Lecture 20 - p-n Junction (cont.) October 21, Non-ideal and second-order effects

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1 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-1 Lecture 0 - p-n Junction (cont.) October 1, 00 Contents: 1. Non-ideal and second-order effects Reading assignment: del Alamo, Ch. 7, Seminar: Oct. - C. Kagan (IBM): Materials for Molecular Devices. Rm , 4 PM.

2 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0- Key questions If the doping distribution in a p-n junction is non-uniform, is the basic operation of the diode changed in a fundamental way? What happens to the C-V characteristics, I-V characteristics, and the dynamics of a p-n diode with non-uniform doping distributions? What happens if there is SCR generation and recombination in a pn diode?

3 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture Non-ideal and second-order effects Non-uniform doping level Real p-n diodes have doping profiles that are highly non-uniform: N(x) N A (x) N D (x) -w p -x p x n W n x p-qnr SCR n-qnr No major differences in operation of pn diode, e.g. rectifying characteristics of p-n diode. Some qualitative differences, e.g., the voltage dependence of the C-V characteristics. Need to revise computation of I, C, and minority carrier time constants.

4 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-4 Depletion capacitance p-n junction electrostatics affected by doping non-uniformity. Treat simple case: linearly graded junction: N(x)=ND-NA p n N(x)=ax a 0 x SCR 0 x SCR x N(x) =N D N A = ax Do depletion approximation with: x n = x p = x SCR

5 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-5 Volume charge density: ρ(x) = qax ρ(x) 0 for x SCR outside x x SCR Electric field: E(x) = qa ɛ [x ( x SCR ) ] for x SCR E(x) 0 outside x x SCR Electrostatic potential distribution (φ(x = 0) = 0): φ(x) = qa 6ɛ [3(x SCR ) x x 3 ] for x SCR x x SCR

6 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-6 x SCR determined by demanding that: Then: With: φ( x SCR ) φ( x SCR )=φ bi V x SCR =[ 1ɛ(φ bi V ) ] 1/3 qa φ bi = kt q ln n o( x SCR ) n o ( x SCR) =kt q ln ax SCR n i These two equations need to be solved iteratively. Capacitance: qaɛ C =[ 1(φ bi V ) ]1/3 = C(V =0) (1 V φ bi ) 1/3 With: 1 C = 1(φ bi V ) 3 qaɛ

7 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-7 Experiments: In general, depletion capacitance well modeled by: C o is value of C at V =0. C = C o (1 V φ bi ) m m =0.5 for ideal abrupt junction m =0.33 for ideal gradual junction

8 + 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-8 Current Non-uniform doping distribution does not affect basic physics of minority carriers. Electric field associated with non-uniform doping distribution affects overall carrier velocity. Computation of current density and dominant minority carrier time constant more complex. Electric field affects current and transit time by aiding or opposing minority carrier diffusion. N(x) N(x) - p o (x) N A (x) N A (x) - p o (x) + ε o ε o -w p -x p x -w p -x p x field aids electron diffusion field opposes electron diffusion Downgoing doping profile aids diffusion I τ t Upgoing doping profile opposes diffusion I τ t

9 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-9 Analytical solution obtainable only in case of transparent or short region: minority carrier recombination takes place at surface. For p-region: Since: J e = qn µ e E o + qd e dn dx E o = kt q 1 dn A N A dx Then: J e = q D e d(n N A ) N A dx Integrate across p-qnr: x p N A J w e dx = x p d(n N A ) dx = n N p qd w A xp n N A wp e p dx

10 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-10 B.C. s: -at junction: n ( x p )= n i qv (exp N A ( x p ) kt 1) -at surface (if S = ): n ( )=0 If recombination mainly takes place at surface, J e independent of x, and: J e ( x p )= qn i qv x p (exp N A D e dx kt 1) qn i <D e > x p N A dx (exp qv kt 1) Since D e is slow function of N A, oftentimes x p N A dx referred to as Gummel number to first order, only integrated doping concentration counts to set the current! Similar equation for n-type side.

11 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-11 Dynamics: calculation of transit time in transparent region. Go back to: J e = q D e d(n N A ) N A dx Integrate up to x: x N A J w e dx = x d(n N A ) dx = n N p qd w A x n N A wp e p dx If S =, n ( ) = 0, and: n (x) = J e( x p ) qn A (x) Total minority carrier charge: x N A D e dx Q p = q x p n (x)dx = J e ( x p ) x p 1 N A ( x N A D e dx)dx Diffusion capacitance: C dp = dq p dv = q kt J e( x p ) x p 1 N A ( x N A D e dx)dx = q kt τ tpj e ( x p ) Then: τ tp = x p 1 N A ( x N A D e dx)dx

12 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-1 Space-charge generation and recombination In real devices, non-ideal I V characteristics often seen: log I e qv/kt e qv/kt I S 0 V Anomalies often due to: recombination through traps in SCR (in forward bias) generation through traps in SCR (in reverse bias)

13 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-13 Simple model for SCR generation and recombination. Starting point: trap-assisted G/R rate equation: In SCR: U tr = np n o p o τ ho (n + n i )+τ eo (p + n i ) np = n i exp qv kt Then: SCR G/R current: U tr SCR = n qv i (exp kt 1) τ ho n + τ eo p +(τ ho + τ eo )n i J SCR = q x n x p U tr dx Since n and p changing quickly with x in SCR, no analytical solution.

14 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-14 Analytical model: U tr SCR = n qv i (exp kt 1) τ ho n + τ eo p +(τ ho + τ eo )n i Since np constant, point of SCR with highest U tr where: τ ho n = τ eo p At that point: U tr SCR,max = n i (exp qv τ eo τ ho kt 1) Upper limit to current: J SCR,max = qn ix SCR τ eo τ ho (exp qv kt 1)

15 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-15 Key dependencies: J SCR,max = qn ix SCR τ eo τ ho (exp qv kt 1) Forward bias: SCR recombination exp qv kt in contrast with QNR recombination exp qv kt In practice, 1 <n< for SCR recombination SCR G/R n i, in contrast with QNR G/R n i E a (SCR) E g /, in contrast with E a (QNR) E g SCR G/R highly process sensitive: small SCR G/R current hallmark of clean process, Reverse bias: SCR generation x SCR I V log I e qv/kt e qv/kt I S 0 V

16 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-16 Two diodes in weblab:

17 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-17 Key conclusions Non-uniformly doped regions do not affect basic operation of pn junction. Exponent of dependence of depletion capacitance with voltage is function of doping distribution: m =0.5 for abrupt junction m = 0.33 for linearly graded junction. Integrated doping concentration sets minority carrier current in transparent or short non-uniformly doped QNR. Minority carrier transit time through non-uniformly doped QNR depends on details of impurity profile. SCR generation and recombination dominates in low forward bias and in reverse bias. Key characteristic of SCR recombination in forward bias: J SCR exp qv nkt with 1 <n< Key characteristic of SCR generation in reverse bias: J SCR V

18 6.70J/3.43J - Integrated Microelectronic Devices - Fall 00 Lecture 0-18 Self study Reverse breakdown High-injection effects