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3.15 Carrier Fundamentals C.A. Ross, Department of Materials Science and Engineering

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3.15 Carrier Fundamentals C.A. Ross, Department of Materials Science and Engineering 3.15 Carrier Fundamentals C.A. Ross, Department of Materials Science and Engineering Reference: Pierret, chapters 1-2. Electron and hole charge: e = 1.6 10-19 C Effective mass: m*, rest mass mo F = -eE = mo dv/dt in vacuum F = -eE = m* dv/dt in solid in Si, mn*/mo = 1.18, mh*/mo = 0.81 at 300K. Intrinsic properties in Si, n = p = 1010 cm- 3 at 300K N = 5 1022 atoms cm-3 Extrinsic properties Donors – group V Acceptors – group III B C N O Al Si P S Ga Ge As Se In Sn Sb Te Tl Pb Bi Po Band diagrams: Ec = conduction band edge, Ev = valence band edge band gaps: Si 1.12 eV diamond 5.4 eV silica 5 eV energies of dopant levels, in meV, in silicon (kT = 26 meV @ 300K) P 45 B 45 As 54 Al 67 Ga 72 Handout 1 Carrier distributions (intrinsic) g(E) dE = density of electron states cm-3 in the interval (E, E+dE), units #/eV.cm-3 2 3 gc (E) dE = mn*√(2mn*(E – Ec)) / (π h ) dE 2 3 gv (E) dE = mp*√(2mp*(Ev – E)) / (π h ) dE In these states, the electrons distribut! e according to Fermi function f(E) = 1/ {1 + exp (E – E )/kT } f ! Number of electrons in the interval (E, E+dE) is therefore f(E)g(E)dE. In a doped semiconductor, the position of Ef with respect to the band gap determines whether there are more electrons or holes. Total number of electrons: by integrating f(E)g(E)dE n = ni exp (Ef - Ei)/kT p = ni exp (Ei - Ef)/kT where ni = Nc exp (Ei - Ec)/kT 2 3/2 Nc = 2{2πmn*kT/h } = ‘effective density of conduction band states’ Ei is the position of the Fermi level in the intrinsic case. Similarly for Nv. Hence 2 np = ni at equilibrium 2 ni = Nc Nv exp (Ev - Ec)/kT = Nc Nv exp (-Eg)/kT Intrinsic case: Ei = (Ev + Ec)/2 + 3/4 kT ln (mp*/ mn*) In a doped material, where n ~ ND or p ~ NA Ef - Ei = kT ln (n/ ni) = - kT ln (p/ ni) ~ kT ln (ND / ni) or - kT ln (NA / ni) n-type p-type Handout 1 Properties Si GaAs SiO2 Ge Atoms/cm3, molecules/cm3 x 1022 5.0 4.42 2.27a Mayer and Structure diamond zincblende amorphous Lau, Lattice constant (nm) 0.543 0.565 Electronic Density (g/cm3) 2.33 5.32 2.27a Materials Science Relative dielectric constant, er 11.9 13.1 3.9 -12 Permittivity, e = ereo (farad/cm) x 10 1.05 1.16 0.34 Expansion coefficient (dL/LdT) x (10-6 K) 2.6 6.86 0.5 Specific Heat (joule/g K) 0.7 0.35 1.0 Thermal conductivity (watt/cm K) 1.48 0.46 0.014 Thermal diffusivity (cm2/sec) 0.9 0.44 0.006 Energy Gap (eV) 1.12 1.424 ~9 0.67 Drift mobility (cm2/volt-sec) Electrons 1500 8500 Holes 450 400 Effective density of states (cm-3) x 1019 Conduction band 2.8 0.047 Valence band 1.04 0.7 Intrinsic carrier concentration (cm-3) 1.45 x 1010 1.79 x 106 Properties of Si, GaAs, SiO2, and Ge at 300 K Figure by MIT OCW. Energy Gap and Lattice Constants 2.6 0.477 2.4 AlP 0.517 GaP 2.2 AlAs 0.563 2.0 0.620 1.8 0.689 AlSb 1.6 0.775 1.4 GaAs (D) 0.885 InP(D) 1.2 1.033 Si Energy gap (eV) Energy 