Stanford University MatSci 152: Principles of Electronic Materials and Devices Spring Quarter, Final Exam, June 8, 2010

Transcription

1 Stanford University MatSci 152: Principles of Electronic Materials and Devices Spring Quarter, Final Exam, June 8, 2010 This is a closed book, closed notes exam. You are allowed two double-sided sheets of paper with your own notes, as well as a calculator and a writing utensil. You should not have any other materials on your desk. The exam consists of 7 problems. You are allowed up to 3 hours to finish the exam (though it shouldn t take that long.) Maximum Score: 100 points Please do not discuss the content of this exam with others until you have received your exam grade. GOOD LUCK, and have a great summer! :0) By signing below, the test-taker agrees that he/she has not violated Stanford s honor code: Signature: Print Name:

2 Problem 1: Conductivity of Metals (15 points) Consider Cu and Ni with their density of states as schematically sketched below. Both have overlapping 3d and 4s bands, but the 3d band is very narrow compared to the 4s band. In the case of Cu the band is full, whereas in Ni, it is only partially filled. a. In Cu, do the electrons in the 3d band contribute to electrical conduction? Explain. b. In Ni, do electrons in both bands contribute to conduction? Explain. c. Do electrons have the same effective mass in the two bands? Explain. d. Can an electron in the 4s band with energy around E F become scattered into the 3d band as a result of a scattering process? Consider both metals. e. Based on the band diagrams, how would you expect the resistivity of Ni to compare with that of Cu? (Note: Ni has two valence electrons and nearly the same density as Cu.) Copper (Cu) Nickel (Ni) g(e) 3d g(e) 3d 4s 4s E E E F E F

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4 Problem 2. Density of States (15 points) a. Derive the density of states g(e) for free electrons in a metal in 2-dimensions. Assume a free particle dispersion relation (E vs. k). Sketch your answer on a plot of g(e) versus E, with energy on the y-axis. b. Using your answer from part (a), sketch the density of states for a 2-dimensional semiconductor. Be sure to include the valence and conduction bands, as well as the bandgap. c. Draw the Fermi-Dirac probability functions for electron and holes in a semiconductor. Be sure to include the appropriate locations of the valence and conduction bands, as well as the Fermi level. d. Assuming that the conduction band is at least a few kt above the Fermi level, we know that Fermi-Dirac statistics can be replaced by Boltzman statistics. Using your answers from parts (a) through (c) above, derive an expression for the number of carriers per unit energy per unit volume in the conduction and valence bands. Sketch your answer. e. The photon energy for the peak emission of a light-emitting diode (LED) corresponds to peak-to-peak transitions in the energy distributions of the electrons and holes in the conduction and valence bands. In a 3D semiconducting LED, the peak emission is at an energy E g +kt. Where is the peak emission energy in a 2D LED?

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6 Problem 3: LEDs and Lasers (10 points) a. Calculate the linewidth of the output radiation from a green LED emitting at 543 nm at T=300K. b. Assume this LED is composed of a GaAs 1-y P y alloy with y>0.45. Generally, these semiconductors are indirect bandgap materials. Why do indirect bandgap semiconductors make poor LEDs? How does nitrogen doping increase radiative recombination in this material? When substituted for P, is nitrogen a donor or acceptor? c. Now calculate the linewidth of a HeNe laser emitting 543 nm light, with an operating gas temperature of 130 o C. This is a so-called GreeNe laser. d. Qualitatively describe the differences in the emission between an LED and a laser.

