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6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 3-1 Lecture 3 - Carrier Statistics in Equilibrium (cont.) February 9, 2007 Contents: 1. Equilibrium electron concentration 2. Equilibrium hole concentration 3. np product in equilibrium 4. Location of Fermi level Reading assignment: del Alamo, Ch. 2, §§2.4-2.6 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 3-2 Key questions • How many electrons and holes are there in thermal equilibrium in a given semiconductor? • How does the equilibrium electron (hole) distribution in the con-duction (valence) band look like? • How can one compute ni? • Where is the Fermi level in a given semiconductor? How does its location depend on doping level? 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 3-3 Carrier statistics in equilibrium Question: how many electrons and holes are there in TE in a given semiconductor? Answer: rigorous model exploiting energy view of semiconductors and concept of Fermi level. Strategy to answer question: 1. derive relationship between no and EF 2. derive relationship between po and EF 3. derive expressions for nopo and ni 4. figure out location of EF from additional arguments (such as charge neutrality) 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 3-4 1. Equilibrium electron concentration • Q: How many electrons are there in a semiconductor in TE? A: It depends on location of EF . • Why? Because EF defines probability that states are occupied by electrons. • The closer EF is to the conduction band edge, the more electrons there are in the conduction band. EEcEF no(E) f(E) gc(E-Ec) Ec E • If EF is not too close to the conduction band edge, what is the relationship between the location of EF and no? EF − Ec no ∝ exp kT Because high energy tail of Fermi-Dirac distribution function is ex-ponential with kT as characteristics energy. 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 3-5 Formally, no obtained by integrating electron concentration in entire conduction band:  ∞ no = no(E) dE Ec At a certain energy, no(E) is the product of CB density of states and occupation probability: no(E)= gc(E) f(E) Then: ⎛ ⎞ √ 2m ∗ 3/2 ∞ E − Ec ⎝ ⎠no =4π de dE h2 Ec 1+ exp E−kTEF Refer energy scale to Ec and normalize by kT . That is, define: E − Ec EF − Ecη = ηc = kT kT Then: ⎛ ⎞ √ 2m ∗ 3/2 ∞ ηdekT ⎝ ⎠no =4π dη h20 1+ eη−ηc 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 3-6 Define also: ⎛ ⎞3/22πm∗ dekT ⎝ ⎠Nc =2 h2 Nc ≡ effective density of states of the conduction band (cm−3) For Si at 300 K, Nc  2.9 × 1019 cm−3 . Then: no = NcF1/2(ηc) with: √ 2 ∞ η F1/2(ηc)= √ dη π 0 1+ eη−ηc F1/2(x)is Fermi integral of order 1/2. 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].6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 3-7 Fermi integral of order 1/2: 1E-05 1E-04 1E-03 1E-02 1E-01 1E+00 1E+01 1E+02 1/2 (x) x3/2 degeneratenon-degenerate e x -10 -8 -6 -4 -2 0 x 2 4 6 8 10 Key result again: no = NcF1/2(ηc) with ηc = EF − Ec kT ηc ↑⇒ the higher EF is with respect to Ec ⇒ no ↑ 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 3-8 Two regimes in no: � Non-degenerate regime Approximation to F1/2(x) for low values of x: F1/2(x −1)  ex Then, if ηc −1, or EC − EF  kT ,or no  Nc: EF − Ec no  Nc exp kT Simple exponential relationship when Fermi level is well below con-duction band edge. no(E) f(E) gc(E-Ec) Ec E EFEc E Can obtain same result with Maxwell-Boltzmann statistics for f(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 3-9 � Degenerate regime More complicated behavior of F1/2(x) for high values of x (see Ad-vanced Topic AT2.3). Degenerate semiconductor if ηc −1, or EF − EC  kT ,or no  Nc . Electron distribution inside conduction band very different from non-degenerate regime: EEc EF f(E) gc(E-Ec) no(E) degenerate Will not deal with degenerate regime in 6.720 because it’s even more complicated [see AT2.6 in notes] 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 3-10 2. Equilibrium hole concen tration • Q: How many holes are there in a semiconductor in TE? A: It depends on location of EF . • Why? Because EF defines probability that states are occupied by electrons. • The closer EF is to the valence band edge, the more holes


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