6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 2-1 Lecture 2 - Carrier Statistics in Equilibrium February 8, 2007 Contents: 1. Conduction and valence bands, bandgap, holes 2. Intrinsic semiconductor 3. Extrinsic semiconductor 4. Conduction and valence band density of states Reading assignment: del Alamo, Ch. 2, §§2.1-2.4 (2.4.1) Announcement: Go to http://ilab.mit.edu and register. Select member-ship in the 6.720 group. You will need this to access the lab for the Device Characterization Projects. 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 2-2 Key questions • What are these ”energy band diagrams”? What are these ”holes”? • • In a perfectly pure semiconductor, how many electrons and holes are there? • Can one engineer the electron and hole concentrations in a semi-conductor? 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 2-3 1. Conduction and valence bands, bandgap, holes energy Ec conduction band valence band Ec bandgap Eg ↓↓Ev Ev space coordinate Conduction and valence bands: • bonding electrons occupy states in valence band • ”free” electrons occupy states in conduction band • holes: empty states in valence band CB electrons and VB holes can move around: ”carriers”• + + ----a)a) b)c)b) c) 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 2-4 ∧ electron energyelectron energyhole energyhole energy EEccEEvv↓↓--------++++++++Elements of energy band diagrams: • at edges of bands, kinetic energy of carriers is zero • electron energies increase upwards • hole energies increase downwards electron energyelectron energyhhυ = Egυ = Eghhυ >υ > EgEg ----++++--++hhυ > Egυ > Ega) b) EEccEEvvCite 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 2-5 2. Intrinsic semiconductor Define intrinsic semiconductor, or ”ideal” semiconductor: • perfectly crystalline (no perturbations to periodic lattice) • perfectly pure (no foreign atoms) no surface effects • Question: How many electrons and holes are there in an intrinsic semiconductor in thermal equilibrium at a given temperature? Answer requires fairly elaborate model [lecture 3], but key dependen- cies can be readily identified. Define: no ≡ equilibrium (free) electron concentration in conduction band [cm−3] po ≡ equilibrium hole concentration in valence band [cm−3] Certainly in intrinsic semiconductor: no = po = ni ni ≡ intrinsic carrier concentration [cm−3] 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 2-6 Key dependencies of ni: • Temperature: T ni↑⇒ • Bandgap: niEg ↑⇒ What is detailed form of dependencies? Use analogy of chemical reactions. 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 2-7 Electron-hole formation can be thought of as chemical reaction: bond��e− + h+ similar to water decomposition reaction: H2O��H+ + OH− Law-of-mass action relates concentration of reactants and reaction products. For water: [H+][OH−] E K =[H2O] ∼ exp(−kT ) E is energy consumed or released in reaction. This is a ”thermally activated” process rate of reaction limited⇒by need to overcome energy barrier E (activation energy). In analogy, for electron-hole formation: nopo EgK =[bonds]∼ exp(−kT ) [bonds] is concentration of unbroken bonds. Note: activation energy is Eg. 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 2-8 In general, relatively few bonds are broken. Hence: [bonds] � no,po and [bonds] � constant Then: Egnopo ∼ exp(− )kT Two important results: • First, Egni ∼ exp(− )2kT As expected: T ni ↑↑⇒ Eg ↑⇒ ni ↓ To get prefactor, need detailed model [lecture 3]. 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 2-9 Arrhenius plot for Si [experiments of Misiakos and Tsamakis, 1993]: 30 50 70 90 110 130 150 1/kT (eV-1) In Si at 300 K, ni � 1.1 × 1010 cm−3 . • Second important result: 2 nopo = ni Equilibrium np product in a semiconductor at a certain temper-ature is a constant specific to the semiconductor. 1E-20 1E-15 1E-10 1E-05 1E+00 1E+05 1E+10 1E+15 ni (cm -3 ) 0.612 eV 300 K 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