1.0 1.240 (microns) Wavelength 0.8 1.550 Ge GaSb (D) 0.6 2.067 Indirect band gap 0.4 InAs (D) 3.100 Direct band gap (D) 0.2 6.200 InSb (D) 0 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 Lattice constant (nm) Handout 1 Figure by MIT OCW. Physical Constants, Conversions, PHYSICALand CONSTANTS, Useful Combinations CONVERSIONS, AND USEFUL COMBINATIONS Physical Constants 23 AvogadroPhysical constant Constants NA = 6.022 x 10 particles/mole Avogadro constant N, = 6.022 x particles/mole Boltzmann constant -5 -23 Boltzmann constant k =k =8.617 8.617 x x10-'eV/K 10 eV/K = =1.38 1.38 x x 10 J/K J/K ElementaryElementary charge charge e =e =1.602 1.602 X x10-l9 10-19 coulomb coulomb Planck constant h = 4.136 x 10-IS eV . s Planck constant -15 . =h =6.626 4.136 x xlo-" 10 joule eV . s Speed of light c = 6.6262.998 xx 1010"-34 cm/s joule .s Permittivity (free space) co = 8.85 X 10-l4 farad/cm Speed of light 10 Electron mass rn c= = 9.1095 2.998 xx 10lo-" cm/skg -14 PermittivityCoulomb (free constant space) kc ε=0 =8.988 8.85 x x lo910 ne~ton-m~/(coulomb)~ farad/cm ElectronAtomic mass mass unit u =m =1.6606 9.1095 x x 10-31kg kg Coulomb constant k = 8.988 x 109 newton-m2/(coulomb)2 Useful Combinations c AtomicThermal mass energyunit (300 K) kT u= = 0.02581.6606 eV x =10 1-27 eV/40 kg Useful CombinationsPhoton energy E = 1.24eVatX = 1 pm Coulomb constant kce2 = 1.44 eV . nm ~_ ThermalPermittivity energy (300(Si) K) E =kT e,e, = 0.0258= 1.05 eV x 10-l21 eV/40 faradlcm PhotonPermittivity energy (free space) q, E= = 55.3e/V 1.24 eV . pmat λ = µm 2 . Coulomb constant kce 1.44 eV nm Prefixes -12 Permittivityk = kilo (Si) = I@; M = mega = εId; = ε Grε 0= = giga 1.05 = x 10'; 10 T =farad/cm tera = 1012 . Permittivitym = milli (free = space)p = micm ε=0 = 55.3e/Vn = nano µm = IW9; Prefixes p = pica = lo-'' k = kilo = 103; M = mega = 106; G = giga = 109; T = tera = 1012 Symbols-3 for Units -6 -9 -12 m = milliAmpere = 10 (A),; µCoulomb = micro (C), = 10Farad; (F)n = nano = 10 ; p = pica = 10 Symbols forGram Units (g), Joule (J), Kelvin (K) Meter (m), Newton (N), Ohm (a) AmpereSecond (A), (s),Coulomb Siemen (C),(S), Tesla Farad (T) (F), Gram (g), Joule (J), Kelvin (K) Meter Volt(m), (V), Newton Watt (W),(N), WeberOhm (Wb)(Ω), Second (s), Siemen (S), Tesla (T) Volt (V), Watt (W), Weber (Wb) Conversions Conversions1 nm = m = 10 A = cm 1 nm =1 10 eV-9 = m 1.602 = 10 xA =10-l9 10-7 joule cm; =1 eV1.602 = 1.602X 10-l2 x 10erg-9 Joule = 1.602 x 10-12 erg; 1 eV/particle1 eV/particle = 23.06 = 23.06 kcal/mol; kcal/mol 1 newton = 0.102 kg ; 1 newton = 0.102 kg,,, force 6 2 7 2 -4 10 newton/mlo6 newton/m2 = 146 = psi146 = psi 10 = dyn/cmlo7 dyn/crn2 ; 1 µm = 10 cm 0.001 inch = 1 mil = 25.4 µm; 1 bar =1 10 Fm6 dyn/cm= 2cm = 105 N/m2; 1 weber/m2 = 104 gauss = 1 tesla; 1 pascal0.001 = 1 inchN/m =2 =1 7.5mil =x 1025.4-3 torr;pm 1 erg = 10-7 joule = 1 dyn-cm 1 bar = lo6 dyn/cm2 = Id ~/m~ 1 weber/m2 = lo4 gauss = 1 tesla 1 pascal = 1 N/m2 = 7.5 X torr Figure by MIT OCW. 1 erg = 10-'joule = 1 dyn-cm .
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