3.15 Electrical, Optical, and Magnetic Materials and Devices Caroline A. Ross Fall Term, 2006

Transcription

1 3.15 Electrical, Optical, and Magnetic Materials and Devices Caroline A. Ross Fall Term, 2006 Exam 2 (6 pages) Closed book exam. Formulae and data are on the last 4 pages of the exam. This takes 80 min and there are 80 points total. Be brief in your answers and use sketches. Assume everything is at 300K unless otherwise noted. 1. [25 points] A JFET is constructed like this: the two gate regions are n+ and the rest of the material is p type. n+ Source p Drain n+ Gate a) Assume that the source is grounded (at zero volts). Draw a sketch of how the current I D flowing out of the drain varies with voltage V D when the gate is at zero volts. Consider both positive and negative values of V D and explain the shape of your graph. (3-4 sentences) [9] b) Now draw another sketch showing again I D vs. V D but this time draw different graphs corresponding to different values of gate voltage V G (both positive and negative). Explain briefly how gate voltage affects the I-V plot. [10] c) Give three reasons why a MOSFET is preferable to a JFET. [6] 2. [33 points] In one of the problem sets we considered two ways of making white light using an LED. Here we will consider a third method. In this new scheme, the device looks like this: cup coated inside with phosphor LED contacts to the LED

2 a) What is the purpose of the phosphor-coated cup in this device? Explain how you can get white light. [6] b) What color LED would you use? Choose a possible material and substrate for the LED, explaining your choice. [7] (some of the data at the end of the exam may be helpful) c) Draw a possible band structure for your LED, in the unbiased case, and explain how bias affects the band structure. [7] d) Explain concisely the differences between the spectral output of an LED, such as the one you just drew, compared with a semiconductor laser. [6] e) If your LED is only 0.1 mm long, how does this affect its output? [7] 3. [22 points] We are designing a photovoltaic system. The solar cell we have available produces an output as shown below: its internal resistance is 0.2 ohms. Current (A) Voltage (V) a) Estimate the maximum power we can produce from the solar cell. [6] b) What load resistance would you use to maximize the output power? [4] c) Mention three methods to improve the efficiency of a solar cell. (1 sentence each). [6] d) Solar cells can be made from amorphous silicon, and are commonly used in devices such as calculators. Explain the reason for using amorphous Si in place of crystal Si (3-4 sentences). [6]

3 Data and Formulae Energy Gap and Lattice Constants GaP AlP AlAs Energy gap (ev) Si GaAs (D) InP(D) AlSb Wavelength (microns) Ge GaSb (D) Indirect band gap Direct band gap (D) InAs (D) InSb (D) Lattice constant (nm) 3.5 ZnS MnSe MnTe Band gap energy (ev) ZnSe AlP GaP GaAs Si CdS lnp CdSe ZnTe AlSb CdTe Ge GaSb InAs InSb Lattice parameter (nm) Band gap energies of II - VI compound and alloy semiconductors.

4 Properties Si GaAs SiO 2 Ge Atoms/cm 3, molecules/cm 3 x Structure Lattice constant (nm) Density (g/cm 3 ) Relative dielectric constant, ε r Permittivity, ε = ε r ε o (farad/cm) x diamond zincblende a amorphous 2.27 a Expansion coefficient (dl/ldt) x (10-6 K) Specific Heat (joule/g K) Thermal conductivity (watt/cm K) Thermal diffusivity (cm 2 /sec) Energy Gap (ev) Drift mobility (cm 2 /volt-sec) Electrons Holes Effective density of states (cm -3 ) x Conduction band Valence band ~ Intrinsic carrier concentration (cm -3 ) 1.45 x x 10 6 Properties of Si, GaAs, SiO 2, and Ge at 300 K Useful equations g c (E) de = m n * {2m n *(E E c )} / (π 2 h 3 ) (h = h-bar) g v (E) de = m p * {2m p *(E v E)} / (π 2 h 3 ) f(e) = 1/ {1 + exp (E E f )/kt } n = n i exp (E f - E i )/kt, p = n i exp (E i - E f )/kt n i = N c exp (E i - E c )/kt where N c = 2{2πm n *kt/h 2 } 3/2 np = n 2 i at equilibrium n 2 i = N c N v exp (E v - E c )/kt = N c N v exp (-E g )/kt E i = (E v + E c )/2 + 3/4 kt ln (m p */ m n *) E f - E i = kt ln (n/ n i ) = - kt ln (p/ n i ) ~ kt ln (N D / n i ) ntype or - kt ln (N A / n i ) ptype Drift: thermal velocity 1/2 mv 2 thermal = 3/2 kt drift velocity v d = μe E = field Current density (electrons) J = n e v d Current density (electrons & holes) J = e (n μ n + p μ h )E Conductivity σ = J/E = e (n μ n + p μ h ) Diffusion J = ed n n + ed p p Einstein relation: D n /μ n = kt/e R and G 2 R = G = rnp = r n i at equilibrium dn/dt = dn/dt drift + dn/dt diffn + dn/dt thermal RG + dn/dt other RG Fick s law dn/dt diffn = 1/e J diffn = D n d 2 n/dx 2 so dn/dt = (1/e) {J drift + J diffn } + G R dn/dt thermal = - n l /τ n or dp/dt thermal = - p l /τ p τ n = 1/rN A, or τ p = 1/rN D L n = τ n D n, or L p = τ p D p. If traps dominate τ = 1/r 2 N T where r 2 >> r

