EquationsBand structureLasersOptical PropertiesMagnetism3.15 Electrical, Optical, and Magnetic Materials and Devices Caroline A. Ross Fall Term, 2005 Final Exam (6 pages) Closed book exam. Formulae and data are on the last 4 pages of the exam. This takes 180 min and there are 180 points total. Be brief in your answers and use sketches. 1. Magnetic materials [36] a) Explain the shape of a M-H loop for a piece of single-crystal cobalt of macroscopic size (e.g. a few mm diameter), for H applied parallel to the c axis. In your answer explain how the magnetization varies inside the material as a function of H. What happens if H is perpendicular to c? [15] b) What would the B-H loop look like for H parallel to c? [3] c) The magnetization of a sufficiently small piece of cobalt becomes thermally unstable. For a spherical particle of Co, estimate the size below which this thermal instability occurs. In a thermally unstable particle, what do you expect the coercivity and remanence to be? [9] Data: Co Ku = 5 x 105 J/m3. d) What is the physical basis of the coercivity for the following three materials (one sentence each)? [Note: coercivity data for these materials are given in the data sheet p6] [9] Alnico SmCo5 amorphous Fe-B-Si alloy 2. Magnetic devices [36] We want to build an electromagnet that can pick up a car in a scrapyard. Assume a car has a mass of 2000 kg of which 25% is made of steel (Bs = 1 T, density 2.5 g/cm3). Assume that the maximum force on a magnetic material of moment M and volume V in a field H is given by µoMHV. Suppose the core has a length of 5 m and the gap length is 1 m and there are 10,000 turns of wire around the core. Choose a core material from the list in the data sheet (on p6) and assess the feasibility of building an electromagnet strong enough to do the job. Hint: Start by calculating how much field you would need to pick up the car. 3. Carriers [36] a) In a pn junction, where is drift, diffusion and R&G occuring when the junction is (i) at equilibrium (ii) in reverse bias [12] b) We have a piece of p-type Si as follows: 1light x=0 x Assume that the light is all absorbed very near the surface. Show how you would derive an expression for the electron density as a function of distance, n(x), explaining your reasoning. You do not have to solve the equation but show where it comes from and which terms it contains. Illustrate with a sketch of n vs. x. [18] c) Explain briefly what happens to n(x) after the light is turned off. (however, you do not need to derive the equation relating n to time) [6] 4. Optics [36] Erbium (Er) at concentrations of ~ 1% in a GaN semiconductor has the following energy levels: Ec = 3.4eV Ev 2.3 eV 1.2 eV 0.8 eV a) If you made it into a LED, what colors of light can this Er-doped GaN produce? Draw a sketch of light intensity vs photon energy. What factors influence how bright each color is and the spectral width of the peaks? [16] b) If the GaN were amorphous instead of crystalline, how would this affect your answer? [4] c) We now want to make the Er-doped crystalline material into a laser. It turns out that the transition from 0.8 eV level to the valence band is the slowest. How would you pump it, and what color light would the laser make? If the active region of the laser is 100 microns long, and the laser light has a spectral width that is 2% of the center frequency, what would the output of the laser look like as a function of frequency? [16] 5. Data storage devices [36] a) Describe briefly the operation of a rewritable optical disk based on phase change material. Identify what materials would be suitable for the data storage layer. [up to 3-4 sentences plus 1-2 figures!] [12] b) Describe briefly the operation of a rewritable optical disk based on magnetooptical material. Identify what materials would be suitable for the data storage layer. [up to 3-4 sentences plus 1-2 figures!] [12] c) What limits the data density of each? [6] d) Why is phase change media now more important than magnetooptical media? [6] 2Equations gc (E) dE = mn*√{2mn*(E – Ec)} / (π2 3)  gv (E) dE = mp*√{2mp*(Ev – E)} / (π2 3)  f(E) = 1/ {1 + exp (E – Ef)/kT } n = ni exp (Ef - Ei)/kT, p = ni exp (Ei - Ef)/kT ni = Nc exp (Ei - Ec)/kT where Nc = 2{2πmn*kT/h2}3/2 np = ni2 at equilibrium ni2 = Nc Nv exp (Ev - Ec)/kT = Nc Nv exp (-Eg)/kT Ei = (Ev + Ec)/2 + 3/4 kT ln (mp*/ mn*) Ef - Ei = kT ln (n/ ni) = - kT ln (p/ ni) ~ kT ln (ND / ni) ntype or - kT ln (NA / ni) ptype Drift: thermal velocity 1/2 mv2thermal = 3/2 kT drift velocity vd = μE E = field Current density (electrons) J = n e vd Current density (electrons & holes) J = e (n μn + p μh)E Conductivity σ = J/E = e (n μn + p μh) Diffusion J = eDn ∇n + eDp ∇p Einstein relation: Dn/μn = kT/e R and G R = G = rnp = r ni2 at equilibrium dn/dt = dn/dtdrift + dn/dtdiffn + dn/dtthermal RG + dn/dtother RG Fick’s law dn/dtdiffn = 1/e ∇Jdiffn = Dnd2n/dx2 so dn/dt = (1/e) ∇{Jdrift + Jdiffn} + G – R dn/dtthermal = - nl/τn or dp/dtthermal = - pl/τp τn = 1/rNA, or τp = 1/rND λn = √(τnDn) or λp = √(τpDp) If traps dominate τ = 1/r2NT where r2 >> r pn junction E = 1/εoεr ∫ ρ(x) dx where ρ = e(p – n + ND - NA) E = -dV/dx eVo = (Ef - Ei)n-type - (Ef - Ei)p-type = kT/e ln (nn/np) or kT/e ln (NAND/ni2) E = NAe dp/εoεr = NDe dp/εoεr at x = 0 Vo = (e /2εoεr ) (NDdn2 + NAdp2) dn = √{(2εoεrVo/e) (NA/(ND(ND + NA))} d = dp + dn = √{(2εoεr(Vo + VA)/e) (ND + NA)/ NAND} J = Jo{exp eVA/kT – 1} where Jo = eni2 {Dp/NDτp + Dn/NAτn} Transistor BJT gain β = IC /IB ~ IE /IB = NA,E / ND,B IE = (eDp/w) (ni2/ND,B) exp(eVEB/kT) JFET VSD, sat = (eNDt2/8εoεr) - (Vo + VG) Photodiode and Photovoltaic: I = Io + IG V = I (RPV + RL) I = Io (exp(eV/kT) - 1) + IG Power = IV Wavelength λ (μm) = 1.24/E (eV) Band structure Effective mass: m* = 2(∂2E /∂k2)−1 3Momentum of an electron typically π/a ~ 1010 m-1 Momentum of a photon = 2π/λ ~ 107 m-1 Uncertainly principle ΔxΔp≥  Lasers probability of absorption = B13, stimulated emission = B31, spontaneous emission = A31 N3 = N1 exp (-hν31 /kT) Planck ρ(ν)dν = {8πhν3/c3 }/{exp (hν /kT) - 1} dν B13 …