1EE105 - Fall 2006Microelectronic Devices and CircuitsProf. Jan M. Rabaey (jan@eecs)Lecture 2: Semiconductor Basics2Your EECS105 Week6pm5pmOffice HourGerald Wang4pmOffice HoursProf. Rabaey511 CoryLab353 CoryDiscussion293 Cory3pmLecture203 McLaughlinOffice HourAsako TodaLecture203 McLaughlinOffice HourNate Pletcher2pm1pmOffice HourRyan Roberts12pm11amLab353 Cory10amLab353 CoryLab353 Cory9amFrThWeTuMoDiscussion293 CoryDiscussion293 Cory23This week Discussion sessions and office hours starting this week Discussions: 293 Cory Office Hours TAs: 477 Cory Labs: 353 Cory Make-up Lecture: Friday September 8 3:30-5pm, 203 McLaughlin4At a glance Last class – Intro and some recap of circuit analysis Today: Review of Semiconductor Basics35Periodic Table of Elements6Electronic Properties of Silicon Silicon is in Group IV (atomic number 14)– Atom electronic structure: 1s22s22p63s23p2– Crystal electronic structure: 1s22s22p63(sp)4– Diamond lattice, with 0.235 nm bond length Very poor conductor at room temperature: why?(1s)2(2s)2(2p)6(3sp)4Hybridized State47The Diamond Structure3sp tetrahedral bondoA43.5oA35.28States of an Atom Quantum Mechanics: The allowed energy levels for an atom are discrete (2 electrons with opposite spin can occupy a state) When atoms are brought into close contact, these energy levels split If there are a large number of atoms, the discrete energy levels form a “continuous” bandEnergyE1E2...E3Forbidden Band GapAllowedEnergyLevelsLattice ConstantAtomic Spacing59Energy Band Diagram The gap between the conduction and valence band determines the conductive properties of the material Metal– negligible band gap or overlap Insulator – large band gap, ~ 8 eV Semiconductor– medium sized gap, ~ 1 eVValence BandConduction BandValence BandConduction Bande-Electrons can gain energy from lattice (phonon) or photon to become “free”band gape-10Model for Good Conductor The atoms are all ionized and a “sea” of electrons can wander about crystal: The electrons are the “glue” that holds the solid together Since they are “free”, they respond to applied fields and give rise to conductions+ + + + + ++ ++ + + + + + + ++ + + + + ++ +On time scale of electrons, lattice looks stationary…611Bond Model for Silicon (T=0K)Silicon Ion (+4 q)Four Valence ElectronsContributed by each ion (-4 q)2 electrons in each bond12Bond Model for Silicon (T>0K) Some bond are broken: free electron Leave behind a positive ion or trap (a hole)+-713Holes Notice that the vacancy (hole) left behind can be filled by a neighboring electron It looks like there is a positive charge traveling around! Treat holes as legitimate particles.+-14More About Holes When a conduction band electron encounters a hole, the process is called recombination The electron and hole annihilate one another thus depleting the supply of carriers In thermal equilibrium, a generation process counterbalances to produce a steady stream of carriers815Thermal Equilibrium (Pure Si) Balance between generation and recombination determines no= po Strong function of temperature: T = 300 oKoptthGTGG += )()(pnkR ×=RG =)()( TGpnkth=×)(/)(2TnkTGpnith==×K300atcm10)(310 −≅TniMass-action law16Doping with Group V Elements P, As (group 5): extra bonding electron … lost to crystal at room temperature+ImmobileChargeLeft Behind917Donor Accounting Each ionized donor will contribute an extra “free”electron The material is charge neutral, so the total charge concentration must sum to zero: By Mass-Action Law:000=++−=dqNqpqnρFree ElectronsFree HolesIons(Immobile))(2Tnpni=×0020=++−diqNnnqqn00220=++− nqNqnqndi18Donor Accounting (cont) Solve quadratic: Only positive root is physically valid: For most practical situations:2402202020iddidnNNnnnNn+±==−−24220iddnNNn++=idnN >>dddiddNNNNnNNn =+≈⎟⎠⎞⎜⎝⎛++=22241201019Doping with Group III Elements Boron: 3 bonding electrons Æ one bond is unsaturated Only free hole … negative ion is immobile!-+20Mass Action Law Balance between generation and recombination: 2ioonnp =⋅• N-type case: • P-type case:)cm10,K300(310 −==inTddNNn ≅=+0aaNNp ≅=−0diNnp20≅aiNnn20≅1121Compensation Dope with both donors and acceptors: – Create free electron and hole!+--+22Compensation (cont.) More donors than acceptors: Nd> NaiadonNNn >>−=adoNNnpi−=2idaonNNp >>−=daoNNnni−=2 More acceptors than donors: Na> Nd1223Disturbing the EquilibriumRapid, random motion of holes and electrons at “thermal velocity” vth= 107cm/s with collisions every τc = 10-13s.Apply an electric field E and charge carriers accelerate … for τc secondszero E fieldvthpositive E vthaτ c (hole case)xkTvmthn212*21=cthvτλ=cm1010/cm106137 −−=×= ssλ(mean free path - 10 nm)24Drift Velocity and MobilityEvpdrμ=EmqmqEmFavpccpcpecdr⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛=⋅=ττττFor electrons: EmqmqEmFavnccncnecdr⎟⎟⎠⎞⎜⎜⎝⎛−=⎟⎟⎠⎞⎜⎜⎝⎛−=⎟⎟⎠⎞⎜⎜⎝⎛=⋅=ττττFor holes: Evndrμ−=1325“default” values: Mobility vs. Doping in Silicon at 300K1000=nμ400=pμ26Speed Limit: Velocity Saturations/m10310×=cThermal VelocityThe field strength to cause velocity saturation may seem very largebut it’s only a few volts in a modern transistor!mmμμV110cmcmV10cmV10444==1427Drift Current Density (Holes)The hole drift current density is: Jpdr = q p μp EHole case: drift velocity is in same direction as Ehole driftcurrent densityxEvdp Jpdr 28Drift Current Density (Electrons)electron driftcurrent densityxEvdn Jndr Electron case: drift velocity is in opposite direction as EThe electron drift current density is:Jndr= (-q) n vdnunits: C/s/cm2= A/cm2EqnEnqJnndrnμμ=−−= )(()EqnqpJJJnpdrdrpnμμ+=+=1529ResistivityBulk silicon: uniform doping concentration, away from surfaces n-type example: in equilibrium, no= NdWhen we apply an electric field, n = NdENqnEqJdnnnμμ==ResistivityConductivity)(, adneffdnnNNqNq −==μμσeffdnnnNq,11μσρ==cm−Ω30Ohm’s Law()RVVLWtLVWtEWttWJJAI =⋅⎟⎠⎞⎜⎝⎛===⋅==σσσWLtWLtRρσ==11631Sheet Resistance (Rs) IC resistors have a specified thickness – not under the control of the circuit designer Eliminate t by absorbing it into a new parameter: the sheet resistance
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