain.

3. At what temperature can we expect a 10% probability that electrons in

silver have an energy which is 1% above the Fermi energy? (EF = 5.5

eV)

4. Calculate the Fermi energy for silver assuming 6.1 x 1022 free electrons

per cubic centimeter. (Assume the effective mass equals the free

electron mass.)

5. Calculate the density of states of 1 m3 of copper at the Fermi level

　(m* = m0, EF = 7 eV). Note: Take 1 eV as energy interval. (Why?)

6. The density of states at the Fermi level (7 eV) was calculated for 1 cm3

of a certain metal to be about 1021 energy states per electron volt.

Someone is asked to calculate the number of electrons for this metal

using the Fermi energy as the maximum kinetic energy which the

electrons have. He argues that because of the Pauli principle, each

energy state is occupied by two electrons. Consequently, there are

2× 1021 electrons in that band.

(a) What is wrong with that argument?

(b) Why is the answer, after all, not too far from the correct numerical

value?

7. Assuming the electrons to be free, calculate the total number of states

below E = 5 eV in a volume of 10-5 m3 .

8. (a) Calculate the number of free electrons per cubic centimeter in

Copper, assuming that the maximum energy of these electrons

equals the Fermi energy (m* = m0).

(b) How does this result compare with that determined directly from

the density and the atomic mass of copper? Hint: Consider equation

(7.5).

(c) How can we correct for the discrepancy?

(d) Does using the effective mass decrease the discrepancy?

9. What fraction of the 3s-electrons of sodium is found within an energy

kBT below the Fermi level? (Take room temperature, i.e., T = 300 K.)

10. Calculate the Fermi distribution function for a metal at the Fermi level for T≠0.

11. Explain why, in a simple model, a bivalent material could be

considered to be an insulator. Also explain why this simple argument

is not true.

12. We stated in the text that the Fermi distribution function can be

approximated by classical Boltzmann statistics if the exponential

factor in the Fermi distribution function is significantly larger than

one.

a) Calculate E-EF=nkBT for various values of n and state at which

value for n,

[pic]

can be considered to be “significantly larger” than 1 (assume T =

300 K). (Hint: Calculate the error in F(E) for neglecting “1” in the

denominator.)

(b) For what energy can we use Boltzmann statistics? (Assume EF =

5eV and E-EF = 4kBT.)

Chapter 7

Problems

1. Calculate the number of free electrons for gold using its density and its atomic mass.

2. Does the conductivity of an alloy change when long-range ordering

takes place? Explain.

3. Calculate the time between two collisions and the mean free path for

pure copper at room temperature. Discuss whether or not this result

makes sense. Hint: Take the velocity to be the Fermi velocity, vF ,

which can be calculated from the Fermi energy of copper EF = 7 eV.

Use otherwise classical considerations and Nf = Na.

4. Electron waves are “coherently scattered” in ideal crystals at T=0.

What does this mean? Explain why in an ideal crystal at T=0 the

resistivity is small.

5. Calculate the number of free electrons per cubic centimeter (and per

atom) for sodium from resistance data (relaxation time 3.1×10-14 s).

6. Give examples for coherent and incoherent scattering.

7. When calculating the population density of electrons for a metal by

using (7.26), a value much larger than immediately expected results.

Why does the result, after all, make sense? (Take σ= 5×105 1/Ω(cm;

vF = 108 cm/s and τ= 3×10-14 s.)

8. Solve the differential equation

[pic] 7.10

and compare your result with (7.11).

9. Consider the conductivity equation obtained from the classical electron theory. According to this equation, a bivalent metal, such as zinc, should have a larger conductivity than a monovalent metal, such as copper, because zinc has about twice as many free electrons as copper. Resolve this discrepancy by considering the quantum mechanical equation for conductivity.

Chapter 8

Problems

Intrinsic Semiconductors

1. Calculate the number of electrons in the conduction band for silicon at

T =300K. (Assume [pic].)

