Lecture #6 OUTLINE • Carrier scattering mechanisms • Drift current • Conductivity and resistivity • Relationship between band diagrams & V, Read: Section 3.1 e Mechanisms of Carrier Scattering Dominant scattering mechanisms: 1. Phonon scattering (lattice scattering) 2. Impurity (dopant) ion scattering Phonon scattering mobility decreases when T increases: phonon phonon 1 1 3 / 2 T phonon density carrier th ermal velocity T T 1/ 2 = q / m Spring 2007 vth EE130 Lecture 6, Slide 2 T Impurity Ion Scattering - Boron _ Ion Electron - + Electron Arsenic Ion There is less change in the electron’s direction of travel if the electron zips by the ion at a higher speed. 3 vth T 3/ 2 impurity N A ND N A ND Spring 2007 EE130 Lecture 6, Slide 3 Matthiessen's Rule • The probability that a carrier will be scattered by mechanism i within a time period dt is dt i 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 dt i 1 i Spring 2007 EE130 Lecture 6, Slide 4 1 1 1 phonon impurity 1 phonon 1 impurity Mobility Dependence on Doping 1600 1400 Electrons 1000 2 -1 -1 Mobility (cm V s ) 1200 800 600 400 Holes 200 0 1E14 1E15 1E16 1E17 1E18 1E19 -3 Total Impurity ConcenrationN(atoms cm -3)) Total Doping Concentration A + ND (cm Spring 2007 EE130 Lecture 6, Slide 5 1E20 Temperature Effect on Mobility 1 1 Spring 2007 EE130 Lecture 6, Slide 6 1 phonon 1 phonon 1 impurity 1 impurity Drift Current vd t A = volume from which all holes cross plane in time t p vd t A = # of holes crossing plane in time t q p vd t A = charge crossing plane in time t q p vd A = charge crossing plane per unit time = hole current Hole current per unit area J = q p vd Spring 2007 EE130 Lecture 6, Slide 7 Conductivity and Resistivity Jn,drift = –qnvdn = qnne Jp,drift = qpvdn = qppe Jdrift = Jn,drift + Jp,drift = e =(qnn+qpp)e Conductivity of a semiconductor is qnn + qpp Resistivity 1 / Spring 2007 (Unit: ohm-cm) EE130 Lecture 6, Slide 8 Resistivity Dependence on Doping For n-type material: 1 qn n p-type For p-type material: 1 qp p n-type Note: This plot does not apply for compensated material! Spring 2007 EE130 Lecture 6, Slide 9 Electrical Resistance I V + _ W t homogeneously doped sample L V L Resistance R I Wt where is the resistivity Spring 2007 EE130 Lecture 6, Slide 10 (Unit: ohms) Example Consider 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 1 1 qn n qp p qp p (1.6 10 Spring 2007 19 16 )(10 )( 450) EE130 Lecture 6, Slide 11 1 1.4 cm Example: Dopant Compensation Consider the same Si sample, doped additionally with 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 1 1 qn n qp p qn n (1.6 10 Spring 2007 19 )(9 10 )(600) 16 EE130 Lecture 6, Slide 12 1 0.12 cm Example: Temperature Dependence of Consider 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 by 770 1.93 400 Spring 2007 EE130 Lecture 6, Slide 13 electron kinetic energy Ec hole kinetic energy Ev increasing hole energy increasing electron energy Potential vs. Kinetic Energy Ec represents the electron potential energy: P.E. Ec Ereference Spring 2007 EE130 Lecture 6, Slide 14 E + – Electrostatic Potential, V Si (a) V(x) 0.7V 0.7V E + – N-Si x 0 (b) (a) E V(x) - Ec(x) Ef(x) • The potential energy of a particle with charge -q is 0.7V x related to the electrostatic potential V(x): 0 (b) E - P.E. qV Ec(x) Ef(x) x 1 V ( Ereference Ec ) q Ev(x) 0.7V + x Spring 2007 (c) EE130 Lecture 6, Slide 15 Ev(x) 0.7V + (c) 0.7V Electric Field, + e E – Si (a) 0.7V V(x) 0.7V E + – N-Si x 0 (b) (a) E V(x) - e 0.7V 0 (b) dV 1 dEc dx q dx Ec(x) Ef(x) Ev(x) x 0.7V + x E - Ec(x) Ef(x) (c) • Variation of Ec with position is called “band bending.” Ev(x) 0.7V + x Spring 2007 (c) EE130 Lecture 6, Slide 16 Carrier Drift (Band Diagram Visualization) Ec Ev Spring 2007 EE130 Lecture 6, Slide 17 Summary • 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 = qnn + qpp • Ec represents the electron potential energy Variation in Ec(x) variation in electric potential V dEc dEv e dx dx • E - Ec represents the electron kinetic energy Electric field Spring 2007 EE130 Lecture 6, Slide 18