EECS 105 Spring 2004 Lecture 11 Lecture 11 P N Diode capacitors intro to small signal models Prof J S Smith Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 11 Prof J S Smith Announcements z z z Reading Finish chapter 3 in the text Next week we will be starting on MOS transistors chapter 4 Don t come to lecture on Monday it s a holiday Department of EECS University of California Berkeley 1 EECS 105 Spring 2004 Lecture 11 Prof J S Smith Context In 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 Berkeley EECS 105 Spring 2004 Lecture 11 Prof J S Smith MOS capacitor z z The PN diode as a variable capacitor varactor is useful in its own right 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 Department of EECS University of California Berkeley 2 EECS 105 Spring 2004 Lecture 11 Prof J S Smith PN diode Under Reverse Bias n VD n n p VD E0 p Xd0 z z z z E0 p X d VD Under thermal equilibrium current is zero If we apply a reverse bias we are increasing the barrier against diffusion current Drift current is low since the field only moves minority carriers across junction 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 Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 11 Prof J S Smith P Type N Type p0 x p0 N a Hole population E0 x p0 n0 x n0 Fixed positive charges Fixed negative charges PN Junction populations ni2 Na E0 J p diff J p drift p0 If we increase the field above its equilibrium value reverse bias the diffusion current will be suppressed but the current will only pull the minority carriers ni2 drift in the reverse direction so the current under backward bias remains very small N d xn 0 J n diff n0 N d Electron population J n drift Depletion Region mobile carrier density fixed charge up to near the edges Department of EECS University of California Berkeley 3 EECS 105 Spring 2004 Lecture 11 Prof J S Smith Plot of Fields In Depletion Region p type NA n type ND Depletion Region E0 x z z z z z qN a s x x po E0 x qN d s xn 0 x E Field zero outside of depletion region Note the asymmetrical depletion widths Which region has higher doping Slope of E Field larger in n region Why Peak E Field at junction Why continuous Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 11 Prof J S Smith Reverse bias z 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 Larger than equilibrium P type N type Department of EECS p x n x ni2 University of California Berkeley 4 Forward or reverse bias abrupt junction full depletion model EECS 105 Spring 2004 Lecture 11 z Forward bias Prof J S Smith qN d x p0 Depletion region narrows xn 0 qN a z qN d Reverse Bias x p 0 xn 0 Department of EECS Depletion region is larger In any case charge must balance so qN a x p qN d xn qN a University of California Berkeley EECS 105 Spring 2004 Lecture 11 Prof J S Smith Electrostatics 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 x d 2 x 2 dx Notice that this says that in a region of no charge the potential will change linearly which is a constant E field Department of EECS University of California Berkeley 5 EECS 105 Spring 2004 Lecture 11 Prof J S Smith Abrupt junction full depletion model For 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 x qN d x p0 xn 0 Department of EECS Where xn0 and xp0 are the widths of the depletion regions extending into their respective doped regions qN a University of California Berkeley EECS 105 Spring 2004 Lecture 11 Prof J S Smith Abrupt junction full depletion model z We find the potential from x d 2 x 2 dx z Integrating twice we find qN a 2 x Ap x B p 2 qN x d x 2 An x Bn 2 x z xp x 0 0 x xn We use the boundary conditions to find xn x p and the values of the constants Department of EECS University of California Berkeley 6 EECS 105 Spring 2004 Lecture 11 Prof J S Smith Boundary conditions z Since the dielectric constant is the same and there are no sheet charges at x 0 we have 0 0 z And qN a 2 qN x Ap x B p d x 2 An x Bn 2 2 B p Bn r r E 0 E 0 d d 0 0 dx dx qN qN 2 d x An 2 a x Ap 2 2 x 0 Notice that the E field at the junction is equal to A x 0 Ap An Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 11 Prof J S Smith More BC s z So we now have qN a 2 x Ax B 2 qN x d x 2 Ax B 2 x z xp x 0 0 x xn We also know that outside the depletion region the E field will be small so r d E x p x 0 dx x x p 2 Department of EECS qN a x A 0 2 x x p qN a xp A University of California Berkeley 7 EECS 105 Spring 2004 Lecture 11 Prof J S Smith Calculating the depletion depth z Depletion depth qN a z x p A A qN a r Since E x 0 A we have xp z xp E x 0 qN a This will be useful later for calculating depletion caused by an E field penetrating into semiconductor regions in general This can also be interpreted as the that the electric field must terminate Esurface qNx on a charge per area of E x …
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