6.012 Spring 2007 Lecture 3 1Lecture 3Semiconductor Physics (II)Carrier TransportOutline• Thermal Motion•Carrier Drift•Carrier DiffusionReading Assignment:Howe and Sodini; Chapter 2, Sect. 2.4-2.66.012 Spring 2007 Lecture 3 21. 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 collisionsτc≡ collison time [s]In between collisions, carriers acquire high velocity:vth≡ thermalvelocity[cms−1]…. but get nowhere!6.012 Spring 2007 Lecture 3 3Characteristic length of thermal motion:λ≡mean free path [cm]λ= vthτcPut numbers for Si at room temperature:τc≈ 10−13svth≈ 107cms−1⇒λ≈ 0.01 µmFor reference, state-of-the-art production MOSFET:Lg≈ 0.1 µm⇒ Carriers undergo many collisions as they travel through devices6.012 Spring 2007 Lecture 3 42. 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,ptE6.012 Spring 2007 Lecture 3 5But there is (on the average) a collision every τcand the velocity is randomized:The average net velocity in direction of the field:v = vd=±qE2mn,pτc=±qτc2mn,pEThis is called drift velocity [cm s-1]Define:µn, p=qτc2mn,p≡ mobility[cm2V−1s−1]Then, for electrons:and for holes:vdn=−µnEvdp=µpEnet velocityin direction of fieldtimeτc6.012 Spring 2007 Lecture 3 6Mobility - is a measure of ease of carrier drift• If τc↑, longer time between collisions ⇒ µ ↑• If m ↓, “lighter” particle ⇒ µ ↑At room temperature, mobility in Si depends on doping:• For low doping level, µ is limited by collisions with lattice. As Temp ->INCREASES; µ-> DECREASES• For medium doping and high doping level, µ limited by collisions with ionized impurities• Holes “ heavier” than electrons– For same doping level, µn> µp101310151014101910201016101710181400120010008006004002000holesNd + Na total dopant concentration (cm−3)electronsmobility (cm2/Vs)6.012 Spring 2007 Lecture 3 7Drift CurrentCheck signs:Net velocity of charged particles ⇒ electric current:Drift current density∝carrier drift velocity∝carrier concentration ∝carrier chargeDrift current densities:Jndrift=−qnvdn= qnµnEJpdrift= qpvdp= qpµpExxvdnvdpJndriftJpdriftEE-+6.012 Spring 2007 Lecture 3 8Total Drift Current Density :Jdrift= Jndrift+ Jpdrift= qnµn+ pµp()EHas the form of Ohm’s LawJ =σE =EρWhere:σ ≡ conductivity [Ω-1• cm-1]ρ≡resistivity [Ω • cm]Then:σ=1ρ= qnµn+ pµp()6.012 Spring 2007 Lecture 3 9Resistivity is commonly used to specify the doping level• In n-type semiconductor:• In p-type semiconductor:ρn≈1qNdµnρp≈1qNaµp1E-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)n-Sip-SiResistivity (ohm.cm)6.012 Spring 2007 Lecture 3 10Numerical Example:Si with Nd= 3 x 1016cm-3at room temperatureµn≈ 1000 cm2/ V • sρn≈ 0.21Ω•cmn ≈ 3X1016cm−3Apply E = 1 kV/cm236/108.4/10cmAJEEEqnqnvJvscmvdriftnndndriftnthdn×≈==µ=≈<<≈ρσTime to drift through L = 0.1 µmtd=Lvdn= 10 psfast!6.012 Spring 2007 Lecture 3 113. 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 gradientnx6.012 Spring 2007 Lecture 3 12Fick’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 ease of carrier diffusion in response to a concentration gradient: D ↑⇒Fdiff↑D limited by vibration of lattice atoms and ionized dopants.For Holes:Fp=−DpdpdxDn≡ electron diffusion coefficient [cm2s-1]Dp≡ hole diffusion coefficient [cm2s-1]Diffusion flux ∝- concentration gradient6.012 Spring 2007 Lecture 3 13Diffusion CurrentCheck signs:Diffusion current density =charge ×carrier fluxJndiff= qDndndxJpdiff=−qDpdpdxnpxxFnFpJndiffJpdiff6.012 Spring 2007 Lecture 3 14Einstein 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 µEinstein relation [will not derive in 6.012]Dµ=kTqIn semiconductors:Dnµn=kTq=DpµpkT/q ≡ thermal voltagekTq≈ 25mVFor example: for Nd= 3 x 1016cm-3µn≈ 1000 cm2/ V • s ⇒ Dn≈ 25cm2/ sµp≈ 400 cm2/ V • s ⇒ Dp≈ 10 cm2/ s6.012 Spring 2007 Lecture 3 15Total Current DensityJn= Jndrift+ Jndiff= qnµnE + qDndndxJp= Jpdrift+ Jpdiff= qpµpE − qDpdpdxJtotal= Jn+ JpIn general, total current can flow by drift and diffusion separately. Total current density:6.012 Spring 2007 Lecture 3 16What 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∝
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