Lecture 6 OUTLINE • Semiconductor Fundamentals (cont’d) – Continuity equations – Minority carrier diffusion equations – Minority carrier diffusion length – Quasi-Fermi levels Reading: Pierret 3.4-3.5; Hu 4.7 Derivation of Continuity Equation • Consider carrier-flux into/out-of an infinitesimal volume: Area A, volume Adx Jn(x) Jn(x+dx) dx 1 n n Adx J n ( x) A J n ( x dx) A Adx q n t EE130/230M Spring 2013 Lecture 6, Slide 2 J n ( x) J n ( x dx) J n ( x) dx x n 1 J n ( x) n t q x n Continuity Equations: EE130/230M Spring 2013 n 1 J n ( x) n GL t q x n p 1 J p ( x) p GL t q x p Lecture 6, Slide 3 Derivation of Minority Carrier Diffusion Equation • The minority carrier diffusion equations are derived from the general continuity equations, and are applicable only for minority carriers. • Simplifying assumptions: 1. The electric field is small, such that n n J n q n n qDn qDn in p-type material x x p p J p q p p qD p qD p in n-type material x x 2. n0 and p0 are independent of x (i.e. uniform doping) 3. low-level injection conditions prevail EE130/230M Spring 2013 Lecture 6, Slide 4 • Starting with the continuity equation for electrons: n 1 J n ( x) n GL t q x n n0 n 1 n0 n n qDn GL t q x x n n 2 n n Dn GL 2 t x n EE130/230M Spring 2013 Lecture 6, Slide 5 Carrier Concentration Notation • The subscript “n” or “p” is used to explicitly denote n-type or p-type material, e.g. pn is the hole (minority-carrier) concentration in n-type mat’l np is the electron (minority-carrier) concentration in n-type mat’l • Thus the minority carrier diffusion equations are n p t n p 2 Dn x 2 n p n GL pn pn pn Dp GL 2 t x p 2 EE130/230M Spring 2013 Lecture 6, Slide 6 Simplifications (Special Cases) • Steady state: • No diffusion current: n p pn 0 0 t t 2 n p 2 pn Dn 0 Dp 0 2 2 x x • No R-G: n p • No light: GL 0 EE130/230M Spring 2013 Lecture 6, Slide 7 n 0 p n p 0 Example • Consider an n-type Si sample illuminated at one end: – constant minority-carrier injection at x = 0 pn (0) pn 0 – steady state; no light absorption for x > 0 Lp is the hole diffusion length: L p D p p EE130/230M Spring 2013 Lecture 6, Slide 8 pn pn 2 2 x Lp 2 The general solution to the equation is pn ( x) Ae x / Lp Be x / Lp where A, B are constants determined by boundary conditions: pn () 0 pn (0) pn 0 Therefore, the solution is pn ( x) pn0e EE130/230M Spring 2013 x / Lp Lecture 6, Slide 9 Minority Carrier Diffusion Length • Physically, Lp and Ln represent the average distance that minority carriers can diffuse into a sea of majority carriers before being annihilated. • Example: ND = 1016 cm-3; p = 10-6 s EE130/230M Spring 2013 Lecture 6, Slide 10 Quasi-Fermi Levels • Whenever n = p 0, np ni2. However, we would like to preserve and use the relations: n ni e ( E F Ei ) / kT p ni e ( Ei E F ) / kT • These equations imply np = ni2, however. The solution is to introduce two quasi-Fermi levels FN and FP such that n ni e ( FN Ei ) / kT EE130/230M Spring 2013 p ni e Lecture 6, Slide 11 ( Ei FP ) / kT Example: Quasi-Fermi Levels Consider a Si sample with ND = 1017 cm-3 and n = p = 1014 cm-3. What are p and n ? What is the np product ? EE130/230M Spring 2013 Lecture 6, Slide 12 • Find FN and FP : n FN Ei kT ln ni p FP Ei kT ln ni EE130/230M Spring 2013 Lecture 6, Slide 13 Summary • The continuity equations are established based on conservation of carriers, and therefore hold generally: n 1 J n ( x) n GL t q x n p 1 J n ( x) p GL t q x p • The minority carrier diffusion equations are derived from the continuity equations, specifically for minority carriers under certain conditions (small E-field, low-level injection, uniform doping profile): n p t DN 2 n p x EE130/230M Spring 2013 2 n p n GL pn 2 pn pn DP GL 2 t x p Lecture 6, Slide 14