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8 Problem 4: Solar Cells (20 points) a) A solar cell under an illumination of 1000 W m -2 has a short circuit current I sc of 50 ma and an open circuit voltage V oc of 0.65 V. What are the short circuit current and open circuit voltage when the light intensity is halved? Assume T=300K, η=1.5, and V oc >>ηkt. b) The intensity of light arriving at a point on earth can be approximated by the Meinel and Meinel equation (cosecα ) I = 1.353(0.7) kw m -2 where α is the solar latitude and cosecα = 1/(sinα). As seen in the figure below, the solar latitude α is the angle between the sun rays and the horizon. Around September 23 and March 22nd, the sun rays arrive parallel to the plane of the equator. What is the maximum power available for a photovoltaic device panel of area 1 m 2 if it s conversion efficiency is 10 percent? c) Laboratory tests on a particular Si pn junction solar cell at 27 C specify an open circuit output voltage of 0.45 V and a short circuit current of 400 ma when illuminated directly with light of intensity 1 kw m -2. The fill factor for the solar cell is This solar cell is to be used in a portable equipment application in Canada at a geographical latitude φ of 63. Calculate the open circuit output voltage and the maximum available power when the solar cell is used at noon on September 23 when the temperature is 10 C. What is the maximum current this solar cell can supply to electronic equipment? How does this current compare with the laboratory tests? (Note: α + φ = π/2, and assume η = 1 and I o = n i2 ) atomic mass (AM) solar illumination Horizon AM0 α Equator φ Sun s Rays Atmosphere θ AM1 α AM(sec α ) Earth 23rd September and 22nd March

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10 Problem 5. Hall Effect Devices (15 points) Hall effect measurements can be used to determine the sign of the carrier in a material. In a semiconductor, the Hall coefficient will involve not only the electron and hole concentrations n and p, but also the electron and hole drift mobilities, μ e and μ h. Assume we are given a piece of Si. Let s pass a current through the sample in the x-direction and apply a magnetic field in the z-direction, as shown below. a. On the diagram, clearly label vectors for the electrostatic force, the Lorentz force, and the velocity of both electrons and holes. Include both the direction and magnitude. B z y x J x hole + - electron J x V In steady state, we know that the net current along the y-direction has to be zero. Additionally, we know that the net force acting on a charge carrier, F net =qv/μ (where q is the carrier charge, v is the particle velocity, and μ is the carrier mobility) can be equated to the sum of the electrostatic and Lorentz forces. b. For intrinsic Si with n i =1x10 10 cm -3, μ e =1350 V -1 s -1, and μ h =450 V -1 s -1, calculate the Hall coefficient, R h =E y /J x B z. c. The Hall coefficient for Cu is -0.55x10-10 m 3 A -1 s -1. Based on your answer from part (b), would you expect typical Hall effect devices to use metals or semiconductors? Why? d. The Hall coefficient of a semiconductor depends both on the carrier concentration and the mobility. i) Sketch the temperature dependence of the electron concentration in an n-type semiconductor. Explain which processes give rise to each region. ii) Sketch the temperature dependence of mobility, explaining the temperature dependence that you find. Hint: it may be easiest to plot log(n) or log(μ) versus 1/T.

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12 Problem 6: Metal-Semiconductor Contacts (15 points) a. Consider a contact between a metal and n-type semiconductor. If Φm< Φn, what type of junction is it (Schottky or Ohmic)? Draw the band diagram, clearly labeling all energy axes. Also sketch the IV characteristics. b. Consider a contact between a metal and p-type semiconductor. If Φm< Φp, what type of junction is it (Schottky or Ohmic)? Draw the band diagram, clearly labelling all energy axes. Also sketch the IV characteristics. c. Sketch the IV curve for the metal/n-semiconductor/metal junctions shown below, when the contacts are i) both Ohmic and ii) both Schottky. metal n-semiconductor metal

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14 Problem 7: Devices and their IV-characteristics (10 points) a. Qualitatively explain the operation of an n-mosfet. Draw the IV curve for this device both below and above threshold. b. Thermoelectric devices or Peltier coolers enable small volumes to be cooled by direct currents. Draw the band diagram for a metal/n-type semiconductor contact that would enable peltier cooling. Be sure to label the Fermi levels of the metal and semiconductor, as well as the conduction and valence bands. Also indicate the direction of the applied current and the direction of electron flow. Draw the IV curve for this junction both in forward and reverse bias.

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