2. Would germanium still be a semiconductor if the band gap was 4 eV

wide? Explain! (Hint: Calculate Ne at various temperatures. Also

discuss extrinsic effects.)

3. Calculate the Fermi energy of an intrinsic semiconductor at T≠0 K. (Hint: Give a mathematical expression for the fact that the probability of finding an electron at the top of the valence band plus the probability of finding an electron at the bottom of the conduction band must be 1.) Let Ne≡Np and me*≡mh*.

4. At what (hypothetical) temperature would all 1022 (cm-3) valence electrons be excited to the conduction band in a semiconductor with Eg = 1 eV? Hint: Use a programmable calculator.

5. The outer electron configuration of neutral germanium in its ground state is listed in a textbook as 4 s24p2. Is this information correct? Someone argues against this configuration stating that the p-states hold six electrons. Thus, the p-states in germanium and therefore the valence band are only partially filled. Who is right?

6. In the figure below, σis plotted as a function of the reciprocal temperature for an intrinsic semiconductor. Calculate the gap energy. (Hint: Use (8.14) and take the ln from the resulting equation.)

[pic]

Extrinsic Semiconductors

7. Calculate the Fermi energy and the conductivity at room temperature for germanium containing 5 x 1016 arsenic atoms per cubic centimeter. (Hint: Use the mobility of the electrons in the host material.)

8. Consider a silicon crystal containing 1012 phosphorous atoms per cubic centimeter. Is the conductivity increasing or decreasing when the temperature is raised from 300° to 350°C? Explain by giving numerical values for the mechanisms involved.

9. Consider a semiconductor with 1013 donors/cm3 which have a binding energy of 10 meV.

(a) What is the concentration of extrinsic conduction electrons at 300 K?

(b) Assuming a gap energy of 1 eV (and m*≡mo)， what is the

concentration of intrinsic conduction electrons?

(c) Which contribution is larger?

10. The binding energy of a donor electron can be calculated by assuming

that the extra electron moves in a hydrogen-like orbit. Estimate the donor binding energy of an n-type impurity in a semiconductor by applying the modified equation (4.18a)

[pic]

where ε = 16 is the dielectric constant of the semiconductor. Assume m* = 0.8 m0. Compare your result with experimental values listed in Appendix 4.

11. What happens when a semiconductor contains both donor and acceptor impurities? What happens with the acceptor level in the case of a predominance of donor impurities?

Semiconductor Devices

12. You are given a p-type doped silicon crystal and are asked to make an

ohmic contact. What material would you use?

13. Describe the band diagram and function of a p-n-p transistor.

14. Can you make a solar cell from metals only? Explain!

* 15. A cadmium sulfide photodetector is irradiated over a receiving area of 4 x 10-2 cm2 by light of wavelength 0.4 x 10-6 m and intensity of 20 W m-2

(a) If the energy gap of cadmium sulfide is 2.4 eV， confirm that electron-hole pairs will be generated.

(b) Assuming each quantum generates an electron-hole pair, calculate the number of pairs generated per second.

16. Calculate the room-temperature saturation current and the forward

current at 0.3 V for a silver/n-doped silicon Schottky-type diode. Take

for the active area 10-8 m2 and C = 1019 A/m2 K2

17. Draw up a circuit diagram and discuss the function of an inverter made with CMOS technology. (Hint: An enhancement-type p-n-p MOSFET needs a negative gate voltage to become conducting; an

enhancement-type n-p-n MOSFET needs for this a positive gate

voltage.)

18. Draw up a circuit diagram for an inverter which contains a normal-on

and a normally-off MOSFET. Discuss its function.

19. Convince yourself that the unit in (8.26) is indeed the ampere.

20. Calculate the thermal energy provided to the electrons at room

temperature You will find that this energy is much smaller than the

band gap of silicon. Thus, no intrinsic electrons should be in the

conduction band of silicon at room temperature. Still, according to

your calculations in Problem 1, there is a sizable amount of intrinsic

electrons in the conduction band at T = 300 K. Why?

