6.012 Spring 2007 Lecture 161Lecture 16The pn Junction Diode (III)Outline• Small-signal equivalent circuit model• Carrier charge storage–Diffusion capacitanceReading Assignment:Howe and Sodini; Chapter 6, Sections 6.4 - 6.56.012 Spring 2007 Lecture 162I-V Characteristics Diode Current equation:I = IoeVVth()−1⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ IVlg |I|V000IoIolinear scalesemilogarithmic scale0.43qkT=60 mV/dec @ 300K6.012 Spring 2007 Lecture 1632. Small-signal equivalent circuit modelIf v small enough, linearize exponential characteristics:i =qIkT• vExamine effect of small signal adding to forward bias:From a small signal point of view. Diode behaves as conductance of value:I + i = IoeqV+v()kT⎛ ⎝ ⎜ ⎞ ⎠ ⎟ −1⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ≈ IoeqV+v()kT⎛ ⎝ ⎜ ⎞ ⎠ ⎟ I + i ≈ IoeqVkT()eqvkT()⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ≈ IoeqVkT()1+qvkT⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = IoeqVkT()+ IoeqVkT()qvkTThen:gd=qIkT6.012 Spring 2007 Lecture 164Small-signal equivalent circuit modelgddepends on bias. In forward bias:gd=qIkTgdis linear in diode current.gd6.012 Spring 2007 Lecture 165Capacitance associated with depletion region:Depletion or junction capacitance:Cj= AqεsNaNd2 Na+ Nd()φB−VD()−xpxnxp-siden-side−qNa(a)−xpxnxρ(x)ρ(x)p-siden-side+qNd+qNd−qNaQJ = −qNaxp qJ = −qNaxp (b)vD = VD + vd > VD-->xp < xp,|qJ| < |QJ| xnxnxn−xp−xp−xpvD = VD (c)x∆ρ(x) = ρ(x) − ρ(x) p-siden-sideqj = qNa∆xp −qj = −qNd∆xn Xdqj = qj − Qj > 0= −qNaxp − (−qNaxp)= qNa (xp −xp) = qNa∆xp + qNd− qNaCj= Cj(VD) =dqJdvDVD6.012 Spring 2007 Lecture 166can rewrite as:Cj= AqεsNaNd2 Na+ Nd()φB•φBφB−VD()or,Cj=Cjo1−VDφBCjo≡ zero-voltage junction capacitanceCj=2CjoCjgdSmall-signal equivalent circuit modelUnder Forward Bias assume VD≈φB26.012 Spring 2007 Lecture 1673. Charge Carrier Storage:diffusion capacitanceWhat happens to majority carriers?Carrier picture thus far:If QNR minority carrier concentration ↑ but majority carrier concentration unchanged? ⇒ quasi-neutrality is violated.ln p, npopnonNdni2Ndx0Nani2Na6.012 Spring 2007 Lecture 168Quasi-neutrality demands that at every point in QNR:pn(x) − pno=nn(x)−nnoexcess minority carrier concentration= excess majority carrier concentrationMathematically:Define integrated carrier charge:qPn= qA12pn(xn) − pno()• Wn− xn()= qAWn− xn2ni2NdeqVkT−1⎡ ⎣ ⎢ ⎤ ⎦ ⎥ =−qNnx−xpxn;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;00(p-type) (n-type)pp(x) = Na + np(x) nn(x) = Nd + pn(x) pn(xn) = pno . eVD/Vth pn(Wn) = pnonp(−xp) = npo . eVD/Vth np(−Wp) = npometalcontact top regionmetalcontact ton regionppo = Na nno = Nd carrier concentrations(cm−3)pn(x)np(x)−WpWn6.012 Spring 2007 Lecture 169Now examine small increase in V:Small increase in V ⇒ small increase in qPn⇒ small increase in |qNn|Behaves as capacitor of capacitance:Cdn=dqPndVVD= qAWn− xn2ni2NdqkTeqVDkT⎡ ⎣ ⎢ ⎤ ⎦ ⎥ x;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;−WpWn−xpxn00metalcontact top regionmetalcontact ton regionp-type n-type−dqNn −dqNp dqPpdqPnnn(xn)[for VD + vd] pn(xn)[for VD + vd] nn(xn)[for VD] pn(xn)[for VD] pp(−xp)[for VD + vd] pp(−xp)[for VD] np(−xp)[for VD + vd] np(−xp)[for VD] carrier concentrations (cm−3)6.012 Spring 2007 Lecture 1610Similarly for p-QNR:Both capacitors sit in parallel ⇒ total diffusion capacitance:Cd= Cdn+CdpComplete small-signal equivalent circuit model for diode:CjCdgdCdp=dqNpdVVD= qAWp− xp2ni2NaqkTeqVDkT⎡ ⎣ ⎢ ⎤ ⎦ ⎥6.012 Spring 2007 Lecture 1611Bias dependence of Cjand Cd:•Cjdominates in reverse bias and small forward bias•Cddominates in strong forward bias∝1φB− V∝ eqVkT⎡ ⎣ ⎢ ⎤ ⎦ ⎥ CV00CdCCj6.012 Spring 2007 Lecture 1612What did we learn today?• Diode Current:• Conductance: associated with current-voltage characteristics–gd∝ I in forward bias,–gdnegligible in reverse bias• Junction capacitance: associated with charge modulation in depletion region• Diffusion capacitance: associated with charge storage in QNRs to maintain quasi-neutrality.Summary of Key ConceptsLarge and Small-signal behavior of diode:Cj∝1φB− VCd∝ eqVkT⎡ ⎣ ⎢ ⎤ ⎦ ⎥ I = Io(eqVkT[]−
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