University:
Massachusetts Institute of Technology
Course:
6.012 | Microelectronic Devices and Circuits
Academic year:

2022
Views:

380

Pages:

20
Author:

Brodie Gonzales

Lecture 15 - The pn Junction Diode (I) I-V Characteristics November 1, 2005 Contents: 1. pn junction under bias 2. I-V characteristics Reading assignment: Howe and Sodini, Ch. 6, §§6.1-6.3 Key questions • Why does the pn junction diode exhibit current rectification? • Why does the junction current in forward bias increase as ∼ exp qV kT ? • What are the leading dependences of the saturation current (the factor in front of the exponential)? 1. PN junction under bias Focus on intrinsic region: p type n type x p type (a) (Na) metal contact to p side p x=0 x (Nd) n metal contact to n side (b) Upon application of voltage: • electrostatics upset: depletion region widens or shrinks • current flows (with rectifying behavior) • carrier charge storage Carrier profiles in thermal equilibrium: log po, no Na Nd po no ni2 Nd ni2 Na x 0 Jhdiff Jhdrift Jediff Jedrift Inside SCR in thermal equilibrium: dynamic balance between drift and diffusion for electrons and holes. |Jdrift| = |Jdiff | Carrier concentrations in pn junction under bias: • for V > 0, φB − V ↓ ⇒ |ESCR | ↓⇒ |Jdrift | ↓ log p, n Na po Nd no p n ni2 Nd ni2 Na x 0 Jhdiff Jhdrift Jediff Jedrift Jh Je Current balance in SCR broken: |Jdrift| < |Jdiff | Net diffusion current in SCR ⇒ minority carrier injection into QNR’s ⇒ excess minority carrier concentrations in QNR’s Lots of majority carriers in QNR’s ⇒ current can be high. • for V < 0, φB − V ↑ ⇒ |ESCR | ↑⇒ |Jdrift | ↑ log p, n Na po no ni2 Na n 0 Jh p Nd ni2 Nd x Jhdiff Jhdrift Je Jediff Jedrift Current balance in SCR broken: |Jdrift| > |Jdiff | Net drift current in SCR ⇒ minority carrier extraction from QNR’s ⇒ deficit of minority carrier concentrations in QNR’s Few minority carriers in QNR’s ⇒ current small. What happens if minority carrier concentrations in QNR change from equilibrium? ⇒ Balance between generation and recombination broken • In thermal equilibrium: rate of break up of Si-Si bonds balanced by rate of formation of bonds generation no + po Si-Si bond recombination • If minority carrier injection: ⇒ carrier concentration above equilibrium ⇒ recombination prevails Si-Si bond recombination n+p • If minority carrier extraction: ⇒ carrier concentrations below equilibrium ⇒ generation prevails Si-Si bond generation n+p Where does generation and recombination take place? In modern devices, recombination mainly takes place at surfaces: • perfect crystalline periodicity broken at a surface ⇒ lots of broken bonds: generation and recombination centers • modern devices are very small ⇒ high area to volume ratio. High generation and recombination activity at surfaces ⇒ carrier concentrations cannot deviate much from equilibrium values: n(s) ' no, p(s) ' po Complete physical picture for pn diode under bias: • Forward bias: injected minority carriers diffuse through QNR ⇒ recombine at semiconductor surface log p, n Na po Nd no p n n i2 Nd n i2 Na x 0 • Reverse bias: minority carriers extracted by SCR ⇒ generated at surface and diffuse through QNR log p, n Na po no ni2 Na Nd ni2 Nd p n 0 x The current view: • Forward bias: p n hole injection and recombination at surface electron injection and recombination at surface I=In+Ip • Reverse bias: p n hole generation at surface and extraction electron generation at surface and extraction I=In+Ip What limits the magnitude of the diode current? • not generation or recombination rate at surfaces • not injection or extraction rates through SCR • diffusion rate through QNR’s p n hole injection and recombination at surface electron injection and recombination at surface -Wp -xp xn Wn x Development of analytical current model: 1. Calculate concentration of minority carriers at edges of SCR, p(xn) and n(−xp ) 2. calculate minority carrier diffusion current in each QNR, In and Ip 3. sum electron and hole diffusion currents, I = In + Ip 2. I-V characteristics 2 Step 1: computation of minority carrier boundary conditions at edges of SCR In thermal equilibrium in SCR, |Jdrift| = |Jdiff |, and no(x1 ) q[φ(x1 ) − φ(x2)] = exp no(x2 ) kT and po (x1 ) −q[φ(x1 ) − φ(x2 )] = exp po (x2 ) kT Under bias in SCR, |Jdrift | 6= |Jdiff |, but if difference small with respect to absolute values of current: n(x1 ) q[φ(x1 ) − φ(x2 )] ' exp n(x2 ) kT and −q[φ(x1 ) − φ(x2 )] p(x1 ) ' exp p(x2 ) kT This is called quasi-equilibrium. p - p-QNR + n SCR n-QNR φ 0 -xp φB-V φB 0 xn x At edges of SCR, then: n(xn) q[φ(xn ) − φ(−xp )] q(φB − V ) = exp ' exp n(−xp ) kT kT and p(xn ) −q[φ(xn ) − φ(−xp )] −q(φB − V ) ' exp = exp p(−xp ) kT kT But: p(−xp ) ' Na and n(xn) ' Nd This is the low-level injection approximation [will discuss in more detail next time]. Then: n(−xp ) ' Nd exp q(V − φB ) kT and p(xn) ' Na exp q(V − φB ) kT Built-in potential: kT NdNa φB = ln 2 q ni Plug in above and get: n2i qV n(−xp ) ' exp Na kT and n2i qV p(xn) ' exp Nd kT Voltage dependence: • Equilibrium (V = 0): n2i n(−xp ) = Na • Forward (V > 0): n2i p(xn ) = Nd n2i n2i n(−xp ) p(xn ) Na Nd Lots of carriers available for injection: ⇒ V ↑→ concentration of injected carriers ↑ ⇒ forward current can be high. • Reverse (V < 0): n2i n2i n(−xp ) p(xn ) Na Nd Few carriers available for extraction: ⇒ reverse current is small. Minority carrier concentration becomes vanishingly small: ⇒ reverse current saturates. Rectification property of pn diode arises from minoritycarrier boundary conditions at edges of SCR. 2 Step 2: Diffusion current in QNR: Diffusion equation (for electrons in p-QNR): dn Jn = qDn dx Inside p-QNR, electrons diffuse to reach and recombine at contact ⇒ Jn constant in p-QNR⇒ n(x) linear. n n(-xp) n(x) ni2 Na -xp -Wp 0 x Boundary conditions: n2i n(x = −Wp) = no = Na n2i qV n(−xp ) = exp Na kT Electron profile: np (x) = np (−xp) + np (−xp ) − np (−Wp ) (x + xp) −xp + Wp np (x) = np (−xp) + np (−xp ) − np (−Wp ) (x + xp) −xp + Wp Electron current density: dn np(−xp ) − np (−Wp) Jn = qDn = qDn dx Wp − xp = qDn n2i Na qV kT exp − Wp − xp n2i Na or n2i Dn qV Jn = q (exp − 1) kT Na Wp − xp Similarly for hole flow in n-QNR: p p(xn) p(x) ni2 Nd 0 xn Wn x Hole current density: n2i Dp qV Jp = q (exp − 1) Nd Wn − xn kT 2 Step 3: sum both current components: J = Jn+Jp = qn2i ( 1 Dn 1 Dp qV + )(exp −1) Na Wp − xp Nd Wn − xn kT Current: I = qAn2i ( 1 Dn 1 Dp qV + )(exp − 1) kT Na Wp − xp Nd Wn − xn often written as: I = Io(exp qV − 1) kT with Io ≡ saturation current [A] B.C.’s contain both forward and reverse bias ⇒ equation valid in forward and reverse bias. [will discuss this result in detail next time] Key conclusions • Application of voltage to pn junction results in disruption of balance between drift and diffusion in SCR: – in forward bias, minority carriers are injected into quasi-neutral regions – in reverse bias, minority carriers are extracted from quasi-neutral regions • In forward bias, injected minority carriers recombine at surface. • In reverse bias, extracted minority carriers are generated at surface. • Computation of boundary conditions across SCR exploits quasi-equilibrium: balance between diffusion and drift in SCR disturbed very little. • Rate limiting step to current flow: diffusion through quasi-neutral regions. • I-V characteristics of p-n diode: qV I = Io(exp − 1) kT

Lecture 15 - The PN Junction Diode (I): I-V Characteristics

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