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EE 535 Mid-Tem Exam Total points: 60

Name: ___________________________

Problem 1: Given that the rock salt crystal (density = 2180 kg/m3, Mol. Wt. of NaCl = 58.5) has a f.c.c. lattice, calculate its lattice constant. (1 pt) a) 5.67 Å b) 5.65 Å c) 5.63 Å d) 5.61Å Problem 2: Fill in the blanks: We know that the {100} family has six equivalent planes. The {110} family has _____ equivalent planes, and the {111} family has _____ equivalent planes. (1 pt) Problem 3: Which type of solid has the weakest bonding force or the least cohesive energy (i.e. energy needed break the crystal into free atoms/ions). (1 pt) a) Hydrogen b) Metallic c) Covalent d) Molecular Problem 4: Suppose there is a discontinuity in the slope of the wavefunction )(xnψ at 0xx = . The contribution of this kink to the expectation value of the particle's kinetic energy is: (1 pt)

..EK∆ = ___________________.

Problem 5: Circle the correct statement/s for 3D reciprocal lattices (RL) and direct lattices (DL).

a) First Brillouin zone is defined by the primitive vectors of the DL. (1 pt) b) The volume of a RL is equal to the inverse of the DL volume. c) The RL basis vectors are perpendicular to the corresponding DL basis vectors. d) X-ray diffraction relates the wave vector to the angle of X-ray reflection from the crystal planes. Problem 6: The diagram shows a 2D square lattice with the coupling between 9 identical atoms represented by 8 force constants (κ1, κ2... κ8). Which of the following is/are true? (1 pt)

a) κ1 = κ5 b) κ1 = κ4 c) κ2 = κ8 d) κ1 = κ7 Problem 7: Circle the correct statement/s:

a) When using the Schrodinger equation, the particle’s energy E can be zero. b) A silicon sample (with no voltages applied) left on planet Jupiter is in a state of equilibrium. c) The Kronig-Penny model can be used to describe the electron motion in amorphous silicon. d) none of the above. (1 pt)

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Problem 8: Consider three semiconductors (A, B, C) having the same valence band but different conduction bands. Circle one or more of the correct answers below: (0.25x8 = 2 pt)

a) Which material has the smallest conduction-band effective mass?

Material A Material B Material C

b) Which material/s requires conservation of energy during optical emission? Material A Material B Material C

c) Which material/s requires momentum conservation during optical absorption? Material A Material B Material C

d) During optical absorption, the effective mass (its absolute value) of an electron increases in: Material A Material B Material C

e) The density of energy states in the conduction band is largest for: Material A Material B Material C

f) The largest offset of the intrinsic energy-level Ei from the energy midgap EG/2 is in: Material A Material B Material C

g) Which material is preferred for making very high-speed digital circuits? Material A Material B Material C

h) Which material is preferred for making high-power devices? Material A Material B Material C

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Problem 9: Roughly sketch any three of the following parameters versus temperature T on the same graph. (no need for values): (3x1 = 3 pt) a) Effective density of states in the valence band Nv

b) The optical energy during photoemission in GaAs c) The thermal velocity of electrons in silicon d) Ratio of the hole concentration to the donor concentration (p/ND) in a n-doped silicon sample.

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PROBLEMS 10 THROUGH 19: ANSWER ANY EIGHT OF THE TEN QUESTIONS 6x8 = 48pt Problem 10: Consider a particle in an infinite one-dimensional potential well of width α. Find the expectation values of position <x> and <x2> for the wave function ( )απαψ /sin/2)( xnxn = . Hence find the uncertainty ∆x in finding the particle's exact position. (2+2+2 = 6 pt)

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Problem 11: The E-k relationship characterizing an electron confined to a 2-D surface layer is of the form

2

22

1

22

22 mk

mk

EE yxC

+=− ....where m1 ≠ m2.

