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ENEE 313, Fall. ’08Homework II - Due March 30, 20091. Examine the dispersion relation (E − k relationship, or band structure) for a semiconductorgiven in the figure above. The dashed line represents the pure parabolic E − k relationship ofa free electron, while the solid lines show the allowed E levels and the corresponding k valuesfor a semiconductor crystal with periodic potential.(a) (8 pts) The effective mass of an electron in any of the bands can be found from thecurvature (second derivative) of the E − k relationship at the maximum and minimumpoints of the allowed energy bands, where the relationships are approximately parabolic:m∗n=1d2Edk2Using the band diagram plots for the free particle and for each band shown in the diagramabove, comment on the effective masses, comparing the case for the free electron to eachof the band minima and maxima and the bands to each other.(b) (7 pts) This E − k diagram results from the solution to the Schroedinger equation undera periodic potential profile. Some energy values in this solution result in imaginary kvalues, and therefore are forbidden. The wavefunction solutionψ(x) = u(x)eikxgives us statistical information about, for instance, the likely location of the electron. Butthanks to the uncertainity principle, no answer to those questions is precise.There is also always some likelyhood that an electron can be in any of a number of givendifferent s tates. In this case, the overall wavefunction associated with the electron will1be a linear superposition of the individual states’ wavefunctions. Such a combination ofwavefunctions is called a wavepacket. The velocity with which the center of the wavepacketis moving is called the group velocity, vg, given asvg=1¯hdEdkIf a force F is applied on the electron with this wavepacket, the wavepacket energy isgoing to change with dE = F dx = F vgdt. By definition of force as the rate of change ofmomentum, we can also write F = d(¯hk)/dt = m∗n(dvg/dt).Show that we can use the definition of vgand the force expressions given above to derivethe effective mass of the electron with a given E − k relationship as defined in part a) ofthe question above.2. A hypothetical semiconductor material has Eg=1.2 eV, NC= 0.8NVand electron effectivemass m∗n= 0.2m0, where m0is free electron mass.(a) (10 pts.) If the electron and hole concentrations are equal for an intrinsic sample of thissemiconductor, find how far Eimust be from the exact center of the bandgap at roomtemperature T = 300◦K such that kT =0.026 eV.(b) (5 pts.) What is the intrinsic carrier concentration niat room temperature (300◦K)in units of 1/c m3? (Hint: You will need to calculate NCfrom m∗n. Make sure you use theBoltzmann constant and Planck’s constant stated in the correct units and convert the result to1/cm3.)(c) (5 pts.) What is the intrinsic carrier concentration niat 400◦K?(d) (5 pts.) The material is doped such that EFis 0.2 eV below the conduction band. Whattype is the material? What are n0and p0at room temperature?(e) (5 pts.) What are n0and p0at 400◦K?3. (10 pts.) Consider a semiconductor material. At room temperature, the mean thermal velocityfor electrons in the conduction band of this material is 2 × 107cm/sec. If the electron mobilityis 2000 cm2/(V.sec), at what applied electric field does the drift velocity equal mean thermalvelocity? What actually happens to the velocity of the electron as the electric field is increasedclose to and then past this “threshold” field?4. (45 pts.) Streetman and Banerjee, 6th edition: Problems 3.3, 3.7a) (5 pts.), 3.8, 3.12 a) andb),


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UMD ENEE 313 - Homework #2

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