# 1.1 Silicon Crystal Structure

Lecture #6 OUTLINE Carrier scattering mechanisms Drift current Conductivity and resistivity Relationship between band diagrams & V, Read: Section 3.1 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 thermal 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 electrons direction of travel if the electron zips by the ion at a higher speed. impurity Spring 2007 vth3 T 3/ 2

N A ND N A ND EE130 Lecture 6, Slide 3 Matthiessen's Rule The probability that a carrier will be scattered by dt mechanism i within a time period dt is 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 i 1 1 1 phonon impurity 1 1

1 phonon impurity Spring 2007 EE130 Lecture 6, Slide 4 Mobility Dependence on Doping 1600 1400 Electrons 1000 2 -1 -1 M obility (cmV s ) 1200 800 600 400

Holes 200 0 1E14 1E15 1E16 1E17 1E18 1E19 -3 TotalDoping Impurity ConcenrationN(atoms cm -3) Total Concentration A + ND (cm ) Spring 2007 EE130 Lecture 6, Slide 5 1E20

Temperature Effect on Mobility 1 1 1 phonon impurity 1 1 1 phonon impurity Spring 2007 EE130 Lecture 6, Slide 6 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 = qnn

Jp,drift = qpvdn = qpp Jdrift = Jn,drift + Jp,drift = =(qnn+qpp) Conductivity of a semiconductor is qnn + qpp (Unit: ohm-cm) Resistivity 1 / Spring 2007 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 9x10x1016/cm3 and p 1.1x103/cm3 1 1 qn n qp p qn n (1.6 10 Spring 2007 19 16 )(9 10 )(600)

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 hole kinetic energy Ec

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 .7 V 0 .7 V

E + N-S i x 0 (b ) (a ) V (x ) E - E c (x ) E f(x ) The potential energy of a particle with charge -q is 0 .7 V x related to the electrostatic potential V(x): 0 (b )

E - P.E. qV E c (x ) E f(x ) 1 V ( Ereference Ec ) q E v (x ) 0 .7 V + x Spring 2007 (c ) E v (x ) EE130 Lecture 6, Slide 15 0 .7 V + x (c )

Electric Field, + E Si (a ) 0 .7 V V (x) 0 .7 V E + N-S i x 0 (b ) (a )

E V (x ) - 0 .7 V 0 (b ) dV 1 dEc dx q dx E c(x) E f(x ) E v(x) x 0 .7 V + E -

E c (x ) E f(x ) x (c ) Variation of Ec with position is called band bending. E v (x ) 0 .7 V + 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 Electric field dx dx E - Ec represents the electron kinetic energy Spring 2007 EE130 Lecture 6, Slide 18

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