5 pn junction E = 1/ε o ε r ρ(x) dx where ρ = e(p n + N D - N A ) E = -dv/dx ev o = (E f - E i ) n-type - (E f - E i ) p-type = kt/e ln (n n /n p ) or kt/e ln (N A N D /n 2 i ) E = N A e d p /ε o ε r = N D e d p /ε o ε r at x = 0 V o = (e /2ε o ε r ) (N D d 2 n + N A d 2 p ) d n = {(2ε o ε r V o /e) (N A /(N D (N D + N A ))} d = d p + d n = {(2ε o ε r (V o + V A )/e) (N D + N A )/ N A N D } J = J o {exp ev A /kt 1} where J o = en 2 i {D p /N D L p + D n /N A L n } Transistor BJT gain β = I C /I B B ~ IE /I B = N A,E / N D,B I E = (ed p /w) (n 2 i /N D,B ) exp(ev EB /kt) JFET V SD, sat = (en D t 2 /8ε o ε r ) - (V o + V G ) Photodiode and photovoltaic I = I o + I G V = I (R PV + R L ) I = I o {exp ev/kt 1} + I G Power = IV Wavelength λ(µm) = 1.24/E g (ev) Band structure Effective mass: m* = 2 ( 2 E / k 2 ) 1 Momentum of an electron typically π/a ~ m -1 Momentum of a photon = 2π/λ ~ 10 7 m -1 Uncertainly principle ΔxΔp Lasers probability of absorption = B 13, stimulated emission = B 31, spontaneous emission = A 31 N 3 = N 1 exp (-hν 31 /kt) Planck ρ(ν)dν = {8πhν 3 /c 3 }/{exp (hν /kt) - 1} dν B 13 = B 31 and A 31 /B 31 = 8πhν 3 /c 3 (Einstein relations) Cavity modes ν = cn/2d, N an integer. Fibers Attenuation (db) = {10/L} log(p in /P out ) L = fiber length Snell s law: n sin φ = n sin φ Dispersion coefft. D λ = {λ o /c}( 2 n / λ 2 ) λ= λo ps/km.nm σ t = σ λ L D λ

6 PHYSICAL CONSTANTS, CONVERSIONS, AND USEFUL COMBINATIONS Physical Constants Avogadro constant Boltzmann constant Elementary charge Planck constant Speed of light Permittivity (free space) Electron mass Coulomb constant Atomic mass unit Useful Combinations Thermal energy (300 K) Photon energy Coulomb constant Permittivity (Si) Permittivity (free space) Prefixes N A = x particles/mole k = x 10-5 ev/k = 1.38 x J/K e = x coulomb h = x ev.s = x joule.s c = x cm/s ε 0 = 8.85 x farad/cm m = x kg k c = x 10 9 newton-m 2 /(coulomb) 2 u = x kg kt = ev ~ _ 1 ev/40 E = 1.24 ev at λ = µm k c e ev. nm ε = ε r ε 0 = 1.05 x farad/cm ε 0 = 55.3e/V. µm k = kilo = 10 3 ; M = mega = 10 6 ; G = giga = 10 9 ; T = tera = m = milli = 10-3 ; µ = micro = 10-6 ; n = nano = 10-9 ; p = pica = Symbols for Units Ampere (A), Coulomb (C), Farad (F), Gram (g), Joule (J), Kelvin (K) Meter (m), Newton (N), Ohm (Ω), Second (s), Siemen (S), Tesla (T) Volt (V), Watt (W), Weber (Wb) Conversions 1 nm = 10-9 m = 10 A = 10-7 cm; 1 ev = x 10-9 Joule = x erg; 1 ev/particle = kcal/mol; 1 newton = kg force ; 10 6 newton/m 2 = 146 psi = 10 7 dyn/cm 2 ; 1 µm = 10-4 cm inch = 1 mil = 25.4 µm; 1 bar = 10 6 dyn/cm 2 = 10 5 N/m 2 ; 1 weber/m 2 = 10 4 gauss = 1 tesla; 1 pascal = 1 N/m 2 = 7.5 x 10-3 torr; 1 erg = 10-7 joule = 1 dyn-cm