21. Explain the “OR” logic circuit.

22. Calculate the lateral dimensions of a quantum well structure made of GaAs. (Hint: Keep in mind that the lateral dimension has to equal the wavelength of the electrons in this material.) Refer to Section 4.2 and Fig. 4.4(a). Use the data contained in the tables of Appendix 4.

23. Calculate the number of electrons and holes per incident photon, i.e., the quantum efficiency, in a transverse photodiode. Take W = 8µm, L=8 mm, andα= 40 cm-1.

Chapter 9

Problems

1. Calculate the mobility of the oxygen ions in UO2 at 700 K. The diffusion coefficient of O2 at this temperature is 10一13 cm 2 /s. Compare this mobility with electron or hole mobilities in semiconductors (see Appendix 4). Discuss the difference!

(Hint: O2- has two charges!)

2. Calculate the number of vacancy sites in an ionic conductor in which the metal ions are the predominant charge carriers. Assume a room-temperature ionic conductivity of 10一17 1/Ω(cm and an ionic mobility of 10一17 m2/V s. Does the calculated result make sense? Discuss how the vacancies might have been introduced into the crystal.

3. Calculate the activation energy for ionic conduction for a metal ion in an ionic crystal at 300 K. Take Do = 10-3 m2/s and D = 10-17 m2/s

4. Calculate the ionic conductivity at 300 K for an ionic crystal. Assume 6×1020 Schottky defects per cubic meter, an activation energy of 0.8 eV and Do = 10-3 m2/s.

5. Show thatε=εvac/ε [Eq. (9.13)] by combining Eqs. (7.3), (9.9), and (9.11) and their equivalents for vacuum.

6. Show that the dielectric polarization is P = (ε－1) ε0ε. What values do P and D have for vacuum?

7. Show that εε0ε= q/A [Eq. (9.14)] by combining some pertinent equations.

Chapter 10

Problems

1. Complete the intermediate steps between (10.5) and (10.6)

2. Calculate the conductivity from the index of refraction and the damping constant for copper (0.14 and 3.35, respectively; measurement at room temperature andλ= 0.6µm). Compare your result with the conductivity of copper (see Appendix 4). You will notice a difference between these conductivities by several orders of magnitude. Why? (Compare only the same units!)

3. Express n and k in terms of ε and σ (or ε1 and ε2) by using ε=n2－k2 and σ=4πε0nkν. (Compare with (10.15) and (10.16).)

4. The intensity of Na light passing through a gold film was measured to be about 15% of the incoming light. What is the thickness of the gold film? (λ= 589 nm; k = 3.2. Note: I=ε2.)

5. Calculate the reflectivity of silver and compare it with the reflectivity of flint glass. (n =1.59). Use λ= 0.6µm.

6. Calculate the characteristic penetration depth in aluminum for Na light (λ= 589 nm; k = 6)

7. Derive the Hagen-Rubens relation from (10.29). (Hint: In the IR region [pic] can be used. Justify this approximation.)

8. The transmissivity of a piece of glass of thickness d = 1 cm was measured at A = 589 nm to be 89%. What would the transmissivity of this glass be if the thickness were reduced to 0.5 cm?

Chapter 11

Problems

1. Calculate the reflectivity of sodium in the frequency ranges v > v 1 and v < v 1using the theory for free electrons without damping. Sketch R versus frequency.

2. The plasma frequency, v 1, can be calculated for the alkali metals by assuming one free electron per atom, i.e., by substituting for Nr the number of atoms per unit volume (atomic density, Na). Calculate v 1 for potassium and lithium.

3. Calculate Neff for sodium and potassium. For which of these two metals is the assumption of one free electron per atom justified?

4. What is the meaning of the frequencies v1 and v2? In which frequency ranges are they situated compared to visible light?

5. Calculate the reflectivity of gold at [pic] from its conductivity. Is the reflectivity increasing or decreasing at this frequency when the temperature is increased? Explain.

