L9: Quasi-Fermi Levels, Continuity Equations

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, quasineutrality 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 Gauss’ law:  · E = q (p − n + ND+ − NA− ) ∇  Electron current equation:  Je = −qn vedrif t + qDe ∇n Hole current equation:  Jh = qpvh drif t − qDh∇p Electron continuity equation: ∂n ∂t  · Je = Gext − U (n, p) + 1q ∇ Hole continuity equation: ∂p ∂t  · Jh = Gext − U (n, p) − 1q ∇ Total current equation: Jt = Je + Jh 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]. Lecture 9-4 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 2. Simplifications of Shockley equations to 1D quasi-neutral situations � One-dimensional approximation In many cases, complex problems can be broken into several 1D subproblems. Example: integrated p-n diode order of tenths of micron order of microns p y n+ n n+ 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: ∂E ∂x Electron current equation: Je = −qnvedrif t (E) + qDe ∂n ∂x Hole current equation: ∂p Jh = qpvhdrif t (E) − qDh ∂x Electron continuity equation: ∂n ∂t e = Gext − U (n, p) + 1q ∂J ∂x Hole continuity equation: ∂p ∂t h = Gext − U (n, p) − 1q ∂J ∂x Total current equation: Jt = Je + Jh = q (p − n + ND − NA ) 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: ρ0 ⇒ ∂E 0 ∂x 2. Carrier concentrations are very low: space-charge and highresistivity 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]. Lecture 9-6 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Overview of simplified carrier flow formulations General drift-diffusion situation (Shockley's equations) 1D approx. Quasi-neutral situation (negligible volume charge) Majority-carrier type situation (V=0, n'=p'=0) Space-charge situation (field independent of n, p) Minority-carrier type situation (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]. Lecture 9-7 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Each formulation uniquely applies to a different region in a device. Example: npn BJT in forward-active regime base contact emitter contact base contact collector contact n+ emitter p base n+ plug n collector n+ buried layer p substrate minority-carrier type situation p electron flow n n space-charge situation 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 approximation 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. QN approximation eliminates Gauss’ law from the set: ρ = q(p − n + ND+ − NA− ) = q(po − no + ND+ − NA− ) + q(p − n ) • Quasi-neutrality in equilibrium: po − no + ND+ − NA− |1 | ND+ − NA− which implies no − po  ND+ − NA− • Additionally, quasi-neutrality outside equilibrium: p − n p − n | || |1 n p which 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: ρ0 ⇒ ∂E 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 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 9-10 2. Subtract one continuity equation from the other: ∂(n − p) ∂ρ ∂Jt =q =− ∂x ∂t ∂t continuity equation for net volume charge: if Jt changes with po­ sition, ρ changes with time. Easier to see in integral form: �  =−∂ t .dS J S ∂t � V ρ dV • In Static case: ∂ρ ∂t =0⇒ ∂Jt ∂x = 0, Jt independent of x • In Dynamic case, we also have in most useful situations: ∂ρ ∂t  0 in times scale of interest [will discuss soon] 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 9-11 Simplified set of Shockley equations for 1D quasi-neutral situations p − n + ND − NA  0 Je = −qnvedrif t + qDe ∂n ∂x ∂p Jh = qpvhdrif t − qDh ∂x ∂n ∂t e = Gext − U + 1q ∂J ∂x or ∂Jt ∂x ∂p ∂t h = Gext − U − 1q ∂J ∂x 0 Jt = Je + Jh 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]. Lecture 9-12 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 3. Majority-carrier type situations Voltage applied to extrinsic quasi-neutral semiconductor without up­ setting the equilibrium carrier concentrations. � Remember what a battery does: ε R r electron flow Ef resisto qV electron flow current flow V V circuit view energy view • Battery picks up electrons from positive terminal, increases their potential energy and puts them at the negative terminal. • If provided with a path (resistor), electrons flow. 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 9-13 � Characteristics of majority carrier-type situations: • electric field imposed from outside • electrons and holes drift • electron and hole concentrations unperturbed from TE Simplifications: • neglect contribution of minority carriers • neglect time derivatives of carrier concentrations ⇒ problem becomes completely quasi-static 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 9-14 � Simplification of majority carrier current (n-type): Must distinguish between internal field in TE (Eo ) and total field outside equilibrium (E). For simplicity, do in low-field limit (exact case done in notes). In equilibrium: Jeo = qμe noEo + qDe dno =0 dx Out of equilibrium: Je  qμe noE + qDe dno dx Hence: Je = qμeno (E − Eo) = qμe noE  In the more general case (see notes): Je = −qno[vde (E) − vde (Eo )] 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]. Lecture 9-15 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 � Equation set for 1D majority-carrier type situations: n-type n  no  ND p-type p  po  NA Je = −qno[vde (E) − vde (Eo )] Jh = qpo [vdh(E) − vdh(Eo )] dJe dx Jt  Je  0, dJh dx  0, dJt dx 0 Jt  Jh 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]. Lecture 9-16 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 � Example 1: Integrated Resistor with uniform doping (n-type) L z W n n+ x n+ p top view x n+ n t n+ y p cross-sectional view Uniform doping ⇒ Eo = 0, then: Jt = −qND vedrif t (E) • If E not too high, Jt  qND μe E I-V characteristics: I = W tqND μe V L 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]. Lecture 9-17 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 • In general (low and high fields): I = W tqND vsat 1 + vμsate VL which for high fields saturates to: Isat = W tqND vsat I Isat 0 0 V 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]. Lecture 9-18 6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Key conclusions • Shockley equations: system of equations that describes carrier phenomena in semiconductors in the drift-diffusion regime. • Quasi-neutral approximation appropriate if semiconductor is sufficiently extrinsic: ρ  0 ⇒ no − po  ND − NA n  p • Consequence of quasi-neutrality: Jt = Jt (x) • Majority carrier-type situations characterized by application of external voltage without perturbing carrier concentrations. • Majority-carrier type situations dominated by drift of majority carriers. • Integrated resistor: – for low voltages, current proportional to voltage across – for high voltages, current saturates due to vsat 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 9-19 Self-study • Integral form of continuity equations and consequences. • Exercises 5.1, 5.2. • Non-uniformly doped resistor • Sheet resistance 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].