Lecture 3 - MIT

1 Spring 2007 Lecture 31 Lecture 3 Semiconductor Physics (II)Carrier TransportOutline Thermal Motion Carrier Drift Carrier DiffusionReading Assignment:Howe and Sodini; Chapter 2, Sect. Spring 2007 Lecture 321. Thermal Motion Undergo collisions with vibrating Si atoms (Brownian motion) Electrostatically interact with each other and with ionized (charged) dopantsIn thermal equilibrium, carriers are not sitting still:Characteristic time constant of thermal motion: mean free time between collisions c collison time[s]In between collisions, carriers acquire high velocity:vth thermal velocity[cms 1].. but get nowhere! Spring 2007 Lecture 33 Characteristic length of thermal motion: mean free path[cm] =vth cPut numbers for Si at room temperature: c 10 13svth 107cms 1 mFor reference, state-of-the-art production MOSFET:Lg m Carriers undergo many collisions as they travel through Spring 2007 Lecture 342. Carrier DriftApply electric field to semiconductor:E electric field[V cm-1] net force on carrierF = qEBetween collisions, carriers accelerate in the direction of the electrostatic field:v(t)=a t= qEmn, Spring 2007 Lecture 35 But there is (on the average) a collision every cand the velocity is randomized:The average net velocity in direction of the field:v =vd= qE2mn,p c= q c2mn,pEThis is called drift velocity[cm s-1]Define: n,p=q c2mn,p mobility[cm2V 1s 1]Then, for electrons:and for holes:vdn= nEvdp= pEnet velocityin direction of fieldtime Spring 2007 Lecture 36 Mobility-is a measure of easeof carrier drift If c , longer time between collisions If m , lighter particle At room temperature, mobility in Si depends on doping: For low doping level, is limited by collisions with lattice.

2 As Temp ->INCREASES; -> DECREASES For medium doping and high doping level, limited by collisions with ionized impurities Holes heavier than electrons For same doping level, n> p101310151014101910201016101710181400120 010008006004002000holesNd + Na total dopant concentration (cm 3)electronsmobility (cm2/Vs) Spring 2007 Lecture 37 Drift CurrentCheck signs:Net velocity of charged particles electric current:Drift current density carrier drift velocity carrier concentration carrier chargeDrift current densities:Jndrift= qnvdn=qn nEJpdrift=qpvdp=qp pExxvdnvdpJndriftJpdriftEE-+ Spring 2007 Lecture 38 Total Drift Current Density :Jdrift=Jndrift+Jpdrift=qn n+p p()EHas the form of Ohm s LawJ= E=E Where: conductivity [ -1 cm-1] resistivity [ cm]Then: =1 =qn n+p p() Spring 2007 Lecture 39 Resistivityis commonly used to specify the doping level In n-type semiconductor: In p-type semiconductor: n 1qNd n p 1qNa p1E-41E-31E-21E-11E+01E+11E+21E+31E+41E+ 12 1E+13 1E+14 1E+15 1E+16 1E+17 1E+18 1E+19 1E+20 1E+21 Doping (cm-3)n-Sip-SiResistivity ( ) Spring 2007 Lecture 310 Numerical Example:Si with Nd= 3 x 1016cm-3at room temperature n 1000cm2/V s n cmn 3X1016cm 3 Apply E = 1 kV/cm236 == = << Time to drift through L = mtd=Lvdn=10psfast!

3 Spring 2007 Lecture 3113. Carrier DiffusionDiffusion= particle movement (flux) in response to concentration gradientElements of diffusion: A medium (Si Crystal) A gradient of particles (electrons and holes) inside the medium Collisions between particles and medium send particles off in random directions Overall result is to erase Spring 2007 Lecture 312 Fick s first law-Key diffusion relationshipFlux number of particles crossing a unit area per unit time [cm-2 s-1]For Electrons:Fn= DndndxD measures the easeof carrier diffusion in response to a concentration gradient: D Fdiff D limited by vibration of lattice atoms and ionized Holes:Fp= DpdpdxDn electron diffusion coefficient [cm2s-1]Dp hole diffusion coefficient [cm2s-1]Diffusion flux - concentration Spring 2007 Lecture 313 Diffusion CurrentCheck signs:Diffusion current density =charge carrier fluxJndiff=qDndndxJpdiff= Spring 2007 Lecture 314 Einstein relationAt room temperature:At the core of drift and diffusion is same physics:collisions among particles and medium atoms there should be a relationship between D and Einstein relation[will not derive in ]D =kTqIn semiconductors:Dn n=kTq=Dp pkT/q thermal voltagekTq 25mVFor example: for Nd= 3 x 1016cm-3 n 1000cm2/V s Dn 25cm2/s p 400cm2/V s Dp 10cm2 Spring 2007 Lecture 315 Total Current DensityJn=Jndrift+Jndiff=qn nE+qDndndxJp=Jpdrift+Jpdiff=qp pE qDpdpdxJtotal=Jn+JpIn general, total current can flow by drift and diffusion separately.

4 Total current Spring 2007 Lecture 316 What did we learn today? Electrons and holes in semiconductors are mobile and charged Carriers of electrical current! Drift current: produced by electric field Diffusion current: produced by concentration gradient Diffusion and drift currents are sizeable in modern devices Carriers move fast in response to fields and gradientsSummary of Key ConceptsJdrift EJdrift d dxJdiffusion dndx,dpdx