Semiconductors School of Physics and Nanotechnology Yachay Tech Professor: Ernesto Medina Homework 1 1) Effective mass: a) A simplified E versus k curve for an electron in the conduction band is given in Fig. I. The value of a is 10 Å. Determine the relative effective mass m*/︎m0 b) Compute the hole mass ratio m*/︎m0 using Fig. II I II 2) Density of states in d dimensions: a) Derive the density of states as a function of the energy for materials with a free dispersion in an arbitrary dimension d. Include the full calculation of the prefactors. b) For a free electron in 3d, calculate the density of quantum states (#/cm3) over the energy range of (i) 0<︎ E < 2.0 eV and (ii) 1<︎ E<︎ 2 eV. c) Determine the number (#/cm3) of quantum states in Si between Ec and Ec ︎+ kT at T ︎ 300 K. 3) Fermi distribution: Assume the Fermi energy level is 0.30 eV below the conduction band energy Ec. Assume T = 300 K. (a) Determine the probability of a state being occupied by an electron at E =︎ Ec ︎+kT/︎4. (b) Repeat part (a) for an energy state at E ︎= Ec ︎+ kT. 4) Band gap I: The bandgap energy in a semiconductor is usually a slight function of temperature. In some cases, the bandgap energy versus temperature can be modeled by αT 2 Eg = Eg(0) − β +T where Eg(0) is the value of the bandgap energy at T = 0 K. For silicon, the parameter values are Eg(0) = ︎1.170 eV, α = ︎ ︎4.73 x︎ 10︎-4 eV/K, and β = ︎636 K. Plot Eg versus T over the range 0 <︎ T <︎ 600 K. In particular, note the value at T ︎ 300 K. 5) Band gap II: (a) The forbidden bandgap energy in GaAs is 1.42 eV. (i) Determine the minimum frequency of an incident photon that can interact with a valence electron and elevate the electron to the conduction band. (ii) What is the corresponding wavelength? (b) Repeat part (a) for silicon with a bandgap energy of 1.12 eV. 6) Effective masses: The E versus k diagrams for a free electron (curve A) and for an electron in a semiconductor (curve B) are shown in the Figure. Sketch (a) dE︎/ dk versus k and (b) d2E/︎dk2 versus k for each curve. (c) What conclusion can you make concerning a comparison in effective masses for the two cases? 4) Effective mass II: The energy-band diagram for silicon is shown in Figure 3.25b. The minimum energy in the conduction band is in the [100] direction. The energy in this one-dimensional direction near the minimum value can be approximated by E︎=E0 ︎+E1 cos︎(k︎-k0) where k0 is the value of k at the minimum energy. Determine the effective mass of the particle at k ︎ k0 in terms of the equation parameters. 5) Kronig-Penney Model: In Ashcroft-Mermin follow through the solution to the general Kronig-Penney model. Derive all the intermediate steps of the calculation, and draw figure 8.6 with your own plotting tools. NOTE: The exercises requested in class will give you additional points. Please add them to your HW. Please send me a note if you find typos in the homework, so I can correct the exercises for all to benefit.