Lecture 10 OUTLINE • pn Junction Diodes (cont’d) – Derivation of the Ideal Diode Equation (for a step junction) Reading: Pierret 6.1; Hu 4.3, 4.6, 4.8-4.9 Current Flow (Qualitative View) Equilibrium (VA = 0) EE130/230A Fall 2013 Forward Bias (VA > 0) Lecture 10, Slide 2 Reverse Bias (VA < 0) R. F. Pierret, Semiconductor Device Fundamentals, pp. 236-237 Carrier Action under Forward Bias • When a forward bias (VA>0) is applied, the potential barrier to diffusion across the junction is reduced – Minority carriers are “injected” into the quasi-neutral regions => Dnp > 0, Dpn > 0 • Minority carriers diffuse in the quasi-neutral regions, recombining with majority carriers EE130/230A Fall 2013 Lecture 10, Slide 3 Ideal Diode Analysis: Assumptions • Non-degenerately doped step junction • Steady-state conditions • Low-level injection conditions in quasi-neutral regions • Recombination-generation negligible in depletion region dJ n 0, dx dJ p dx 0 i.e. Jn & Jp are constant inside the depletion region EE130/230A Fall 2013 Lecture 10, Slide 4 Components of Current Flow • Current density J = Jn(x) + Jp(x) dn d ( Dn ) J n ( x ) qn n qDn qn n qDn dx dx dp d ( Dp ) J p ( x ) q p p qD p q p p qD p dx dx • J is constant throughout the diode, but Jn(x) and Jp(x) vary with position: Example: p+n junction under forward bias: J JN JP -xp EE130/230A Fall 2013 Lecture 10, Slide 5 xn x “Game Plan” for Obtaining Diode I-V 1. Solve minority-carrier diffusion equations in quasi-neutral regions to obtain excess carrier distributions Dnp(x,VA),Dpn(x,VA) – boundary conditions: • p side: Dnp(-xp), Dnp(-) • n side: Dpn(xn), Dpn() 2. Find minority-carrier current densities in quasi-neutral regions d ( Dn p ) d ( Dpn ) J p ( x,VA ) qD p J n ( x,VA ) qDn dx dx 3. Evaluate Jn at x=-xp & Jp at x=xn to obtain total current density J: J (VA ) J n ( x p ,VA ) J p ( xn ,VA ) EE130/230A Fall 2013 Lecture 10, Slide 6 Carrier Concentrations at –xp, xn Consider the equilibrium (VA = 0) carrier concentrations: p side n side p p 0 ( x p ) N A nn 0 ( xn ) N D 2 i n n p 0 ( x p ) NA 2 i n p n 0 ( xn ) ND If low-level injection conditions hold in the quasi-neutral regions when VA 0, then nn ( xn ) N D p p ( x p ) N A EE130/230A Fall 2013 Lecture 10, Slide 7 “Law of the Junction” The voltage applied to a pn junction falls mostly across the depletion region (assuming low-level injection in the quasi-neutral regions). We can draw 2 quasi-Fermi levels in the depletion region: p ni e( Ei FP ) / kT n ni e ( FN Ei ) / kT 2 ( FN FP ) / kT i pn n e pn ni2eqVA / kT EE130/230A Fall 2013 Lecture 10, Slide 8 Excess Carrier Concentrations at –xp, xn p side n side p p ( x p ) N A nn ( xn ) N D ni2 e qVA / kT n p ( x p ) NA ni2 e qVA / kT p n ( xn ) ND pn 0 e qVA / kT n p 0 e qVA / kT 2 i n Dn p ( x p ) e qVA / kT 1 NA EE130/230A Fall 2013 ni2 qVA / kT Dpn ( xn ) e 1 ND Lecture 10, Slide 9 Carrier Concentration Profiles under Forward Bias EE130/230A Fall 2013 Lecture 10, Slide 10 R. F. Pierret, Semiconductor Device Fundamentals, Fig. 6.8a Example Consider a pn junction with NA=1018 cm-3 and ND=1016 cm-3, under a forward bias of 0.6 V. (a) What are the minority carrier concentrations at the edges of the depletion