PN Junction. Ang M.S. October 8, Maxwell s Eqautions Review : Poisson s Equation for PNJ. Q encl S. E ds. σ = dq ds. ρdv = Q encl.

Transcription

1 PN Junction Ang M.S. October 8, 0 Reference Sedra / Smith, M icroelectronic Circuits Maxwell s Eqautions Review : Poisson s Equation for PNJ. Gauss Law for E field The total enclosed charge Q encl. insde closed surface S is given by Q encl Φ E,S i.e. With the proportional constant Charge Density ρ dq d Q encl S E ds Q Encl E ds S σ dq ds λ dq dl. Poisson s Equation ρd Q encl. E ds S Recall, the Gauss s Divergence Theorem A ds Ad Thus Equalize the integrand Rearrange Recall, the relation of E and ρd Ed ρ E ρ E E Combine the equations ρ P oisson s Eqaution)

2 .3 Poisson s Equation for PNJ Simplify to one dimensional case x With carrier concentration in for PN-Junction ρ x q ρ PN-Junction carrier concentration Affected by dopping ) ρ p n + With assumption that donor, acceptor 00% ionized The final Poisson s Equation for PN-Junction x q [p n + ] Analysis of the Depletion Region of PN-Junction. Approximations n, p, : Depletion region is depleted of carriers. No net charge outside depletion region : Q-neutral Thus in the P side : d p x 0 ) n p, p p ρ In the N side : 0 x d n ) n n, p n ρ Outside : x > d n, x < d p ) ρ 0 Slope of E-x diagram x q N-side x q P-side * E-field continuous at x d p, 0, d n Build-in E-field dopant ions in depletion region E field Stop further carrier flow

3 . Build-in Potential In equilibrium, carrier stop flowing across. Fermi Dirac statistics f E) e E E F )/kt Electrons in conduction band n N C fe C ) N C e E C E F )/kt As E C is few level higher than E F, so apply approximation e x e x Remark. E C, E n N C e E C E F )/kt e E C E F )/kt, E F are energy of the specific energy bands, not E-field At equilibrium : n p NC e E CP E F )/kt n n NC e E CN E F )/kt n p n i Neutral P-side) n n Netural N-side) Recall, the T hermal oltage n n n p e E CP E CN )/kt e q bi/kt bi kt q ln ND ) T kt q Build-in voltage of PN-Junction bi kt q ln ND ) ) ND T ln.3 E-field in PNJ x q ρ x q 0 x d n N side x q d p x 0 P side E N q x + C E P q x + C N side P side 3

4 Apply Boundary Condition Thus E P 0 x d p E N 0 x d n E N q C q d n C q d p x d n ) N side E P q x + d p ) P side Max. E-field E j E p x 0) E n x 0) qd p qd n d p d n Q neutrality).4 Build-in Potential Barrier ˆ Edx E P q x + d p ) N side E N q P E p dx q N E n dx q x d n ) P side x + d p ) dx q x d n ) dx q x + dp ) dx x dn ) dx P q N q [ ] x + d px [ ] x d nx C 0 By 0) 0 C 0 By 0) 0 P d p ) q d p N d n ) q d n Build-in Potential Barrier np N d n ) P d P ) q ND d n + d p) 4

5 .5 Width of Depletion Region Apply Q-neutrality Into Build-in Potential Barrier np q ) ND d n + d p d p d p d n d n d p np q [ ] ) NA d p + d p q ) NA + d p np q [ ] ) d ND n + d n q NA + ) np ND np N j or N j np q + q qn j ) d n d n ) np NA np N j or N j np q + q qn j Where N j Thus, the width w d n + d p.6 Junction Capacitance +, N j +, { np Nj + N } j qn j N }{{ D } N j Q q d p q d n ) Q qn j np qn j + np np + ) qn j q qn j np With external voltage source R, the PN-Junction barrier voltage will become : np Q qn j v np + R ) qn j v np + R C j dq d R R very small qn j For C j0, with no applied voltage C j0 v np + R ) q NA + v np Therefore qn j C j v np + R ) qn j v np v np + R ) v np qn j v np qn j v np + R ) vnp v np + R R np + R C j0 + v R v np 5

6 3 Summary Name / Description Equation General Poisson s Equation ρ D Poisson s Eqaution for PN-Junction x q [p n + ] 0 x < d p Carrier Concentrations in PNJ ρ d p x 0 0 x d n 0 x > d n Fermi-Dirac Statistic f E) e E E F )/kt Intrinsic Carrier Concentration B T 3 e E g kt Thermal oltage T kt q e- in depletion region n n N C e E CN E F )/kt n p N C e E CP E F )/kt N-side SCR P-side SCR e in neutral rehion Build-in oltage across SCR E-field in PNJ Charge Neutrality Build-in Potential Barrier SCR distance bi E CP E CN q n n n p n i kt q ln NA E N q x d n ) N-side E P q x + d p ) P-side E j E N 0) E P 0) q d n q d p Max d n Edx d p N j d p P qn [ ] A x + d px N qn [ ] D x d nx np d n N j qn j ) np qn j P-side N-side N j N j Space Charge Region Width w 0 d p + d n q 0 + ) N D SCR Width with external source w w 0 + v R PNJ Capacitance C j END v j qn j v np + R ) 6