1Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8Lecture 8: Electrons and hole currents, IC Resistors Prof. J. S. SmithDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithAnnouncementsz The midterm is scheduled for March 10, 6-8 pm, in Sibley Auditorium z The third homework is due Wednesday 2/112Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithContext:In the last lecture, we discussed how atoms come together to form insulators, metals and semiconducorsIn this lecture, we will cover:z Electron and hole densities z Carrier Driftz Velocity Saturation z IC Process Flow z Resistor Layoutz Diffusion Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithThermal Equilibriumz Balance between generation and recombination determines no= poz Generation is a function of temperature G(T), but recombination only depends on the number of electrons and holes n(r,t) × p(r,t), because electrons and holes are rare.optthGTGG+=)()( pnkR⋅=3Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithElectron and Hole densitiesz But at thermal equilibrium, generation and recombination must be equal:z This holds true for doped as well as intrinsic silicon, and we know:RG=)()( TGpnkth=⋅)(/)(2TnkTGpnith==⋅K300atcm10)(310 −≅TniDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithLaw of Mass Actionz This is called the law of Mass action (the name is borrowed from a similar thermal equilibrium law from chemistry)z This wouldn’t be of much use, except for the fact that we can vary the number of electrons and holes by adding fixed charges to the crystal—by adding nuclei which have an extra proton, or one fewer that silicon.z In thermal equilibrium, if we increase the number of electrons, the number of holes goes down, and visa versa)(/)(2TnkTGpnith==⋅4Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithPeriodic Table of ElementsExtra proton, and therefore extra electronDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithDoping with Group V Elementsz P, As (group 5): extra bonding electron … lost to crystal at room temperature+ImmobileChargeLeft Behind5Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithDonor Accountingz Each ionized donor will contribute an extra electronz If the material is charge neutral, the total charge concentration must sum to zero: z By Mass-Action Law:000=++−=dqNqpqnρFree ElectronsFree HolesIons(Immobile))(2Tnpni=⋅0020=++−diqNnnqqn00220=++− nqNqnqndiDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithDonor Accounting (cont)z Solve quadratic:z Only positive root is physically valid:z For most practical situations:2402202020iddidnNNnnnNn+±==−−24220iddnNNn++=idnN >>dddiddNNNNnNNn =+≈⎟⎠⎞⎜⎝⎛++=22241206Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithPeriodic Table of ElementsOne less proton, and Therefore short an electronDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithDoping with Group III Elementsz Boron: 3 bonding electrons Æ one bond is unsaturatedz Only free hole … negative ion is immobile!-7Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithMass Action Lawz Balance between generation and recombination: 2ioonnp =⋅• N-type case: • P-type case:)cm10,K300(310 −==inTddNNn ≅=+0aaNNp ≅=−0diNnn20≅aiNnp20≅Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithCompensationz Dope with both donors and acceptors: – Create free electron and hole!+--+8Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithCompensation (cont.)z More donors than acceptors: Nd> NaiadonNNn >>−=• More acceptors than donors: Na> NdadoNNnpi−=2idaonNNp >>−=daoNNnni−=2Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithThermal EquilibriumRapid, random motion of holes and electrons at “thermal velocity” vth≈ 107cm/s with an average time between collisions of τc = 10-13s .kTvmthn212*21=9Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithApplied E fieldApply an electric field E and charge carriers accelerate … for τc secondszero E fieldvthpositive E vthaτ c (hole case)xcthvτλ=cm1010/cm106137 −−=×= ssλDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithDrift Velocity and MobilityEvpdrµ=EmqmqEmFavpccpcpecdr⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛=⋅=ττττFor electrons: EmqmqEmFavpccpcpecdr⎟⎟⎠⎞⎜⎜⎝⎛−=⎟⎟⎠⎞⎜⎜⎝⎛−=⎟⎟⎠⎞⎜⎜⎝⎛=⋅=ττττFor holes: Evndrµ−=10Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. Smith“default” values: Mobility vs. Doping in Silicon at 300 oK1000=nµ400=pµDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithSpeed 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==11Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithDrift 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 Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S. SmithDrift 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: Ccm-2s-1 = Acm-2EqnEnqJnndrnµµ=−−= )(()EqnqpJJJnpdrdrpnµµ+=+=12Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 8 Prof. J. S.
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