An electric field ξ is applied in the x-y plane at a 120° to the positive x-axis as shown in the figure. Taking the electron to be initially at rest, derive an expression for its radial motion r in the x-y plane at a time t. The answer should be of the form zyxr {}{}{} ++=

. (2+2+2 = 6 pt)

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Problem 12: Suppose a particle of mass m* is confined in 3D potential box has impenetrable walls (quantum box) and the dimensions of the box are such that cba 2== . a) Write an expression for the energy level lmnE , where l, m, n are integers. b) What is the value of the energy level 411E ? c) What is the degeneracy of the energy level 411E (i.e. number of states have the same energy value)? (1+2+3 = 6 pt)

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Problem 13: Consider a linear diatomic chain of atoms (masses M and m) with nearest (spring constant α) and next-nearest neighbors (spring constant β) coupling. Select the distances as X2n = 2na for mass M and X2n+1 = (2n+1)a for mass m, as shown in the figure. Assuming that the displacements are of the form u2n = Aexp{i(2nka-ωt)} for mass M and u2n+1 = Aexp{i((2n+1)ka-ωt)} for mass m, show that the frequency of lattice vibration ω(k) can be written in the following form: (1+1+4 = 6 pt) Mω2-2α-2β.{1-cos(2ka)} 2αcos(ka) A = 0 2αcos(ka) mω2-2α-2β.{1-cos(2ka)} B

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Problem 14: The figure shows the six constant energy ellipsoids for the silicon conduction band. The k-values at the conduction-band minima are:

{2π/a}(0.85, 0, 0); {2π/a}(0, 0.85, 0); {2π/a}(0, 0, 0.85) {2π/a}(-0.85, 0, 0); {2π/a}(0, -0.85, 0); {2π/a}(0, 0, -0.85)

The heavy-hole (hh) and light-hole (lh) valence-band maxima are at k =0. Assume that ml

* is the longitudinal effective mass, mt* is the

effective transverse mass, mhh* is the effective heavy-hole mass, and

mlh* is the effective light-hole mass.

Write the energy-momentum relationships for the valence-band (hh and lh) and the conduction band (any 4 of the 6 minima given). The answer should have 6 E-k expressions. (1x6 = 6 pt)

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Problem 15: a) Find the transmission probability T for a square well potential with the particle energy E > 0 : 0)( UxU −= , for α≤x

0)( =xU , for α>x

b) In the limiting case of a one-dimensional delta potential well, )()( 0 xVxU δ−= , what does the above solution of the transmission probability T reduce to?

c) What is the square well width 2α for which complete transmission (T =1) occurs? (3+2+1 = 6 pt)

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Problem 16: The figures on the left {1(a) and 2(a)} show the conduction and valence band quantum wells in a semiconductor heterojunction, respectively. By applying an electric field, we can bend these bands and change the properties of the device (figure 1(b) and 2(b)). Figure 3 shows a triangular potential well created by applying a uniform electric field to a heterojunction. The first three energy levels are labeled as E1, E2, and E3. In all the five figures, please draw the wave functions at the first three energy levels. (2+2+2 = 6 pt)

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Problem 17: a) The figure on the left shows a doped semiconductor. Draw the profile of the carrier distributions in the conduction and valence bands (n(E) and p(E)).

b) The figure on the right shows a typical electron distribution n(E) in the conduction band as a function of energy E. Write an expression for this n(E) at a particular energy level E.

c) Show that the kinetic energy at which the peak electron concentration occurs is at Emax = EC - 0.5kT and the average kinetic energy is at <E> = EC - 1.5kT. (1+1+4 = 6 pt)

dEEfEg

dEEfEgEEE

TOP

C

TOP

C

E

Ec

E

EcC

∫

∫ −

>=<

)()(

)()()(, ∫

∞

++− =

+Γ=

011

!)1(nn

axn

an

andxex , )()1( nnn Γ=+Γ , π=Γ )2/1(

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Problem 18: Find the potential energy U(x) and the particle energy E, given the following wave function:

xnn eAxx βψ −=)( for x ≥ 0

0)( =xnψ for x ≥ 0 (3+3 = 6 pt)

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Problem 19: Consider a n-type nondegenerate Si sample where AD NN > but 0≠AN . Assume that the system temperature is restricted to the freeze-out and extrinsic ranges. a) Show that AA NN ≅− and AD NNn −≅ + given that n>>p for this n-type sample. b) What is the value of +

DN as the temperature approaches zero, 0→T ? c) What is the limiting position of the Fermi level FE as 0→T ? Briefly explain. (2+2+2 =6 pt)

The end