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Semiconductor Device Physics Lecture 3 Dr. Gaurav Trivedi, EEE Department, IIT Guwahati Boltzmann Approximation of Fermi Function Boltzmann Approximation of Fermi Function The Fermi Function that describes the probability that a state at energy E is filled with an electron, under equilibrium conditions, is already given as: f (E) 1 1 e( E EF ) / kT Fermi Function can be approximated as: f ( E) e( E EF ) / kT if E – EF > 3kT 1 f ( E) e( EF E )/ kT if EF – E > 3kT Nondegenerately Doped Semiconductor The expressions for n and p will now be derived in the range Boltzmann Approximation ofapplied: Fermi Function where the Boltzmann approximation can be Ev 3kT EF Ec 3kT 3kT Ec EF in this range 3kT Ev The semiconductor is said to be nondegenerately doped (lightly doped) in this case. Degenerately Doped Semiconductor Degenerately Doped Semiconductor Degenerately Doped Semiconductor If a semiconductor is very heavily doped, the Boltzmann approximation is not valid. For Si at T = 300 K, EcEF < 3kT if ND > 1.6 1018 cm–3 EFEv < 3kT if NA > 9.1 1017 cm–3 The semiconductor is said to be degenerately doped (heavily doped) in this case. • ND = total number of donor atoms/cm3 • NA = total number of acceptor atoms/cm3 Equilibrium Carrier Concentrations Integrating n(E) over all the energies in the conduction band to Boltzmann Approximation of Fermi Function obtain n (conduction electron concentration): Etop n g c ( E ) f ( E )dE Ec By using the Boltzmann approximation, and extending the integration limit to , 3/ 2 n NCe ( EF Ec ) kT mn* kT where N C 2 2 2 h • NC = “effective” density of conduction band states • For Si at 300 K, NC = 3.22 1019 cm–3 Equilibrium Carrier Concentrations Integrating p(E) over all the energies in the conduction band to Boltzmann Approximation of Fermi Function obtain p (hole concentration): p EV g v ( E ) 1 f ( E ) dE Ebottom By using the Boltzmann approximation, and extending the integration limit to , 3/ 2 p N V e( Ev EF ) kT mp* kT where N V 2 2 2 h • NV = “effective” density of valence band states • For Si at 300 K, NV = 1.83 1019 cm–3 Intrinsic Carrier Concentration Relationship between EFApproximation and n, p : Boltzmann of n NCe( EF Ec ) kT ( Ev EF ) kT p NVe For intrinsic semiconductors, where n = p = ni, np n ni NC N V e EG 2 i 2 kT • EG : band gap energy Fermi Function Intrinsic Carrier Concentration kT npBoltzmann ( NCe( EF Ec )Approximation ) ( NVe( Ev EF ) kT of ) Fermi Function ( Ev Ec ) kT NC NVe NC NVe EG ni NC NV e kT EG 2 kT Alternative Expressions: n(ni, Ei) and p(ni, Ei) In an intrinsic semiconductor, n = p = ni and EF = Ei, where Ei denotesBoltzmann the intrinsic Fermi level. Approximation of Fermi Function n NCe( EF Ec ) kT p NVe( Ev EF ) kT ( Ei Ec ) kT pi NVe( Ev Ei ) kT NV ni e( Ev Ei ) kT ni NCe ( Ei Ec ) kT NC nie n ni e( Ei Ec ) kT e( EF Ec ) kT n ni e( EF Ei ) kT n EF Ei kT ln ni p nie( Ev Ei ) kT e( Ev EF ) kT p nie( Ei EF ) kT p EF Ei kT ln ni Intrinsic Fermi Level, Ei To find EF for an intrinsic semiconductor, we use the fact that n = p. Boltzmann Approximation of Fermi Function NCe( Ei Ec ) kT NVe( Ev Ei ) kT Ei EG = 1.12 eV Ec Ev Ec Ev kT N V Ei ln 2 2 NC * m Ec Ev 3kT p Ei ln * m 2 4 n Ec Ev Ei • Ei lies (almost) in the middle 2 between Ec and Ev Si Example: Energy-Band Diagram Boltzmann Approximation 17of Fermi Function –3 For Silicon at 300 K, where is EF if n = 10 Silicon at 300 K, ni = 1010 cm–3 n EF Ei kT ln ni 17 10 5 0.56 8.62 10 300 ln 10 eV 10 0.56 0.417 eV 0.977 eV cm ? Charge Neutrality and Carrier Concentration Boltzmann Approximation ND: concentration of ionized donor (cm–3) of Fermi NA: concentration of ionized acceptor (cm–3)? Charge neutrality condition: p n N D N A 0, 2 i n n ND NA 0 n n2 n( ND NA ) ni2 0 ni2 p n • Ei quadratic equation in n Function Charge-Carrier Concentrations The solution of the previous quadratic equation for n Function is: Boltzmann Approximation of Fermi 12 N D N A N D N A 2 n ni 2 2 2 New quadratic equation can be constructed and the solution for p is: 12 2 NA ND NA ND 2 p ni 2 2 • Carrier concentrations depend on net dopant concentration ND–NA or NA–ND Dependence of EF on Temperature Ec Boltzmann Approximation of Fermi Function Ei Ev 1013 1014 1015 1016 1017 1018 1019 1020 n EF Ei kT ln , donor-doped ni p EF Ei kT ln , acceptor-doped ni Net dopant concentration (cm–3) Carrier Concentration vs. Temperature Phosphorus-doped Si –3 Boltzmann Approximation NofD =Fermi Function 1015 cm • n : number of majority carrier • ND : number of donor electron • ni : number of intrinsic conductive electron Carrier Action