region? (b) What are the excess minority carrier concentrations at the edges of the depletion region? EE130/230A Fall 2013 Lecture 10, Slide 11 Excess Carrier Distribution (n side) • From the minority carrier diffusion equation: d 2 Dpn Dpn Dpn dx 2 Dp p Lp 2 • We have the following boundary conditions: Dpn ( xn ) pno (e qVA / kT 1) Dpn () 0 • For simplicity, use a new coordinate system: NEW: x’’ 0 0 x’ • Then, the solution is of the form: Dpn ( x' ) A1e EE130/230A Fall 2013 Lecture 10, Slide 12 x '/ L p A2e x '/ L p Dpn ( x' ) A1e x ' / Lp A2e x ' / Lp From the x = boundary condition: From the x = xn boundary condition: x ' / Lp , x' 0 x ''/ Ln , x' ' 0 Therefore Dpn ( x' ) pno (eqVA / kT 1)e Similarly, we can derive Dn p ( x' ' ) n po (e EE130/230A Fall 2013 qVA / kT 1)e Lecture 10, Slide 13 Total Current Density p side: dDn p ( x' ' ) J n qDn dx' ' Dn q n p 0 (e qVA Ln Dp dDpn ( x' ) qVA n side: J p qD p q pn 0 (e dx' Lp J J n x x J p p x xn J n x0 J p Dn D p qVA J qn ( e Ln N A Lp N D 2 i EE130/230A Fall 2013 Lecture 10, Slide 14 kT x 0 1) kT 1)e x '' Ln kT x' Lp 1)e Ideal Diode Equation I I 0 (e qVA kT 1) Dp D n I 0 Aqni L N L N n A p D 2 C. C. Hu, Modern Semiconductor Devices for Integrated Circuits, Figure 4-22 EE130/230A Fall 2013 Lecture 10, Slide 15 Diode Saturation Current I0 • I0 can vary by orders of magnitude, depending on the semiconductor material and dopant concentrations: Dp D n I 0 Aqni L N L N p D n A 2 • In an asymmetrically doped (one-sided) pn junction, the term associated with the more heavily doped side is negligible: Dp – If the p side is much more heavily doped, I 0 Aqni L N p D 2 Dn – If the n side is much more heavily doped, I 0 Aqni L N n A 2 EE130/230A Fall 2013 Lecture 10, Slide 16 Carrier Concentration Profiles under Reverse Bias R. F. Pierret, Semiconductor Device Fundamentals, Fig. 6.8b • Depletion of minority carriers at edges of depletion region • The only current which flows is due to drift of minority carriers across the junction. This current is fed by diffusion of minority carriers toward junction (supplied by thermal generation). EE130/230A Fall 2013 Lecture 10, Slide 17 Alternative Derivation of Formula for I0 “Depletion approximation”: • I0 is the rate at which carriers are thermally generated within one diffusion length of the depletion region: Dn p ni / N A n t n n -LN -x p x -x p Dp n / ND p n i t p p xn x xn LP 2 2 ni 2 / N D ni 2 / N A qALP I 0 qALN n p EE130/230A Fall 2013 Lecture 10, Slide 18 R. F. Pierret, Semiconductor Device Fundamentals, Fig. E6.4 Summary • Under forward bias (VA > 0), the potential barrier to carrier diffusion is reduced minority carriers are “injected” into the quasi-neutral regions. – The minority-carrier concentrations at the edges of the depletion region change with the applied bias VA, by the factor e qVA / kT – The excess carrier concentrations in the quasi-neutral regions decay to zero away from the depletion region, due to recombination. Dn D p qVA pn junction diode current I qAn (e Ln N A L p N D 2 i kT • I0 can be viewed as the drift current due to minority carriers generated within a diffusion length of the depletion region EE130/230A Fall 2013 Lecture 10, Slide 19 1)