Here are the details of the grading. All the rubrics used for the grading of this exam are listed here.
Part 1 (10 points) : kB T for RT? 26 meV or 25.85 meV.
100 %: practically perfect
75 %: missing unit, otherwise correct
50 %: 1000 times too small an answer.
50 %: 10 times too large an answer.
25 %: correct starting formula, but no or little development from it
Part 2 (10 points) : photon wave vector for 1 eV = 1/(1973 Angstrom)
100 %: practically perfect
80 %: incorrect unit, otherwise correct
50 %: correct formula, but did not know where to plug in 1 eV
0 %: answer missing, or completely incorrect
Part 3p1 (5 points) : KE identified correctly as p^2/(2m), not mc^2 or mc^2 + p^2/(2m)
100 %: practically perfect
0 %: answer missing, or completely incorrect
Part 3p2 (5 points) : p^2/(2m) = 1 eV: k = p/hbar; k = 0.512 Angstrom^{-1}
100 %: practically perfect
90 %: correct except for a (small) numerical error
25 %: incorrectly handled formula, no numeric answer
0 %: answer missing, or completely incorrect
Part 4 (10 points) : Definitions of metal, insulator, semi-metal, semi-conductor
100 %: practically perfect
75 %: metal and semi-metal are not distinguished properly
75 %: practically perfect except for semi-metal
65 %: semi-metal not described or incorrectly described, semiconductor vaguely described
50 %: semi-metal and semi-conductor incorrectly defined
Part 5 (10 points) : Degenerate and non-degenerate semi-conductors
100 %: practically perfect
75 %: definitions correct but rationale missing
75 %: rationale good (quantum vs. classical statistics) but definitions in terms of EF pos missing
60 %: definitions with an error (3kT or O(kT) missing) and rationale missing
0 %: answer missing, or completely incorrect
Part 6 (10 points) : np = ni pi = ni^2 --> n = 3.33 x 10^5 cm^{-3}
100 %: practically perfect
90 %: correct except for a (small) numerical error
75 %: missing unit, otherwise correct
25 %: correct starting formula, but no or little development from it
0 %: answer missing, or completely incorrect
Part 7a (10 points) : fcc lattice
100 %: practically perfect
Part 7b (10 points) : Si crystal = two fcc lattices with relative displacement = (a,a,a)/4
100 %: practically perfect
95 %: would have been perfect, were it not for some errors in the notation
90 %: correct except for a (small) numerical error
90 %: correctly identified tetrahedron motif but did not clearly describe the crystal
50 %: mentioned "tetrahedron" but did not show it (or incorrectly showed it) or explain the crystal
50 %: Said "made up of fcc lattices" without specifying the relative displacement.
33 %: simply said that Si forms an fcc, or things to that effect
25 %: simply said "cubic structure" with no mor details
0 %: answer missing, or completely incorrect
Part 7c (10 points) : (4 + 4)/a^3 = 8/a^3
100 %: practically perfect
80 %: counting slightly incorrect
50 %: correct answer but incorrect or missing explanation
50 %: dimensionally correct, counting incorrect
25 %: correct starting formula, but no or little development from it
0 %: answer missing, or completely incorrect
Part 7d (10 points) : (111) plane identification
100 %: practically perfect
0 %: answer missing, or completely incorrect
Part 7e (10 points) : (111) plane 2D diagram -- hexagonal lattice
100 %: practically perfect
33 %: hexagonal-like diagram with superflous or missing atoms
0 %: answer missing, or completely incorrect
Part 7f (10 points) : (2 x 4) / (1/2 * sqrt(2)a * sqrt(3/2)a) = 16/(sqrt(3) a^2)
50 %: numerical error, double counting missing
0 %: answer missing, or completely incorrect
Part 8a (10 points) : Se is a donor when replacing In or Sb
100 %: practically perfect
50 %: Logic OK, Se was mistaken as Si!
Part 8b (15 points) : Se replacing Sb, E_D = -0.66 meV relative to E_C
80 %: Derivation partial, answer correct
70 %: Derviation missing, answer correct
50 %: Description missing, answer incorrect due to an incorrect Z value used
50 %: Derivation partial, anwer incorrect
25 %: correct starting formula, but no or little development from it
0 %: answer missing, or completely incorrect
Part 8c (12 points) : E_v, E_c, E_D, E_F (T = 0), E_F (T = 300 K), E_i in a diagram
84 %: All correct except one (EF at T = 0 or EF at T = 300 or missing E_i)
80 %: incorrect unit, otherwise correct
67 %: Two of them are incorrect (E_F (T = 0) and E_i, for example)
50 %: E_v, E_c, E_i correct, others incorrect
33 %: E_c and E_v correct. All others are incorrect.