6. Calculate v1 and v2 for silver ([pic]free electrons per cubic centimeter).

7. The experimentally found dispersion of NaCl is as follow:

|λ[μm] 0.3 0.4 0.5 0.7 1 2 5 |

|n 1.607 1.568 1.552 1.539 1.532 1.527 1.519 |

Plot these results along with calculated values obtained by using the equations of the“bound electron theory”assuming small damping. Let

[pic] and [pic].

8. The optical properties of an absorbing medium can be characterized by various sets of parameters. One such set is the index of refraction and the damping constant. Explain the physical significance of those parameters, and indicate how they are related to the complex dielectric constant of the medium. What other sets of parameters are commonly used to characterize the optical properties? Why are there always “sets”of parameters?

9. Describe the damping mechanisms for free electrons and bound electrons.

10. Why does it make sense that we assume one free electron per atom for the alkali metals?

11. Derive the Drude equations from (11.45) and (11.46) by setting v0→0.

12. Calculate the effective number of free electrons per cubic centimeter and per atom for silver from its optical constants ([pic]and [pic] at 600 nm) (Hint: Use the free electron mass.) How many free electrons per atom would you expect? Does the result make sense? Why may we use the free electron theory for this wavelength?

13. Computer problem. Plot (11.26), (11.27), and (10.29) for various values of v1 and v2. Start with [pic]and[pic].

14. Computer problem. Polt (11.45), (11.46), and (10.29) for various values Na,[pic], and[pic]and[pic]and vary[pic] between 100 and 0.1.

15. Computer problem. Plot (11.51), (11.52), and (10.29) by varying the parameters as in the previous problems. Use one, two or three oscillators. Try to“fit”an experimental curve such as the ones in Figs. 13.10 or 13.11.

Chapter12

Problems

1. What information can be gained from the quantum mechanical treatment of the optical properties of metals which cannot be obtained by the classical treatment?

2. What can we conclude from the fact that the spectral reflectivity of a metal (e.g., copper) has “structure”?

3. Below the reflection spectra for two materials A and B are given.

(a) What type of material belongs to reflection spectrum A, what type

to B? (Justify). Note the scale difference!

(b) For which colors are these (bulk) materials transparent?

(c) What is the approximate threshold energy for interband transitions

for these materials?

(d) For which of the materials would you expect intraband transitions

in the infrared region? (Justify.)

(e) Why do these intraband transitions occur in this region?

[pic]

4. What is the smallest possible energy for interband transitions for aluminum? (Hint: Consult the band diagram in Fig. 5.21.)

5. Are intraband transitions possible in semiconductors at high temperatures?

Chapter 13

Problems

1. Calculate the difference in the refractive indices which is necessary in order that an asymmetric waveguide operates in the zeroth mode. Take λ0 = 840 nm, t = 800 nm, and n2 = 3.61.

2 .How thick is the depletion layer for an electro-optical waveguide when the index of refraction (n3= 3.6) increases in Medium 2 by 0.l %? Take nl = 1, λ0 =1.3µm and zeroth-order mode.

3. Calculate the angle of total reflection in (a) a GaAs waveguide (n = 3.6), and (b) a glass waveguide (n =1.5) against air

4. Of which order of magnitude does the doping of an electro-optical waveguide need to be in order that the index of refraction changes by one-tenth of one percent? Take n3= 3.6, m* = 0.067 m0, and A =1.3µm.

5. Calculate the free carrier absorption loss in a semiconductor assuming n = 3.4, m* = 0.08 mo,λ0 =1.15µm, Nf = 1018 cm-3, and µ= 2*103 cm2/Vs.

6. Show that the energy loss in an optical device, expressed in decibels per centimeter, indeed equals 4.3α.

7. Calculate the necessary step height of a “bump” on a compact disk in order that destructive interference can occur. (Laser wavelength in air, 780 nm; index of refraction of transparent polymeric materials, 1.55)

8. Calculate the gap energy and the emitting wavelength of a GaAs laser that is operated at 100(C. Take the necessary data from the tables in Appendix 4 and Table 19.2.

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