6.012 Recitation 6: P-N Junction

Recitation 6 p-n junction under T.E. & reverse bias 6.012 Spring 2009 Recitation 6: p-n junction Today’s agenda: 1. p-n junction under thermal equilibrium (T.E.): • space charge region (depletion layer) • quasi-neutral region 2. p-n junction under reverse bias 3. depletion capacitance p-n junction under T.E. Yesterday we talked about p-n junction. We continue here: 1. When we bring the p and n interface together, what will happen? Holes move from p-side to n-side and electrons diffuse from n-side to p-side, leaving behind space charges. The result is an electric field counteracting diffusion. 2. Space Charge Region (SCR): is a depletion region. xno , xpo , xno + xpo = xdo . What is the charge density in SCR? q Na (−) and q Nd (+) xpo Nd And q Na · xpo = q Nd · xno =⇒ = . xno Na Quasi-Neutral Region (QNR) 1 Recitation 6 p-n junction under T.E. & reverse bias 3. Summaries of equations:  xno =  2s φB Na q(Na + Nd ) · Nd 2s φB Nd q(Na + Nd ) · Na xpo = 2 6.012 Spring 2009 Recitation 6 p-n junction under T.E. & reverse bias 6.012 Spring 2009 xno , xpo are determined by doping on both sides! kT N a Nd ln q n2i    2s φB 1 1 = + q N a Nd Built-in potential φB = φn − φp = xdo = xno + xpo  Maximum field |Emax | = |Eo | = = 2qφB Na · Nd s Na + Nd q Na q Nd xpo = xno s s 4. If strongly asymmetric, the lowly doped side controls the electrostatics.   2qφB 2s φB 1 + p n : xpo  xno  xdo = |Eo |  Nd q Nd s Examples Now let us do some exercises before moving on to the next topic: Nd [cm−3 ] Na [cm−3 ] xno xpo Eo [V/cm] φB 1016 1016 216 nm 216 nm 3.3 × 104 V/cm 720 mV 1019 1016 3.14 Å 341 nm 5.26 × 104 V/cm 900 mV φB is easier to calculate first. 1. φn = 360 mV, φp = −360 mV φB = φn − φp = 720 mV  2s φB  N a xno =  q(Na + Nd )  Nd  √ 2 × 1 × 10−12 (F/cm) × 0.72 (V) = = 4.66 × 10−5 cm = 216 nm 1.6 × 10−19 (C) × 2 × 1016 (cm−3 ) xpo = xno (symmetric)  2qφB Na · Nd |Eo | = s Na + Nd  1.6 × 10−19 (C) × 0.72 (V) 1016 × 1016 (cm−3 )2 × = 2× = 3.3 × 104 V/cm 2 × 1016 (cm−3 ) 1 × 10−12 (F/cm) 3 Recitation 6 p-n junction under T.E. & reverse bias 6.012 Spring 2009 2. φB = φn − φp = 540 mV + 360 mV = 900 mV  2s φB Na = 3.41˚ A (really thin) xno = q(Na + Nd ) Nd   2s φB  N 2s φB 1 d xpo = = = 344 nm    q Na (N + Nd ) Na q a |Eo | = qNa 1.6 × 10−19 (C) · 1016 (cm−3 ) × 3.41 × 10−7 cm = 5.26 × 104 (V/cm) = 1 × 10−12 (F/cm) s xpo Reverse Bias Now what happens when we apply a reverse bias? First: Bias convention for pn junction: use n-side as reference: V>0: forward bias V<0: reverse bias In your weblab experiment, when you have V < 0, what current do you measure? 6×10−15 A (very small). With such a small current, the voltage drop across the QNRs will be really small, safely to ignore. Then, where is the voltage drop? =⇒ across SCR 4 Recitation 6 p-n junction under T.E. & reverse bias 6.012 Spring 2009 Forward bias means V > 0, moving “φp ” up and reverse bias means V < 0, moving “φp ” down. Reverse bias V: barrier becomes φB − V > φB (V is negative, so barrier increases). Now, what happened to the electric field? Still zero in QNRs. Larger barriers means higher E-field, and ∴ |Emax | increases. This will mean we need more space charges to support the larger E-field. However, the charge density qNa or qNd can not change, =⇒ xn , xp widens. 5 Recitation 6 p-n junction under T.E. & reverse bias 6.012 Spring 2009 It turns out a straightforward way to do the new calculations is: replace φB with φB − V . So:   V φrmB − V xn = xno = xno 1 − φB φB  2s (φB − V ) Na = q(Na + Nd ) Nd  V xp = xpo 1 − φB  V xd = xdo 1 − φB  V Emax = Eo 1 − φB The concept of depletion capacitance Now let us think about the junction region a bit further. 1. It stores charges 2. By changing the voltage (bias) across the junction, the amount of charges stored there changes =⇒ Capacitor! 3. The charges that it stores:  Qjo = | − q Na · xpo | = |q Nd · xno | = 2qs Na Nd · φB Na + Nd As voltage changes, Qj becomes:  Qj (V ) =  2qs Na Nd (φB − V ) = Qjo Na + Nd 6 1− V . φB Recitation 6 p-n junction under T.E. & reverse bias 6.012 Spring 2009 Q Previously, we had C = , but that was for constant capacitance, a more rigorous definition V should be: dQ C= dV Therefore,   dQ    d( 1 − φVB )    j   Cj = Cj (VD ) =    = Qjo  dV VD dV VD +qNa xpo 1  = · 2 1 − VD φB φB = qNa xpo 1 · 2φB 1− VD φB qNa xpo Cj (V = 0) = Cjo = 2φB   qNa 2s φB Nd qs Na Nd = = × 2φB q(Na + Nd ) Na 2φB Na + Nd  2s φB Na + Nd ∵ xdo = xno + xpo = q N a Nd =⇒ Cjo = Cj (VD ) = Cjo s , Cj (VD ) =  xdo 1 − VφDB  s s = xd x 1 − VD do φB Can be considered as “parallel plate capacitor”. 7