0 %: answer missing, or completely incorrect
Part 8d (13 points) : r = 640 Angstroms
100 %: practically perfect
80 %: Derivation partial, answer correct
60 %: Correct answer, but r was not derived.
25 %: (nearly) correct starting formula, but no or little development from it
0 %: answer missing, or completely incorrect
Part 8e (10 points) : High mobility since mass is small
100 %: practically perfect
50 %: correct answer, but incorrect reason (low binding energy, e.g.)
0 %: answer missing, or completely incorrect
Part 8f (10 points) : N_D = 4.8e14 cm^{-3} --> impurity band starts to form
70 %: N_D was indicated correctly, but the consequence not discussed.
50 %: correct qualitative statement (hopping) but no quantitative estimate of N_D
0 %: answer missing, or completely incorrect
Part 9 (10 points) : 1d tight binding band -- k=0 is min (bonding) while k=pi/a is max (anti-bonding).
50 %: sketched wave functions correctly but did not discuss the reason why the energy is lower or higher.
50 %: argued based on the epsilon(k), but without explaining why t > 0.
33 %: argument based on epsilon(k), which is presented incorrectly.
0 %: answer missing, or completely incorrect
Part 10 (10 points) : Fermi level aligns because there should not be any net particle current.
100 %: practically perfect
75 %: Analogy with the pressure equilbriation of two gas systems, but the exact correspondence (p <-> EF, V <-> N) is missing.
70 %: Good overall argument, but energy = energy (T) only (energy = energy (T,N,V)) and the "chemical potential" was never mentioned.
60 %: EF will become equalized but without mentioning "chemical potential" or "(free) energy"
60 %: mentioned chemical potential, but some misconception ("until the concentration is equivalent" e.g.)
50 %: Definition of the Fermi level approximately correct. But no discussion regarding the chemical potential nature.
50 %: described energy and entropy but did not clearly define the chemical potential
50 %: Gave the correct definition of the Fermi level, or the chemical potential. No more progress though.
33 %: Did not mention "chemical potential" and a simple restatement "Fermi levels must be the same" or something like it
33 %: Did not mention "chemical potential" and expressed misconception "particle flows until the concentration is equal."
0 %: answer missing, or completely incorrect
Part 11 (10 points) : resistance, conductance, resistivity, conductivity
80 %: vague or incorrect definitions for two, otherwise perfect
65 %: vague but correct definitions for two quantities and correct units for all
50 %: Resistance and resisitivity are good, but not the other two.
40 %: two good definitions, two vague definitions, no units given.
40 %: two good definitions and one good unit
35 %: vague definitions (how easy current flows or something like it) and two units correct
25 %: no or incorrect definitions, two units correct
25 %: one good definition, two vague definitions, no units given
20 %: one good definition, one vague defintion, no units given
20 %: one unit correct, with incorrect or vague definitions
0 %: answer missing, or completely incorrect
Part 12 (10 points) : electron mass -- light -- wave + infinite reflections
95 %: crystal momentum, dispersion relation, and effective mass all good, except not ever mentioning "reflection" or "Bragg diffraction."
50 %: mentioned the correlation between "energy dispersion/function" and "effective mass"
50 %: wave nature or "quality very different from free electron" invoked, but reflected waves (Bragg-diffracted waves) not mentioned
0 %: answer missing, or completely incorrect
Part 13 (10 points) : hole -- absence of electron of an otherwise filled band -- Hall, or thermo-power
100 %: practically perfect
90 %: all good except no mentioning of a "filled band" (or at least a "filled something")
50 %: good qualitative definition of hole, but no discussion of how to distinguish hole from electron
33 %: hole is the absence of an electron, with no mentioning of referencing to the filled band and no good valid examples of how to distinguish hole from electron
0 %: answer missing, or completely incorrect
Part 14 (10 points) : tunneling -- through the Coulumb potential barrier, makes the electron go explore all atoms in a crystal very fast providing a prerequisite condition for conduction
100 %: practically perfect
80 %: all good except did not specify the nature of the energy barrier.
33 %: tunneling is "going through a barrier," with no, or incorrect, explanation of what the barrier is and subsequently why the tunneling is essential for conduction
0 %: answer missing, or completely incorrect