6.720J/3.43J Integrated Microelectronic Devices - Spring 2007 Lecture 10-1 Lecture 10 - Carrier Flow (cont.) February 28, 2007 Contents: 1. Minority-carrier type situations Reading assignment: del Alamo, Ch. 5, §5.6 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 10-2 Key questions • What ch a racterizes minority-carrier type situations? • What is the length scale for minority- carrier type situations? • What do majority carriers do in minority-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 10-3 Overview of simplified carrier flow f ormulations General drift-diffusion� situation� (Shockley's equations) 1D approx. Quasi-neutral� situation� (negligible volume� charge) Majority-carrier � type situation� (V=0, n'=p'=0) Space-charge� situation� (field independent� of n, p) Minority-carrier � type situation� (V=0, n'=p'=0, LLI) 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 10-4 Simplified set of Shockley e quations for 1D quasi-neutral situations p − n + ND −NA � 0 Je = −qnvdrif t + qDe∂n e ∂x Jh = qpv drif t − qDh∂p h ∂x 1∂p = ∂Jh∂n ∂t = ∂Je1 q Gext − U + Gext −U − or ∂x ∂t q ∂x ∂Jt ∂x � 0 Jt = Je + Jh 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 10-5 1. Minority-carrier type situations Situations characterized by: excess carriers over TE • • no external electric field applied (but small internal field gener-ated by carrier injection: E = Eo + E�) Example: electron transport through bas e of npn BJT . Two approximations: 1. E small ⇒ |vdrif t| ∝ |E| 2. Low -level injection for n-type: ⇒• n � no • p � p� • U �pτ � • negligib l e minority carrier drift due to E� (but can’t say the same about majo rity carriers) 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 10-6 Shockley equations for 1D quasi-neutral situations p − n + ND −NA � 0 Je = −qnvdrif t + qDe∂n e ∂x Jh = qpv drif t − qDh∂p h ∂x ∂n 1 ∂Je ∂p 1∂Jh= Gext − U + or = Gext −U −∂t q ∂x ∂t q ∂x ∂Jt ∂x � 0 Jt = Je + Jh � Further simplifications for n-type minority-carrier-type situations • Majority- carrier current equation: ∂no ∂n�Je � q(no + n�)µe(Eo + E�) + qDe(∂x + ∂x) but in TE: ∂noJeo = qnoµeEo + qDe = 0 ∂x Then: ∂n�Je � qnoµeE�+ qn�µeEo + qDe ∂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 10-7 • Minority-carrier current equation: Jh � q(po + p�)µh(Eo + E�) − qDh(∂po + ∂p�)∂x ∂xIn TE, Jho = 0, and: ∂p� ∂p�Jh � qp�µhEo + qp�µhE�− qDh ∂x � qp�µhEo − qDh ∂x • Minority-carrier continuity equation: ∂p� p� 1 ∂Jh ∂t = Gext −τ −q∂x Now plug in Jh from above: ∂2p� ∂p� p� ∂p�Dh ∂x2 − µhEo ∂x −τ + Gext = ∂t One differential equation with one u nknown : p�. If Gext and BC’s are specified, problem can be solved. 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 10-8 Shockley equations for 1D minority-carrier type situations n-type p-type nopo − + ND − NA � 0 p�� n� ∂n�Je = qnoµeE�+ qn�µeEo + qDe∂x Jh = qp�µhEo − qDh∂p� ∂x ∂2p� ∂p� p� ∂p�Dh∂x2 − µhEo∂x −τ + Gext = ∂t ∂n�Je = qn�µeEo + qDe∂x Jh = qpoµhE�+ qp�µhEo − qDh∂p� ∂x ∂2n� ∂n� n� ∂n�De∂x2 + µeEo∂x −τ + Gext = ∂t ∂Jt ∂x � 0 Jt = Je + Jh 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 10-9 Example 1: Diffusion and bulk recombination in a ”long” bar Uniform doping : Eo = 0; static cond i tion s: ∂ = 0∂t hυ n gl 0x Minority carrier profile: x p' p'(0) 0 Majority carrier profile? n�= p�exactly? 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 10-10 x p' p'(0) 0 Far away Jt = 0 Jt = 0 everywhere. ⇒Jt = Je + Jh � qnoµeE�+ q(De − Dh)dp� = 0 dx If De = Dh diffu sion term = 0⇒drift term = 0⇒⇒ E�= 0 n�= p� ⇒But, typically De > Dhdiffu sion term < 0 (for x > 0) ⇒drift term > 0⇒⇒ E�> 0 (for x > 0) ⇒ and E�∝ De − Dh ⇒ n��= p�(but still n�� p�) ⇒