6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 9-1 Lecture 9 - Carrier Flow (cont.) February 23, 2007 Contents: 1. Shockley’s Equations 2. Simplifications of Shockley equations to 1D quasi-neutral situations 3. Majority-carrier type situations Reading assignment: del Alamo, Ch. 5, §§5.3-5.5 Quote of the day: ”If in discussing a semiconductor problem, you cannot draw an energy band diagram, then you don’t know what you are talking about.” -H. Kroemer, IEEE Spectrum, June 2002. Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 9-2 Key questions • How can the equation set that describes carrier flow in semicon-ductors be simplified? • In regions where carrier concentrations are high enough, quasi-neutrality holds in equilibrium. How about out of equilibrium? • What characterizes majority-carrier type situations? Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 9-3 1. Shockley’s Equations qGauss’ law: E∇·=  (p − n + ND + − NA −) drif t Electron current equation: Je = −qnve + qDe∇n drif t Hole current equation: Jh = qpvh − qDh∇p ∂n Electron continuity equation: ∂t = Gext − U(n, p)+ 1 q ∇·Je ∂p Hole continuity equation: ∂t = Gext − U(n, p) −1 q ∇·Jh Total current equation: Jt = Je + Jh System of non-linear, coupled partial differential equations. Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 9-4 2. Simplifications of Shockley equations to 1D quasi-neutral situations � One-dimensional approx imation In many cases, complex problems can be broken into several 1D subproblems. Example: integrated p-n diode order of microns n+ n+ n porder of tenths of microny x p 1D approximation: ∂x∇⇒ ∂ Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 9-5 Shockley’s equations in 1D: Gauss’ law: ∂∂x E = q(p − n + ND − NA) Electron current equation: Je = −qnvdrif t (E)+ qDe∂n e ∂x Hole current equation: Jh = qpv drif t (E) − qDh∂p h ∂x Electron continuity equation: ∂n = Gext − U(n, p)+ 1 ∂Je ∂t q ∂x Hole continuity equation: ∂p = Gext − U(n, p) − 1 ∂Jh ∂t q ∂x Total current equation: Jt = Je + Jh Equation set difficult because of coupling through Gauss’ law. Two broad classes of important situations break Gauss’law coupling: 1. Carrier concentrations are high: quasi-neutral situation: ∂E ρ  0 ⇒  0 ∂x 2. Carrier concentrations are very low: space-charge and high-resistivity situations: E independent of n, p Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 9-6 Overview of simplified carrier flow formulations General drift-diffusion situation (Shockley's equations) 1D approx. Quasi-neutral situation (negligible volume charge) Space-charge situation (field independent of n, p) Majority-carrier Minority-carrier type situation type situation (V=0, n'=p'=0) (V=0, n'=p'=0, LLI) Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 9-7 Each formulation uniquely applies to a different region in a device. Example: npn BJT in forward-active regime base emitter base collector contactcontact contact contact n+ emitter p base n collector n+ buried layer n+ plug p substrate situation electron flow n n p minority-carrier type situation space-charge majority-carrier type situation Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 9-8 � Quasi-neutral approx imation At every location, the net volume charge that arises from a dis-crepancy of the concentration of positive and negative species is negligible in the scale of the charge density that is present. AQN approximation eliminates Gauss’ law from the set: + D+ N − N−− N− • Quasi-neutrality in equilibrium: A− n ) + D ρ = q(p − n + N )= q(po − no )+ q(p −−  −N NnpAo o −− NA −−N NA + D+ D po − no + N | |1 which implies + D • Additionally, quasi-neutrality outside equilibrium: p− n p− n ||||1 n pwhich implies: p  n  • QN approximation good if n, p high ⇒ carriers move to erase ρ. • QN holds if length scale of problem  Debye length Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 9-9 � Consequence of quasi-neutrality 1. Uncouple Gauss’ law from rest of system: ∂E ρ  0 ⇒  0 ∂x If, in general define: E = Eo + E Then, in equilibrium: ∂Eo q = (po − no + ND + − NA −)∂x and out of equilibrium: ∂E q = (p − n )∂x Eo computed as in Ch. 4. Here will learn to compute E . Cite as: Jesús del Alamo, course materials for