EECS 105 Spring 2004 Lecture 8 EECS 105 Spring 2004 Lecture 8 Prof J S Smith Context In the last lecture we discussed how atoms come together to form insulators metals and semiconducors In this lecture we will cover z Electron and hole densities z Carrier Drift z Velocity Saturation z IC Process Flow z Resistor Layout z Diffusion Lecture 8 Electrons and hole currents IC Resistors Prof J S Smith Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 8 Prof J S Smith Department of EECS EECS 105 Spring 2004 Lecture 8 Announcements z z University of California Berkeley Prof J S Smith Thermal Equilibrium The midterm is scheduled for March 10 6 8 pm in Sibley Auditorium The third homework is due Wednesday 2 11 z z Balance between generation and recombination determines no po 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 G Gth T Gopt R k n p Department of EECS University of California Berkeley Department of EECS University of California Berkeley 1 EECS 105 Spring 2004 Lecture 8 Prof J S Smith EECS 105 Spring 2004 Lecture 8 Prof J S Smith Periodic Table of Elements Extra proton and therefore extra electron Electron and Hole densities But at thermal equilibrium generation and recombination must be equal z G R k n p Gth T 2 n p Gth T k ni T This holds true for doped as well as intrinsic silicon and we know z ni T 1010 cm 3 at 300 K Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 8 Prof J S Smith Department of EECS EECS 105 Spring 2004 Lecture 8 Prof J S Smith Doping with Group V Elements Law of Mass Action z University of California Berkeley This is called the law of Mass action the name is borrowed from a similar thermal equilibrium law from chemistry z P As group 5 extra bonding electron lost to crystal at room temperature 2 n p Gth T k ni T z 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 In thermal equilibrium if we increase the number of electrons the number of holes goes down and visa versa Department of EECS University of California Berkeley Immobile Charge Left Behind Department of EECS University of California Berkeley 2 EECS 105 Spring 2004 Lecture 8 Prof J S Smith EECS 105 Spring 2004 Lecture 8 Donor Accounting z Periodic Table of Elements Each ionized donor will contribute an extra electron If the material is charge neutral the total charge concentration must sum to zero One less proton and Therefore short an electron z Prof J S Smith qn0 qp0 qN d 0 Free Electrons z Free Holes Ions Immobile By Mass Action Law 2 n p ni T qn0 q 2 i n qN d 0 n0 qn02 qni2 qN d n0 0 Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 8 Prof J S Smith Department of EECS EECS 105 Spring 2004 Lecture 8 Donor Accounting cont z Solve quadratic n02 N d n0 ni2 0 n0 z z Boron 3 bonding electrons one bond is unsaturated Only free hole negative ion is immobile Only positive root is physically valid n0 z Prof J S Smith Doping with Group III Elements z N d N d2 4ni2 2 University of California Berkeley N d N d2 4ni2 2 For most practical situations N d ni 2 n N d N d 1 4 i N N N n0 d d Nd 2 2 2 Department of EECS University of California Berkeley Department of EECS University of California Berkeley 3 EECS 105 Spring 2004 Lecture 8 Prof J S Smith EECS 105 Spring 2004 Lecture 8 Mass Action Law z Compensation cont Balance between generation and recombination po no ni N type case 2 z T 300 K ni 10 cm n0 P type case p0 N a N a Department of EECS 2 i n Nd p0 More donors than acceptors Nd Na no N d N a ni 3 10 n0 N d N d ni2 Na po N a N d ni University of California Berkeley Nd Na Prof J S Smith no n i2 Na Nd Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 8 Prof J S Smith Thermal Equilibrium Compensation Dope with both donors and acceptors po n i2 More acceptors than donors Na Nd EECS 105 Spring 2004 Lecture 8 z Prof J S Smith Create free electron and hole Rapid random motion of holes and electrons at thermal velocity vth 107 cm s with an average time between collisions of c 10 13 s 1 2 mn vth2 12 kT Department of EECS University of California Berkeley Department of EECS University of California Berkeley 4 EECS 105 Spring 2004 Lecture 8 Prof J S Smith EECS 105 Spring 2004 Lecture 8 Prof J S Smith Mobility vs Doping in Silicon at 300 oK Applied E field Apply an electric field E and charge carriers accelerate for c seconds vth c zero E field 10 7 cm s 10 13 s 10 6 cm v th positive E a c hole case default values vth x Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 8 Prof J S Smith Drift Velocity and Mobility For holes p 400 n 1000 Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 8 Prof J S Smith Speed Limit Velocity Saturation c 3 1010 m s qE F q c E vdr a c e c m c m m p p p Thermal Velocity vdr p E For electrons 10 4 qE F q c E vdr a c e c m m c m p p p V V cm V 10 4 1 cm cm 10 4 m m vdr n E The field strength to cause velocity saturation may seem very large but it s only a few volts in a modern transistor Department of EECS University of California Berkeley Department of EECS University of California Berkeley 5 EECS 105 Spring 2004 Lecture 8 Prof J S Smith EECS 105 Spring 2004 Lecture 8 Drift Current Density Holes Prof J S Smith Resistivity Hole case drift velocity is in same direction as E hole drift current density Bulk silicon uniform doping concentration away from surfaces n type example in equilibrium no Nd Jpdr When we apply an electric field vdp E x n Nd J n q n nE q n N d E The hole drift current density is Conductivity n q n N d eff q n N d N a Jp dr q p p E Department of EECS Resistivity University of California Berkeley EECS 105 Spring 2004 Lecture 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