lecture9

# lecture9 - 6.720J/3.43J Integrated Microelectronic Devices...

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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].

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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 q Gauss’ law: ± E ∇· ± = ± ( p n + N D + N A ) ± drift ± Electron current equation: J e = qn± v e + qD e n ± ± Hole current equation: J h = qp± v h h p ∂n ± Electron continuity equation: ∂t = G ext U ( n, p )+ 1 q J ± e ∂p ± Hole continuity equation: = G ext 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].

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6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 9-4 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 microns n + n + n p order of tenths of micron y 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 + N D N A ) Electron current equation: J e = qnv drift ( E )+ qD e ∂n e Hole current equation: J h = qpv ( E ) h ∂p h Electron continuity equation: = G ext U ( n, p 1 ∂J e ∂t q ∂x Hole continuity equation: = G ext U ( n, p ) 1 h Total current equation: J t = J e + J h Equation set diﬃcult because of coupling through Gauss’ law.

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## lecture9 - 6.720J/3.43J Integrated Microelectronic Devices...

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