EECS 105 Spring 2004 Lecture 8 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 Announcements z z The midterm is scheduled for March 10 6 8 pm in Sibley Auditorium The third homework is due Wednesday 2 11 Department of EECS University of California Berkeley 1 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 Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 8 Prof J S Smith Thermal Equilibrium 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 2 EECS 105 Spring 2004 Lecture 8 Prof J S Smith 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 Law of Mass Action z This is called the law of Mass action the name is borrowed from a similar thermal equilibrium law from chemistry 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 3 EECS 105 Spring 2004 Lecture 8 Prof J S Smith Extra proton and therefore extra electron Periodic Table of Elements Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 8 Prof J S Smith Doping with Group V Elements z P As group 5 extra bonding electron lost to crystal at room temperature Immobile Charge Left Behind Department of EECS University of California Berkeley 4 EECS 105 Spring 2004 Lecture 8 Prof J S Smith Donor Accounting z z Each ionized donor will contribute an extra electron If the material is charge neutral the total charge concentration must sum to zero qn0 qp0 qN d 0 Free Electrons z Free Holes Ions Immobile By Mass Action Law 2 n p ni T ni2 qn0 q 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 Donor Accounting cont z Solve quadratic n02 N d n0 ni2 0 n0 z N d N d2 4ni2 2 Only positive root is physically valid N d N d2 4ni2 n0 2 z 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 5 EECS 105 Spring 2004 Lecture 8 Prof J S Smith One less proton and Therefore short an electron Periodic Table of Elements Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 8 Prof J S Smith Doping with Group III Elements z z Boron 3 bonding electrons one bond is unsaturated Only free hole negative ion is immobile Department of EECS University of California Berkeley 6 EECS 105 Spring 2004 Lecture 8 Prof J S Smith Mass Action Law z Balance between generation and recombination po no ni N type case 2 T 300 K ni 1010 cm 3 n0 N d N d P type case p0 N a n0 Na ni2 Nd ni2 p0 Na Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 8 Prof J S Smith Compensation z Dope with both donors and acceptors Create free electron and hole Department of EECS University of California Berkeley 7 EECS 105 Spring 2004 Lecture 8 Prof J S Smith Compensation cont z More donors than acceptors Nd Na no N d N a ni po n i2 Nd Na More acceptors than donors Na Nd po N a N d ni no n i2 Na Nd Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 8 Prof J S Smith Thermal Equilibrium 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 Department of EECS mn vth2 12 kT University of California Berkeley 8 EECS 105 Spring 2004 Lecture 8 Prof J S Smith 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 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 qE F c q c E vdr a c e c m m m p p p vdr p E For electrons F qE c q c E vdr a c e c m m m p p p vdr n E Department of EECS University of California Berkeley 9 EECS 105 Spring 2004 Lecture 8 Prof J S Smith Mobility vs Doping in Silicon at 300 default values oK 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 Thermal Velocity 10 4 V V cm V 10 4 1 4 m cm 10 m cm 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 10 EECS 105 Spring 2004 Lecture 8 Prof J S Smith Drift Current Density Holes Hole case drift velocity is in same direction as E hole drift current density Jpdr v dp E x The hole drift current density is Jp dr q p p E Department of EECS University of California Berkeley EECS 105 Spring 2004 Lecture 8 Prof J S Smith Drift Current Density Electrons Electron case drift velocity is in opposite direction as E electron drift current density Jndr v dn E J ndr q n n E qn n E x The electron drift current density is Jndr q n vdn units Ccm 2 s 1 Acm 2 J J pdr J ndr …
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