Boltzmann Approximation of Fermi Function Three primary types of carrier action occur inside a semiconductor: Drift: charged particle motion in response to an applied electric field. Diffusion: charged particle motion due to concentration gradient or temperature gradient. Recombination-Generation: a process where charge carriers (electrons and holes) are annihilated (destroyed) and created. Carrier Scattering Mobile electrons and atoms in the Si lattice are always in random thermal motion. Boltzmann Approximation of Fermi Function Electrons make frequent collisions with the vibrating atoms. “Lattice scattering” or “phonon scattering” increases with increasing temperature. Average velocity of thermal motion for electrons: ~1/1000 x speed of light at 300 K (even under equilibrium conditions). Other scattering mechanisms: Deflection by ionized impurity atoms. Deflection due to Coulombic force between carriers or “carrier-carrier scattering.” Only significant at high carrier concentrations. The net current in any direction is zero, if no electric field is applied. 2 3 1 electron 4 5 Carrier Drift When an electric field (e.g. due to an externally applied of Fermi Function voltage)Boltzmann is applied to aApproximation semiconductor, mobile charge-carriers will be accelerated by the electrostatic force. This force superimposes on the random motion of electrons. 3 F = –qE 2 1 4 electron 5 E Electrons drift in the direction opposite to the electric field Current flows. • Due to scattering, electrons in a semiconductor do not achieve constant velocity nor acceleration. • However, they can be viewed as particles moving at a constant average drift velocity vd. Drift Current Boltzmann Approximation of Fermi Function vd t All holes this distance back from the normal plane vd t A All holes in this volume will cross the plane in a time t p vd t A Holes crossing the plane in a time t q p vd t A Charge crossing the plane in a time t q p vd A Charge crossing the plane per unit time I (Ampere) Hole drift current q p vd Current density associated with hole drift current J (A/m2) Hole and Electron Mobility For holes, oftoFermi Function IP|driftBoltzmann qpvd A Approximation • Hole current due drift J P|drift qpvd • Hole current density due to drift In low-field limit, vd pE JP|drift qp pE • μp : hole mobility Similarly for electrons, J N|drift qnvd vd n E J N|drift qn nE • Electron current density due to drift • μn : electron mobility Drift Velocity vs. Electric Field Boltzmann Approximation of Fermi Function vd pE vd n E • Linear relation holds in low field intensity, ~5103 V/cm Hole and Electron Mobility Boltzmann Approximation of Fermi Function has the dimensions of v/E : cm/s cm2 V/cm V s Electron and hole mobility of selected intrinsic semiconductors (T = 300 K) n (cm2/V·s) p (cm2/V·s) Si 1400 Ge 3900 GaAs 8500 InAs 30000 470 1900 400 500 Temperature Effect on Mobility RL RI Boltzmann Approximation of Fermi Function Impedance to motion due to lattice scattering: • No doping dependence • Decreases with decreasing temperature Impedance to motion due to ionized impurity scattering: • increases with NA or ND • increases with decreasing temperature Temperature Effect on Mobility Boltzmann Approximation of Fermi Function Carrier mobility varies with doping: Decrease with increasing total concentration of ionized dopants. Carrier mobility varies with temperature: Decreases with increasing T if lattice scattering is dominant. Decreases with decreasing T if impurity scattering is dominant. Conductivity and Resistivity Boltzmann Approximation of Fermi Function JN|drift = –qnvd = qnnE JP|drift = qpvd = qppE Jdrift = JN|drift + JP|drift =q(nn+pp)E = E Conductivity of a semiconductor: = q(nn+pp) Resistivity of a semiconductor: = 1 / Resistivity Dependence on Doping ForFermi n-typeFunction material: Boltzmann Approximationof 1 q n N D For p-type material: 1 q p N A Example Consider a Si sample at 300 K doped with 1016/cm3 Boron. What is Boltzmann its resistivity? Approximation of Fermi Function NA = 1016/cm3 , ND = 0 (NA >> ND p-type) p 1016/cm3, n 104/cm3 1 qn n qp p 1 qp p (1.6 10 )(470)(10 ) 1.330 cm 19 16 1 Example Consider a Si sample doped with 1017cm–3 As. How will its resistivity change when the temperature isof increased from Boltzmann Approximation Fermi Function T = 300 K to T = 400 K? The temperature dependent factor in (and therefore ) is n. From the mobility vs. temperature curve for 1017cm–3, we find that n decreases from 770 at 300 K to 400 at 400 K. As a result, increases by a factor of: 770/400 = 1.93 Assignment Boltzmann Approximation of Fermi Function