1Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11Lecture 11: P-N Diode capacitors, intro to small signal modelsProf. J. S. SmithDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithAnnouncementsz Reading: Finish chapter 3 in the textz Next week we will be starting on MOS transistors, chapter 4z Don’t come to lecture on Monday, it’s a holiday!2Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithContextIn the last lecture, we looked at the PN diode at equilibrium, and under bias, and several applications for PN diodes. We discussed a simple model for the PN diode, for an abrupt junction, and for a sharp edge depletion In the this lecture, we will look at the reverse biased PN diode as a variable capacitor, and solve for the fields, voltages, and currents.We will also use this device to introduce the concept of a small signal model.Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithMOS capacitorz The PN diode as a variable capacitor (varactor) is useful in its own right: z And we will also draw on this analysis to model field effect transistors, where the action of the gate on the channel is similar to this analysis.3Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithPN diode Under Reverse Biasz Under thermal equilibrium current is zero z If we apply a reverse bias, we are increasing the barrier against diffusion currentz Drift current is low since the field only moves minority carriers across junctionz The small current under a reverse bias is due to minority carriers which diffuse into the depletion region from either side, and from generation (thermal generation, or light)pn+−DV−pφnφDnV+φpφ0dX)(DdVX0E0EDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithPN Junction populations)(0xpaNp =0diNnp20=diffpJ−0EaiNnn20=Depletion Region: mobile carrier density << fixed charge up to near the edgesdiffnJ−dNn =0––+ +0E0px−0nxHole populationElectron populationP Type N TypeFixed negative chargesFixed positive chargesIf we increase the field above itsequilibrium value, (reverse bias) the diffusion current will be suppressed, but thedrift current will only pull the minority carriers in the reverse direction, so the current under backward bias remains very smalldriftpJ−driftnJ−)(0xn4Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithPlot of Fields In Depletion Regionz E-Field zero outside of depletion regionz Note the asymmetrical depletion widthsz Which region has higher doping?z Slope of E-Field larger in n-region. Why?z Peak E-Field at junction. Why continuous?n-typep-typeNDNA––––––––––––––––––––+ + + + ++ + + + ++ + + + ++ + + + +DepletionRegion)()(00xxqNxEnsd−−=ε)()(0 posaxxqNxE +−=εDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithReverse biasz Under a reverse bias, a voltage is applied which increases the built in field, pulling the mobile carriers out of the depletion region. The drift current rises only slightly, because only the minority carriers and generated carriers get pulled across the region.P typeN typeφ∆Larger than equilibrium2)()(inxnxp <5Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithForward or reverse bias (abrupt junction, full depletion model)z Forward biasz Reverse BiasaqN−dqN+{{0nx0pxDepletion region narrowsaqN−dqN+{{0nx0pxDepletion region is largerndpaxqNxqN=In any case, charge must balance so:Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithElectrostatics (1-D)From Maxwell’s equations and the definition of the potential (voltage) we have a differential equation for the fields due to the charge distribution Notice that this says that in a region of no charge, the potential will change linearly, which is a constant E field. ερφ)()(22xdxxd−=6Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithAbrupt junction, full depletion modelFor the Abrupt junction, full depletion model of the PN diode, we can find the potential as a function of position by integrating over the charge distribution)(xρaqN−dqN+{{0nx0pxWhere xn0and xp0are thewidths of the depletion regionsextending into their respectivedoped regionsDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithAbrupt junction, full depletion modelz We find the potential from:z Integrating twice, we find:z We use the boundary conditions to find and the values of the constants.ερφ)()(22xdxxd−=nndppaBxAxqNxBxAxqNx++−=++=222)(2)(εφεφnpxxxx<<<<−00pnxx ,7Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithBoundary conditionsz Since the dielectric constant is the same, and there are no sheet charges at x=0, we have:z And:npnndppaBBBxAxqNBxAxqN=++−=++=+−2222)0()0(εεφφnpxndxpaAAAxqNAxqNdxddxdEE=+−=+−=−===+−+−002222)0()0()0()0(εεφφrrNotice that the E field at thejunction is equal to ADepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithMore BC’sz So we now havez We also know that outside the depletion region, the E field will be small, soBAxxqNxBAxxqNxda++−=++=222)(2)(εφεφ0220)()(=+=−=−=−=ppxxaxxpAxqNxdxdxEεφrAxqNpa−=εnpxxxx<<<<−008Department of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithCalculating the depletion depthz Depletion depth:z Since we have:z This can also be interpreted as the that the electric field must terminate on a charge per area of soAxqNpa−=− )(εapqNAxε−=−AxE =− )0(rapqNxExε)0( =−=−)0(=xEεqNxEsurface=εThis will be useful later, for calculatingdepletion caused by an E field penetrating into semiconductor regions in generalDepartment of EECS University of California, BerkeleyEECS 105 Spring 2004, Lecture 11 Prof. J. S. SmithDepletion depth, n region (Nd)z The depletion depth into the n region is calculated the same way:and:AxqNnd=εdnqNAxε=dnqNxExε)0( ==9Department of
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