6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-1Lecture 3 - Semiconductor Physics (II)Carrier TransportSeptember 15, 2005Contents:1. Thermal motion2. Carrier drift3. Carrier diffusionReading assignment:Howe and Sodini, Ch. 2, §§2.4-2.66.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-2Key questions• What are the physical mechanisms responsible for cur-rent flow in semiconductors?• How do electrons and holes in a semiconductor behavein an electric field?• How do electrons and holes in a semiconductor behaveif their concentration is non-uniform in space?6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-31. Thermal MotionIn thermal equilibrium, carriers are not sitting still:• undergo collisions with vibrating Si atoms (Brownianmotion)• electrostatical ly interact with charged dopants andwith each otherCharacterist ic time constant of thermal motion - meanfree time between collisions:τc≡ collision time [s]In between collisions, carriers acquire high velocity:vth≡ thermal velocity [cm/s]...but get nowhere!6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-4Characterist ic length of thermal motion:λ ≡ mean free path [cm]λ = vthτcPut numbers for Si at 300 K:τc' 10−14∼ 10−13svth' 107cm/s⇒ λ ' 1 ∼ 10 nmFor reference, state-of-the-art MOSFETs today:Lg' 50 nm⇒ carriers undergo many collisions in modern devices6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-52. Carrier DriftApply electric field to semiconductor:E ≡ electric field [V/cm]⇒ net force on carrierF = ±qEE-Between collisions, carriers accelerate in direction of field:v( t)=at = −qEmnt for electronsv( t)=qEmpt for holes6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-6But velocity randomized every τc(on average):net velocityin direction of fieldaveragenet velocitytimeτcThen, average net velocity in direction of field:v = vd= ±qE2mn,pτc= ±qτc2mn,pEThis is called drift velocity [cm/s].Define:µn,p=qτc2mn,p≡ mobility [cm2/V · s]Then, for electrons:vdn= −µnEfor holes:vdp= µpE6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-7Mobility is measure of ease of carrier drift:• if τc↑, longer time between collisions → µ ↑• if m ↓, ”lighter” particle → µ ↑Mobility depends on doping. For Si at 300K:101310151014101910201016101710181400120010008006004002000holesNd+ Natotal dopant concentration (cm−3)electronsmobility (cm2/Vs)• for low doping level, µ limited by collisions with lattice• for medium and high doping level, µ limited by colli-sions with ionized impurities• holes ”heavier” than electrons:→ for same doping level, µn>µp6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-8Drift currentNet velocity of charged particles ⇒ electric current:Drift current density ∝ carrier drift velocity∝ carrier concentration∝ carrier chargeDrift currents:Jdriftn= −qnvdn= qnµnEJdriftp= qpvdp= qpµpECheck signs:xxvdnvdpJndriftJpdriftEE-+6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-9Total drift current:Jdrift= Jdriftn+ Jdriftp= q(nµn+ pµp)EHas the shape of Ohm’s Law :J = σE =EρWhere:σ ≡ conductivity [Ω−1· cm−1]ρ ≡ resistiviy [Ω · cm]Then:ρ =1σ=1q(nµn+ pµp)6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-10Resistivity commonly used to specify doping level.• In n-type semiconductor:ρn'1qNdµn• In p-type semiconductor:ρp'1qNaµpFor Si at 300K:1E-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+21Doping (cm-3)Resistivity (ohm.cm)n-Sip-Si6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-11Numerical example:• Si with Nd=3× 1016cm−3at 300 Kµn' 1000 cm2/V · sρn' 0.21 Ω · cm• apply |E| =1kV/cm|vdn|'106cm/s vth|Jdriftn|'4.8 × 103A/cm2• time to drift through L =0.1 µm:td=Lvdn=10psfast!6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-123. Carrier diffusionDiffusion: particle movement in response to concentrationgradient.nxElements of diffusion:• a medium (Si crystal)• a gradient of particles (electrons and holes) insidethe medium• collisions between particles and medium send particlesoff in random directions:→ overall, particle movement down the gradient6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-13Key diffusion relationship (Fick’s first law):Diffusion flux ∝ - concentration gradientFlux ≡ number of particles crossing unit area per unittime [cm−2· s−1]For electrons:Fn= −DndndxFor holes:Fp= −DpdpdxDn≡ electron diffusion coefficient [cm2/s]Dp≡ hole diffusion co efficient [cm2/s]D measures the ease of carrier diffusion in response to aconcentration gradient: D ↑⇒ Fdiff↑.D limited by vibrating lattice atoms and ionized dopants6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-14Diffusion current density = charge × carrier fluxJdiffn= qDndndxJdiffp= −qDpdpdxCheck signs:npxxFnFpJndiffJpdiff6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-15Einstein relationAt the core of diffusion and drift is same physics: collisionsamong particles and medium atoms⇒ there should be a relationship between D and µEinstein relation [don’t derive in 6.012]:Dµ=kTqIn semiconductors:Dnµn=Dpµp=kTqkTq≡ thermal voltage [V ]At 300 K:kTq' 25 mVFor example: for Nd=3× 1016cm−3:µn' 1000 cm2/V · s → Dn' 25 cm2/sµp' 400 cm2/V · s → Dp' 10 cm2/s6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-16Total currentIn general, current can flow by drift and diffusion sepa-rately. Total current:Jn= Jdriftn+ Jdiffn= qnµnE + qDndndxJp= Jdriftp+ Jdiffp= qpµpE − qDpdpdxAndJtotal= Jn+ Jp6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-17Summary: relationship between v, F , and JIn semiconductors: charged particles move⇒ particle flux ⇒ electrical current densityParticle flux: number of particles that cross surface ofunit area placed normal to particle flow every unit timeFnvn dtRelationship between particle flux and velocity:Fn= nvnFp= pvpCurrent density: amount of charge that crosses surfaceof unit area placed normal to particle flow every unit timeJn= −qFn= −qnvnJp= qFp= qpvpwhether carriers move by drift or diffusion.6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 3-18Key conclusions• Electrons and holes in semiconductors are
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