PN Junction: Physical and Mathematical Description of Operation Dragica Vasileska Professor Arizona State University • • • • • PN Junctions – General Considerations Ideal Current-Voltage Characteristics Generation and Recombination Currents Breakdown Mechanisms AC-Analysis.
Download ReportTranscript PN Junction: Physical and Mathematical Description of Operation Dragica Vasileska Professor Arizona State University • • • • • PN Junctions – General Considerations Ideal Current-Voltage Characteristics Generation and Recombination Currents Breakdown Mechanisms AC-Analysis.
PN Junction: Physical and Mathematical Description of Operation Dragica Vasileska Professor Arizona State University • • • • • PN Junctions – General Considerations Ideal Current-Voltage Characteristics Generation and Recombination Currents Breakdown Mechanisms AC-Analysis and Diode Switching 1. PN-junctions - General Consideration: • PN-junction is a two terminal device. • Based on the doping profile, PN-junctions can be separated into two major categories: - step junctions - linearly-graded junctions ND N A ND N A ax n-side p-side Step junction p-side n-side Linearly-graded junction (A) Equilibrium analysis of step junctions EC qVbi Ei EF EV p-side (x) n-side p p 0 ni expEi EF k BT W + qND -qNA - x V (x) Emax N AND k BT p p 0 nn0 Vbi ln V ln T n2 n2 q i i (b) Majority- minority carrier relationship: Vbi E (x) xp (a) Built-in voltage Vbi: qVbi Ei EF p EF Ei n nn0 ni exp EF Ei k BT x xn x pn0 p p 0 exp Vbi / VT n p 0 nn0 exp Vbi / VT (c) Depletion region width: Solve 1D Poisson equation using depletion charge approximation, subject to the following boundary conditions: V ( x p ) 0, V ( xn ) Vbi , E ( xn ) E ( x p ) 0 qN A p-side: V p ( x) x x p 2 2k s 0 qN D n-side: Vn ( x) xn x 2 Vbi 2k s 0 Use the continuity of the two solutions at x=0, and charge neutrality, to obtain the expression for the depletion region width W: xn x p W 2k s 0 ( N A N D )Vbi V p (0) Vn (0) W qN A N D N A x p N D xn (d) Maximum electric field: The maximum electric field, which occurs at the metallurgical junction, is given by: Emax dV dx x 0 qN A N DW k s 0 ( N A N D ) (e) Carrier concentration variation: 15 N A N D 1015 cm 3 Wcalc 1.23 mm Emax( DC ) 9.36 kV / cm Concentration [cm-3] 10 13 10 11 10 -3 n [cm ] -3 Emax( sim) 8.93 kV / cm p [cm ] 9 10 7 10 5 10 0 0.5 1 1.5 2 2.5 Distance [mm] 3 3.5 (f) Analytical vs. numerical data N A N D 1015 cm3 1.23 mm, Emax( DC ) 9.36 kV / cm, Emax( sim) 8.93 kV / cm Wcalc 5x10 15 0 Electric field [kV/cm] -3 (x)/q [cm ] 10 14 0 -5x10 14 -10 15 0 0.5 1 1.5 2 2.5 Distance [mm] 3 3.5 -2 -4 -6 -8 -10 0 0.5 1 1.5 2 2.5 Distance [mm] 3 3.5 N A 1016 cm3 , N D 1018 cm3 0.328 mm, Emax( DC ) 49.53 kV / cm, Emax( sim) 67 kV / cm Wcalc 10 Electric field [kV/cm] 5x10 17 16 -3 (x)/q [cm ] 10 0 -5x10 -10 16 17 0.6 0.8 1 1.2 Distance [mm] 1.4 0 -10 -20 -30 -40 -50 -60 -70 0.6 0.8 1 1.2 Distance [mm] 1.4 (g) Depletion layer capacitance: Consider a p+n, or one-sided junction, for which: 2ks 0 Vbi V W qN D The depletion layer capacitance is calculated using: dQc qN D dW qN D ks 0 1 2(Vbi V ) C 2 dV dV 2(Vbi V ) C qN D ks 0 1 C2 Measurement setup: 1 slope ND Reverse bias vac ~ Forward bias Vbi V W V V dW (B) Equilibrium analysis of linearly-graded junction: 12k s 0 Vbi V W qa 1/ 3 (a) Depletion layer width: (c) Maximum electric field: Emax (d) Depletion layer capacitance: qaW 2 8k s 0 1/ 3 C 12 V V bi qaks202 Based on accurate numerical simulations, the depletion layer capacitance can be more accurately calculated if Vbi is replaced by the gradient voltage Vg: a 2k s 0VT 2 Vg VT ln 3 3 8qni (2) Ideal Current-Voltage Characteristics: Assumptions: • Abrupt depletion layer approximation • Low-level injection injected minority carrier