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lecture8 - 6.720J/3.43J Integrated Microelectronic Devices...

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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 8-1 Lecture 8 - Carrier Drift and Diffusion (cont.) , Carrier Flow February 21, 2007 Contents: 1. Quasi-Fermi levels 2. Continuity equations 3. Surface continuity equations Reading assignment: del Alamo, Ch. 4, § 4.6; Ch. 5, §§ 5.1, 5.2 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 8-2 Key questions Is there something equivalent to the Fermi level that can be used outside equilibrium? How do carrier distributions in energy look like outside equilib- rium? In the presence of carrier flow in the bulk of a semiconductor, how does one formulate bookkeeping relationships for carriers? How about at surfaces? 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 8-3 1. Quasi-Fermi levels Interested in energy band diagram representations of complex sit- uations in semiconductors outside thermal equilibrium . In TE, Fermi level makes statement about energy distribution of carriers in bands E F relates n o with N c and p o with N v : E F E c E v E F n o = N c F 1 / 2 ( ) p o = N v F 1 / 2 ( ) kT kT Outside TE, E F cannot be used. Define two ”quasi-Fermi levels” such that: E fe E c E v E fh n = N c F 1 / 2 ( ) p = N v F 1 / 2 ( ) kT kT Under Maxwell-Boltzmann statistics ( n N c , p N v ): E fe E c n = N c exp kT E v E fh p = N v exp kT What are quasi-Fermi levels good for? 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 8-4 Take derivative of n = f ( E fe ) with respect to x : E fe E c n = N c exp kT dn n dE fe dE c n dE fe q = ( ) = n E dx kT dx dx kT dx kT Plug into current equation: dn J e = e n E + qD e dx To get: dE fe J e = μ e n dx Similarly for holes: dE fh J h = μ h p dx Gradient of quasi-Fermi level: unifying driving force for carrier 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].
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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 8-5 Physical meaning of E f For electrons, dE fe J e = μ e n = qnv e dx Then: dE fe q = v e dx μ e E fe linearly
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