Lecture #6Mechanisms of Carrier ScatteringImpurity Ion ScatteringMatthiessen's RuleMobility Dependence on DopingTemperature Effect on MobilityDrift CurrentConductivity and ResistivityResistivity Dependence on DopingElectrical ResistanceExampleExample: Dopant CompensationExample: Temperature Dependence of rPotential vs. Kinetic EnergyElectrostatic Potential, VElectric Field, eCarrier Drift (Band Diagram Visualization)SummaryLecture #6OUTLINE• Carrier scattering mechanisms• Drift current• Conductivity and resistivity• Relationship between band diagrams & V, Read: Section 3.1EE130 Lecture 6, Slide 2Spring 2007Dominant scattering mechanisms:1. Phonon scattering (lattice scattering)2. Impurity (dopant) ion scattering2/32/11 velocityermalcarrier thdensityphonon 1 TTTphononphononPhonon scattering mobility decreases when T increases: = q / mMechanisms of Carrier ScatteringTvthEE130 Lecture 6, Slide 3Spring 2007_+ - -Electron Boron Ion ElectronArsenic IonDADAthimpurityNNTNNv2/33There is less change in the electron’s direction of travel if the electron zips by the ion at a higher speed.Impurity Ion ScatteringEE130 Lecture 6, Slide 4Spring 2007Matthiessen's Rule•The probability that a carrier will be scattered by mechanism i within a time period dt is where i is the mean time between scattering events due to mechanism i The probability that a carrier will be scattered within a time period dt is impurityphononimpurityphonon111111 iidtidtEE130 Lecture 6, Slide 5Spring 20071E14 1E15 1E16 1E17 1E18 1E19 1E2002004006008001000120014001600HolesElectrons M obility (cm2 V-1 s-1)Total Impurity Concenration (atoms cm-3)Total Doping Concentration NA + ND (cm-3)Mobility Dependence on DopingEE130 Lecture 6, Slide 6Spring 2007Temperature Effect on Mobilityimpurityphononimpurityphonon111111EE130 Lecture 6, Slide 7Spring 2007vd t A = volume from which all holes cross plane in time tp vd t A = # of holes crossing plane in time tq p vd t A = charge crossing plane in time tq p vd A = charge crossing plane per unit time = hole current Hole current per unit area J = q p vdDrift CurrentEE130 Lecture 6, Slide 8Spring 2007Jp,drift = qpvdn = qppJn,drift = –qnvdn = qnnJdrift = Jn,drift + Jp,drift = =(qnn+qpp)Conductivity of a semiconductor is qnn + qppResistivity 1 / Conductivity and Resistivity(Unit: ohm-cm)EE130 Lecture 6, Slide 9Spring 2007n-typep-typeResistivity Dependence on DopingFor n-type material:nqn1For p-type material:pqp1Note: This plot does not apply for compensated material!EE130 Lecture 6, Slide 10Spring 2007Electrical Resistancewhere is the resistivity Resistance WtLIVR(Unit: ohms)V+_LtWIhomogeneously doped sampleEE130 Lecture 6, Slide 11Spring 2007Consider a Si sample doped with 1016/cm3 Boron.What is its resistivity?Answer: NA = 1016/cm3 , ND = 0 (NA >> ND p-type) p 1016/cm3 and n 104/cm3 Example cm 4.1)450)(10)(106.1(1111619ppnqpqpqnEE130 Lecture 6, Slide 12Spring 2007Example: Dopant CompensationConsider the same Si sample, doped additionallywith 1017/cm3 Arsenic. What is its resistivity?Answer: NA = 1016/cm3, ND = 1017/cm3 (ND>>NA n-type) n 9x1016/cm3 and p 1.1x103/cm3 cm 12.0)600)(109)(106.1(1111619npnqnqpqnEE130 Lecture 6, Slide 13Spring 2007Consider a Si sample doped with 1017cm-3 As.How will its resistivity change when the temperature is increased from T=300K to T=400K?Solution: 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 300K to 400 at 400K. As a result, increases byExample: Temperature Dependence of 93.1400770EE130 Lecture 6, Slide 14Spring 2007Potential vs. Kinetic Energyelectron kinetic energyincreasing electron energyEcEvhole kinetic energyincreasing hole energyreferencecP.E. EE Ec represents the electron potential energy:EE130 Lecture 6, Slide 15Spring 2007N-+– 0 . 7 VS iExEc(x)Ef(x)Ev(x) E0 . 7 V - +V(x)0 . 7 V x0 ( a )( b ) ( c )+– 0 . 7 VS iExEc(x)Ef(x)Ev(x) E0 . 7 V - +V(x)0 . 7 V x0 ( a )( b ) ( c )Electrostatic Potential, V•The potential energy of a particle with charge -q is related to the electrostatic potential V(x):)(1creferenceEEqV qVP.E.EE130 Lecture 6, Slide 16Spring 2007N-+– 0 . 7 VS iExEc(x)Ef(x)Ev(x) E0 . 7 V - +V(x)0 . 7 V x0 ( a )( b ) ( c )+– 0 . 7 VS iExEc(x)Ef(x)Ev(x) E0 . 7 V - +V(x)0 . 7 V x0 ( a )( b ) ( c )Electric Field, dxdEqdxdVc1•Variation of Ec with position is called “band bending.”EE130 Lecture 6, Slide 17Spring 2007Carrier Drift (Band Diagram Visualization)EcEvEE130 Lecture 6, Slide 18Spring 2007Summary•Carrier mobility varies with doping–decreases w/ increasing total concentration of ionized dopants •Carrier mobility varies with temperature–decreases w/ increasing T if lattice scattering is dominant–decreases w/ decreasing T if impurity scattering is dominant •The conductivity of a semiconductor is dependent on the carrier concentrations and mobilities•Ec represents the electron potential energyVariation in Ec(x) variation in electric potential V Electric field•E - Ec represents the electron kinetic energy = qnn +
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