R. T. HoweEECS 105 Spring 2005 Lecture 3Dept. of EECSUniversity of California, BerkeleyLecture 3• Last time:– Drift current density– Ohm’s and resistivity and IC resistors• Today :– Applied electrostatics• Gauss’s Law, boundary conditions– IC capacitors: metal-metal and pn junctionR. T. HoweEECS 105 Spring 2005 Lecture 3Dept. of EECSUniversity of California, BerkeleyElectrostatics:a Tool for Device ModelingPhysics approach: vector calculus, highly symmetrical problemsεE()∇• ρ=E φ∇–=εφ∇–()()∇•ε–φ∇2ρ==R. T. HoweEECS 105 Spring 2005 Lecture 3Dept. of EECSUniversity of California, BerkeleyOne-Dimensional Electrostaticsd εE()dx-------------- ρ=Edφdx------–=ddx------εdφdx------–εd2φdx2---------– ρ==Gauss’s LawPotential Def.Poisson’s Eqn.R. T. HoweEECS 105 Spring 2005 Lecture 3Dept. of EECSUniversity of California, BerkeleyBoundary Conditionsxφ(x)012φ x=0−()φ x=0+()=ε1Ex=0−()Q+ ε2Ex=0+()=Q is a surface charge (C/cm2) located at x = 0R. T. HoweEECS 105 Spring 2005 Lecture 3Dept. of EECSUniversity of California, BerkeleySilicon/Silicon Dioxide InterfacexE(x)0ε1= 3ε221common materials: silicon, εs= 11.7 εosilicon dioxide (SiO2), εox=3.9 εo12R. T. HoweEECS 105 Spring 2005 Lecture 3Dept. of EECSUniversity of California, BerkeleyGetting Past the Math• Electric field vector points from positive to negative charge• Electric field points “downhill” on a plot of potential• Electric field is confined to a narrow charged region, in which the positive charge is balanced by an equal and opposite negative charge• Boundary conditions on potential or electric field can “patch” together solutions from regions of differing material properties• Gauss’s law in integral form relates the electric field at the edges of a region to the charge inside. Often, the field on one side is known to be zero (e.g., because it’s on the outside of the charged region), which allows the electric field at an interface to be solved for directlyR. T. HoweEECS 105 Spring 2005 Lecture 3Dept. of EECSUniversity of California, BerkeleyIC CapacitorsMetal layers separated by insulators Æ get intentional (or parasitic) capacitor dielectricMetal 2Metal 1ddtCε=R. T. HoweEECS 105 Spring 2005 Lecture 3Dept. of EECSUniversity of California, BerkeleySurface Charge and Electric Field x 0 td Q (C/cm2) + - V x td V QtdExR. T. HoweEECS 105 Spring 2005 Lecture 3Dept. of EECSUniversity of California, BerkeleyMetal-Metal Capacitor Layout A A’ A’A Overlap area A12 1 2R. T. HoweEECS 105 Spring 2005 Lecture 3Dept. of EECSUniversity of California, BerkeleyCircuit Model• Capacitance between metal 1 and metal 2:• Other capacitors: what is terminal 3?1212AtCdd=εR. T. HoweEECS 105 Spring 2005 Lecture 3Dept. of EECSUniversity of California, BerkeleyComplete Circuit
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