density much smaller than the majority carrier density • No generation-recombination within the space-charge region (SCR) (a) Depletion layer: W EC qV E Fn E Fp np ni2 exp V / VT n p ( x p ) n p 0 exp V / VT pn ( xn ) pn 0 exp V / VT EV xp xn (b) Quasi-neutral regions: • Using minority carrier continuity equations, one arrives at the following expressions for the excess hole and electron densities in the quasi-neutral regions: V / VT pn ( x ) pn 0 ( e V / VT n p ( x ) n p 0 ( e 1)e ( x xn ) / L p 1)e n p (x) ( x x p ) / Ln pn (x ) Forward bias Space-charge region W pn 0 n p0 xp xn x Reverse bias • Corresponding minority-carriers diffusion current densities are: diff J p ( x) diff J n ( x) qD p pn 0 qDn n p 0 Lp Ln V / VT (e V / VT (e 1)e 1)e ( x xn ) / L p ( x x p ) / Ln Shockley model diff diff J tot J p ( xn ) J n ( x p ) diff majority J p majority J ndiff J ndrift drift Jp J tot diff minority J p minority J ndiff xp xn No SCR generation/recombination x (c) Total current density: • Total current equals the sum of the minority carrier diffusion currents defined at the edges of the SCR: I tot I diff diff I p ( xn ) I n ( x p ) Ge Si GaAs D p pn 0 Dn n p 0 V / V e T 1 qA L L p n • Reverse saturation current IS: D p pn 0 Dn n p 0 Dp D 2 n qAni I s qA L L N L L N p n n A p D V (d) Origin of the current flow: Reverse bias: Forward bias: W EC Ln qVbi V qV E Fp EC E Fn qV qVbi V E Fp EV E Fn EV Lp W Reverse saturation current is due to minority carriers being collected over a distance on the order of the diffusion length. (e) Majority carriers current: • Consider a forward-biased diode under low-level injection conditions: Quasi-neutrality requires: nn (x ) nn 0 nn ( x ) pn ( x ) This leads to: pn (x ) pn 0 x xn diff J n ( x) Dn diff J p ( x) Dp • Total hole current in the quasi-neutral regions: tot J p ( x) diff drift J p ( x) J p ( x) diff J p ( x) • Electron drift current in the quasi-neutral region: diff J n ( x) J tot Dn diff 1 diff 1 J p ( x ), E ( x ) J n ( x) D qn( x )m n p J ndrift (x ) J tot J ntot ( x ) J ndiff ( x ) J ndrift ( x ) diff diff J n ( x) J p ( x) diff J p (x ) x J ndiff (x ) (f) Limitations of the Shockley model: • The simplified Shockley model accurately describes IVcharacteristics of Ge diodes at low current densities. • For Si and Ge diodes, one needs to take into account several important non-ideal effects, such as: Generation and recombination of carriers within the depletion region. Series resistance effects due to voltage drop in the quasi-neutral regions. Junction breakdown at large reverse biases due to tun- neling and impact ionization effects. (3) Generation and Recombination Currents J scr Continuity equation for holes: p 1 J p Gp Rp t q x Steady-state and no light genera- tion process: p t 0 , G p 0 • Space-charge region recombination current: xn xn xp xp dJ p ( x ) J p ( xn ) J p ( x p ) q R p dx xn J scr q R p dx xp Reverse-bias conditions: • Concentrations n and p are negligible in the depletion region: ni2 ni E t Ei Ei E t n exp R , g p exp p n1 n p1 g k BT k BT Generation lifetime • Space-charge region current is actually generation current: J scr J gen qniW qniW J gen Vbi V g g • Total reverse-saturation current: V / VT J Js e 1 J scr J s J gen V VT • Generation current dominates when ni is small, which is always the case for Si and GaAs diodes. I (log-scale) AJ s V (log-scale) AJ gen IV-characteristics under reverse bias conditions EC E Fp EV E Fn W Generated carriers are swept away from the depletion region. Forward-bias conditions: • Concentrations n and p are large in the depletion region: np 2 V / VT ni e 2 V /V ni e T 1 R p n n1 n p p1 • Condition for maximum recombination rate: V / 2VT Recombination lifetime n p ni e V /V ni2 e T ni V / 