1. Carrier Concentration a) Intrinsic Semiconductors - Pure single-crystal material For an intrinsic semiconductor, the concentration of electrons in the conduction band is equal to the concentration of holes in the valence band. We may denote, ni : intrinsic electron concentration pi : intrinsic hole concentration However, ni = pi Simply, ni :intrinsic carrier concentration, which refers to either the intrinsic electron or hole concentration Commonly accepted values of ni at T = 300°K Silicon 1.5 x 1010 cm-3 Gallium arsenide 1.8 x 106 cm-3 Germanium 2.4 x 1013 cm-3 b) Extrinsic Semiconductors - Doped material The doping process can greatly alter the electrical characteristics of the semiconductor. This doped semiconductor is called an extrinsic material. n-Type Semiconductors (negatively charged electron by adding donor) p-Type Semiconductors (positively charged hole by adding acceptor) c) Mass-Action Law n0 : thermal-equilibrium concentration of electrons p0 : thermal-equilibrium concentration of holes n0p0 = ni2 = f(T) (function of temperature) The product of n0 and po is always a constant for a given semiconductor material at a given temperature.d) Equilibrium Electron and Hole Concentrations Let, n0 : thermal-equilibrium concentration of electrons p0 : thermal-equilibrium concentration of holes nd : concentration of electrons in the donor energy state pa : concentration of holes in the acceptor energy state Nd : concentration of donor atoms Na : concentration of acceptor atoms Nd+ : concentration of positively charged donors (ionized donors) Na- : concentration of negatively charged acceptors (ionized acceptors) By definition, Nd+ = Nd - nd Na- = Na – pa by the charge neutrality condition, n0 + Na- = p0 + Nd+ or n0 + (Na - pa) = p0 + (Nd – nd) assume complete ionization, pa = nd = 0 then, eq # becomes, n0 + Na = p0 + Nd by eq # and the Mass-Action law (n0p0 = ni2) n0 = ½{(Nd - Na) + ((Nd - Na)2 + 4ni2)1/2}, where Nd > Na (n-type) p0 = ½{(Na - Nd) + ((Na - Nd)2 + 4ni2)1/2}, where Na > Nd (p-type) n0 = p0 = ni, where Na = Nd (intrinsic) If Nd - Na >> ni, then n0 = Nd - Na, p0 = ni2 / (Nd - Na) If Na – Nd >> ni, then p0 = Na – Nd, n0 = ni2 / (Na – Nd) Example 1) Determine the thermal equilibrium electron and hole concentrations for a given doping concentration.Consider an n-type silicon semiconductor at T = 300°K in which Nd = 1016 cm-3 and Na = 0. The intrinsic carrier concentration is assumed to be ni = 1.5 x 1010 cm-3. - Solution The majority carrier electron concentration is no = ½{(Nd - Na) + ((Nd - Na)2 + 4ni2)1/2} ≅ 1016 cm-3 The minority carrier hole concentration is p0 = ni2 / n0 = (1.5 x 1010)2/1016 = 2.25 x 104 cm-3 - Comment Nd >> ni, so that the thermal-equilibrium majority carrier electron concentration is essentially equal to the donor impurity concentration. The thermal-equilibrium majority and minority carrier concentrations can differ by many orders of magnitude. Example 2) Determine the thermal equilibrium electron and hole concentrations for a given doping concentration. Consider an germanium sample at T = 300°K in which Nd = 5 x 1013 cm-3 and Na = 0. Assume that ni = 2.4 x 1013 cm-3. - Solution The majority carrier electron concentration is no = ½{(5 x 1013) + ((5 x 1013)2 + 4(2.4 x 1013)2)1/2} = 5.97 x 1012 cm-3 The minority carrier hole concentration is p0 = ni2 / n0 = (2.4 x 1013)2/(5.97 x 1012) = 9.65 x 1012 cm-3 - Comment If the donor impurity concentration is not too different in magnitude from the intrinsic carrier concentration, the thermal-equilibrium majority carrier electron concentration is influenced by the intrinsic concentration. Example 3) Determine the thermal equilibrium electron and hole concentrations in a compensated n-type semiconductor. Consider a silicon semiconductor at T = 300°K in which Nd = 1016 cm-3 and Na = 3 x 1015 cm-3. Assume that ni =1.5 x 1010 cm-3. - Solution The majority carrier electron concentration is no = ½{(1016 – 3 x 1015) + ((1016 – 3 x 1015)2 + 4(1.5 x 1010)2)1/2} ≅ 7 x 1015 cm-3 The minority carrier hole concentration is p0 = ni2 / n0 = (1.5 x 1010)2/(7 x 1015) = 3.21 x 104 cm-3 - CommentIf we assume complete ionization and if Nd - Na >> ni, the the majority carrier electron concentration is, to a very good approximation, just the difference between the donor and acceptor concentrations. 2. Carrier Transport The net flow of the electrons and holes in a semiconductor will generate currents. The process by which these charged particles move is called transport. The two basic transport mechanisms in a semiconductor crystal: - Drift: the movement of charge due to electric fields - Diffusion: the flow of charge due to density gradients a) Carrier Drift - Drift Current Density Let, Jdr : drift current density ρ : positive volume charge density vd : average drift velocity then, Jdr = ρvd Jpdr = (qp)vdp (hole) Jndr = (-qn)vdn (electron) Jdr = Jpdr + Jndr = (qp)vdp + (-qn) vdn for low electric field, vdp = µpE (µp : proportionality factor, hole mobility) vdn = -µnE (µn : proportionality factor, electron mobility) thus, Jdr = Jpdr + Jndr = q(pµp + nµn)E Example 1) Calculate the drift current density in a semiconductor for a given electric field. Consider a germanium sample at T = 300°K with doping concentration of Nd = 0 and Na = 1016 cm-3. Assume complete ionization and electron and hole mobilities are 3900 cm2/V⋅sec and 1900 cm2/V⋅sec. The applied electric field is E = 50 V/cm. - Solution Since Na > Nd, the semiconductor is p-type and the majority carrier hole concentration,p = ½{(Na - Nd) + ((Na - Nd)2 + 4ni2)1/2} ≅ 1016 cm-3 The minority carrier electron concentration is n = ni2 / p = (2.4 x 1013)2/1016 = 5.76 x 1010 cm-3 For this extrinsic p-type semiconductor, the drift current density is Jdr = Jpdr + Jndr = q(pµp + nµn)E ≅ qNaµpE Then Jdr = (1.6 x 10-19)(1900)(1016)(50) = 152 A/cm2 - Comment Significant drift current densities can be obtained in a semiconductor applying relatively small electric fields. The drift current will be due primarily to the majority carrier in an extrinsic
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