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].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].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 ⇒ EF relates no with Nc and po with Nv: EF − Ec Ev − EF no = NcF1/2( ) po = NvF1/2()kT kT Outside TE, EF cannot be used. Define two ”quasi-Fermi levels” such that: Efe − Ec Ev − Efh n = NcF1/2( ) p = NvF1/2()kT kT Under Maxwell-Boltzmann statistics (n Nc, p Nv): Efe − Ec n = Nc exp kT Ev − Efh p = Nv 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].6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 8-4 � Take derivative of n = f (Efe) with respect to x: Efe − Ec n = Nc exp kT dn n dEfe dEc n dEfe q = ( − )= − nE dx kT dx dx kT dx kT Plug into current equation: dn Je = qμenE + qDedx To get: dEfeJe = μen dx Similarly for holes: dEfhJh = μhp 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].6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007 Lecture 8-5 � Physical meaning of ∇Ef For electrons, dEfeJe = μen = −qnvedx Then: dEfe q = − vedx μe ∇Efe linearly proportional to electron velocity! Similarly for holes: dEfh q = vhdx μh 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 8-6 � Quasi-Fermi levels: effective way to visualize carrier phenomena outside equilibrium in energy band diagram 1. Visualize carrier concentrations and net recombination 2 Efe − Efh np = ni exp kT • If Efe >Efh ⇒ np > ni ⇒ U> 02 • If Efe <Efh ⇒ np < ni ⇒ U< 02 • If Efe = Efh ⇒ np = ni ⇒ U = 0 (carrier conc’s in TE) 2 Examples (same semiconductor): Ec Ev EF Efe EF Efh Efe Efh Efh Efe But can’t visualize Gext. 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 8-7 2. Visualize currents dEfeJe = μen dx • ∇Efe =0 ⇒ Je =0 • ∇Efe  ⇒ Je =0=0 • if n high, ∇Efe small to maintain a certain current level • if n low, ∇Efe large to maintain a certain current level Examples: thermal equilibrium under bias n-type p-type Ec EF Ec Efe Ev Ev Ec Ev EF Ec Ev Efh 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 8-8 � The concept of Quasi-Fermi level hinges on notion of: Quasi-equilibrium: carrier distributions in energy never depart too far from TE in times scales of practical interest. Quasi-equilibrium appropriate if: scattering time dominant device time constant ⇒ carriers undergo many collisions and attain thermal quasi-equilibrium with the lattice and among themselves very quickly. In time scales of interest, carrier distribution is close to Maxwellian (i.e., well described by a Fermi level). hν hν Ec Ev (1) generation (2) thermalization (3) recombination 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 8-9 2. Continuity Equations Semiconductor physics so far: qGauss’ law: ∇·E=  (p − n + ND + − NA −) drift Electron current equation: Je = −qnve + qDe∇n drift Hole current equation: Jh = qpvh − qDh∇p Total current equation: Jt = Je + Jh dn dpCarrier dynamics: dt = dt = G − R Still, can’t solve problems like this: hυ n S S -L/2 0 L/2x Equation system does not capture: • impact of carrier movement on carrier concentration (i.e. when carriers move away from a point, their concentration drops!) • boundary conditions (surfaces are not infinitely far away) 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 8-10 Need ”book-keeping relationships” for particles: n ΔV Fe rate of increase of number of electrons in ΔV = rate of