2VT Rmax e , rec p n n p p n rec • Estimate of the recombination current: max J scr qniW V / 2VT e rec • Exact expression for the recombination current: J scr qni V / 2VT 1 qN D 2Vbin V e , VT , Enp rec 2 Enp k s 0 • Corrections to the model: J scr qni V / mrVT e rec • Total forward current: V / VT J Js e qni V / mrVT V / VT 1 e J s,eff e 1 rec ideality factor. Deviations of from unity represent an important measure for the recombination current. • Importance of recombination effects: Low voltages, small ni recombination current dominates Large voltages diffusion current dominates log(I) AJ scr AJ V AJ d (4) Breakdown Mechanisms • Junction breakdown can be due to: tunneling breakdown avalanche breakdown • One can determine which mechanism is responsible for the breakdown based on the value of the breakdown voltage VBD : VBD < 4Eg/q tunneling breakdown VBD > 6Eg/q avalanche breakdown 4Eg/q < VBD < 6Eg/q both tunneling and avalanche mechanisms are responsible Tunneling breakdown: • Tunneling breakdown occurs in heavily-doped pnjunctions in which the depletion region width W is about 10 nm. Zero-bias band diagram: EF EC W EV Forward-bias band diagram: EFn EFp EC EV W • Tunneling current (obtained by using WKB approximation): Reverse-bias band diagram: It EF p EFn EC EV * 3 2m q FcrVA 2 2 1/ 2 4 E g 4 2m* E 3 / 2 g exp 3qFcr Fcr average electric field in the junction • The critical voltage for tunneling breakdown, VBR, is estimated from: I t (VBR ) 10 I S • With T, Eg and It . Avalanche breakdown: • Most important mechanism in junction breakdown, i.e. it imposes an upper limit on the reverse bias for most diodes. • Impact ionization is characterized by ionization rates an and ap, defined as probabilities for impact ionization per unit length, i.e. how many electron-hole pairs have been generated per particle per unit length: Ei ai exp qlFcr - Ei critical energy for impact ionization to occur - Fcr critical electric field - l mean-free path for carriers Avalanche mechanism: EF p EFn EC EV Generation of the excess electron-hole pairs is due to impact ionization. Expanded view of the depletion region • Description of the avalanche process: Jn J p an J n dx dx J n an J n dx Jp Impact ionization initiated by electrons. Jn J p a p J p dx dx J n a p J p dx Jp Impact ionization initiated by holes. dJ p dJ n 0, 0 dx dx dJ p dJ n dx dx J J n J p const. Multiplication factors for electrons and holes: J p (0) J n (W ) Mn , Mp J n (0) J p (W ) • Breakdown voltage voltage for which the multiplication rates Mn and Mp become infinite. For this purpose, one needs to express Mn and Mp in terms of an and ap: x an a p dx ' W 1 1 0 dJ n a e dx a J a J n M n n p p dx 0 n dJ x an a p dx ' W p an J n a p J p 1 dx a pe 0 dx 1 M p 0 The breakdown condition does not depend on which type of carrier initiated the process. • Limiting cases: (a) an=ap (semiconductor with equal ionization rates): W 1 1 1 M a n dx M n W 0 n 1 a n dx 0 W 1 1 1 a p dx M p W Mp 0 1 a p dx 0 (b) an>>ap (impact ionization dominated by one carrier): W an dx Mn e 0 W 1 a n dx 0 Breakdown voltages: (a) Step p+n-junction p • For one sided junction we can make the following approximation: W Wn Wp Wn • Voltage drop across the depletion region on the n-side: n Wp Fmax 1 1 Vn FmaxWn VBD FmaxW 2 2 Wn • Maximum electric field: k s 0 2 qN DW Fmax VBD Fmax k s 0 2qN D • Empirical expression for the breakdown voltage VBD: F (x) x VBD Eg 60 1.1 3/ 2 ND 16 10 -3 / 4 kV cm (b) Step p+-n-n+ junction • Extension of the n-layer large: p n n VBD 1 FmaxWm 2 • Extension of the n-layer small: Wp Fmax 1 1 VP FmaxWm F1 Wm W1 2 2 W1 Wm F (x) • Final expression for the punchthrough voltage VP: F1 x W1 W1 2 VP VBD Wm Wm • Doping-dependence of the breakdown voltage VBD: