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 n d 0 then eq becomes n0 Na p0 Nd by eq and the Mass Action law n0p0 ni2 n0 Nd Na Nd Na 2 4ni2 1 2 where Nd Na n type p0 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 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 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 ntype 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 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 Comment If 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 positive volume charge density vd average drift velocity then Jdr vd Jpdr qp vdp hole Jndr qn vdn electron Jdr Jpdr Jndr qp vdp qn vdn for low electric field vdp pE p proportionality factor hole mobility vdn nE n proportionality factor electron mobility thus Jdr Jpdr Jndr q p p n n 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 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 p p n n E qNa pE 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 semiconductor
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