Width of the n-layer W1 increases One-sided abrupt junction Tunneling breakdown Log-scale VBD p+-n-n+ Log-scale ND • Temperature dependence: As temperature increases, lattice scattering increases which makes impact ionization less probable. As a result of this, the breakdown voltage increases. (c) Plane vs. planar or cylindrical junction • Plane junction: Maximum electric field: p+ n • Planar junction: Fmax Except for surface effects, this is an ideal junction. Maximum electric field: rj p+ W qN DW Q k s 0 k s 0 Fmax n qN DW W 1 k s 0 2r j The smaller the radius rj, the larger the electric field crowding. (5) AC-Analysis and Diode Switching (a) Diffusion capacitance and small-signal equivalent circuit • This is capacitance related to the change of the minority carriers. It is important (even becomes dominant) under forward bias conditions. • The diffusion capacitance is obtained from the device impedance, and using the continuity equation for minority 2 carriers: dp d p p n dt Dp n dx 2 n p • Applied voltages, currents and solution for pn: it V (t ) V0 V1e , V1 V0 it J (t ) J 0 J 1e , J 1 J 0 pn ( x, t ) pns ( x ) pn1 ( x )eit • Equation for pn1(x): d 2 pn1 1 i p d 2 pn1 pn1 ( x ) pn1 ( x ) 0 2 0 2 2 Dp p dx dx L p' • Boundary conditions: pn (, t ) pn 0 pn1 () 0 V0 V1eit pn 0V1 V0 pn (0, t ) pn 0 exp pn1 (0) exp V VT VT T • Final expression for pn1(x): x pn 0V1 V0 pn1 ( x, t ) exp exp L VT VT p' • Small-signal hole current: AqD p pn 0V1 dpn1 V0 I1 AqD p 1 i p exp YV1 dx x 0 L pVT VT • Low-frequency limit for the admittance Y: Y AqD p pn 0 L pVT V0 1 exp 1 i p Gd iCdif VT 2 V0 I s eV0 / VT I dI Gd exp , I Forward current L pVT VT VT dV VT V0 1 I 1 AqD p pn 0 Cdif p exp p 2 L pVT VT 2 VT AqD p pn 0 • RC-constant: Rd Cdif p 2 The characteristic time constant is on the order of the minority carriers lifetime. • Equivalent circuit model for forward bias: Cdepl Rs Cdif Rd • Bias dependence: Ls 1 Gd C Cdif Cdepl Va (b) Diode switching • For switching applications, the transition from forward bias to reverse bias must be nearly abrupt and the transit time short. • Diode turn-on and turn-off characteristics can be obtained from the solution of the continuity equations: dpn 1 1 J p pn 1D J p R p dt q q x p dQ p Qp dQ p Q p I p (t ) I (t ) I p (t ) dt p dt p Qp(t) = excess hole charge Valid for p+n diode Diode turn-on: • For t<0, the switch is open, and the excess hole charge is: Q p (t 0) Q p (0 ) 0 • At t=0, the switch closes, and we have the following boundary condition: p+ t=0 IF Q p (0 ) Q p (0 ) 0 • Final expression for the excess hole charge: Q p (t ) A Be t / p t / p pIF 1 e n • Graphical representation: Q p (t ) pn ( x, t ) pIF Slope almost constant t increasing pn 0 x t • Steady state value for the bias across the diode: Va / VT pn ( x ) pn 0 e IF Va VT ln 1 IS 1 e x / Lp Va / VT Q p Aqpn 0 L p e 1 Diode turn-off: • For t<0, the switch is in position 1, and a steady-state situation is established: VF IF R p+ t=0 1 • At t=0, the switch is moved to position 2, and up until time t=t1 we have: pn (0, t ) pn 0 Va 0 • The current through the diode until time t1 is: VR IR R VF R 2 VR R n • To solve exactly this problem and find diode switching time, is a rather difficult task. To simplify the problem, we make the crucial assumption that IR remains constant even beyond t1. • The differential equation to be solved and the initial condition are, thus, of the form: IR dQ p dt Qp p , Q p (0 ) Q p (0 ) p I F • This gives the following final solution: Q p (t ) p I R p I F I R e t / p • Diode switching time: Q p (trr ) 0 trr IF p ln 1 IR • Graphical representation: Va (t ) pn ( x, t ) t Slope almost constant t=0 pn 0 VR t=ts ttrr ts switching time trr reverse recovery time IF x 0.1I R IR ts t rr t