Avalanche multiplication

Transcription

1 Avalanche multiplication E (-) (+) 0 x L 1

2 Ionization coefficients α, β Ge β α Si α β α β GaAs α, β (cm -1 ) 4 GaAsSb α β α Ga.94 Al.06Sb β α β Ga In As.47. β InP α E ( V/ cm) 7

3 APD Gain di n /dx = αi n +βi p, di p /dx = - (αi n +βi p ) with: i n (0) = I ph, i n (L)= I, i p (L) = 0 then, d i n /dx =(α β)di n /dx, i n =[C 1 /(α β)]exp(α β)x+c, βi p =-[βc 1 /(α β)]exp(α β)x-αc. (α - β) exp(α - β)l M = α - β exp(α - β)l special cases: for β<< α: M = exp αl, for β= α: M = 1/(1- αl)

4 APD gain vs αl 8 GAIN M = α/β 0 00 M o =α/β is the optimal gain number of multiplication lengths α L 4

5 Frequency response β/α =1 β/α =0.1 β/α =0.01 β/α =0 0 GAIN M (ω ) c M (0)=0 c NORMALIZED FREQUENCY ωl/v*

6 -db cutoff frequency NORMALIZED CUTOFF FREQUENCY f d τ d M (0)=00 c IONIZATION RATIO α/β M o =α/β is the optimal gain f d 0.6 /τ d (M<<M o ), f d M 0.6(α/β)/τ d (M>M o ) 6

7 Delta response normalized current i (t) L/qv* β/α =0 β/α = M(0)= M(0)= β/α = M(0)= normalized time - t v*/l normalized time - t v*/l 7

8 noise relative variance of output charge for 1 pair at x=0: σ M (0)/M c (0) = 1+β M c (0)/(α β)-αm c (L)/M c (0)(α β) = 1+M c (0){β/(α β)-[α/(α β)] exp-(α β)l} for moderate gains M c : σ M (0)/M c (0) 1 M c (0)β/α for M c (0)<α/β for M c (0)>α/β once again, M o =α/β is the optimal gain. Noise current: σ I = eb I ph (M +σ M ) = eb I ph M F F = 1+σ M /M 8

9 Noise figure 00 NOISE FIGURE - F 0 0 M (0)=00 c IONIZATION RATIO - α/β 9

10 Optimum gain and beyond σ I = eb (I ph +I o ) M F+4kTB/R σ in = eb (I ph +I o ) F + 4kTB/RM but, S=I ph, N = σ in, and F= Mβ/α (S/N) = I ph / [eb(i ph +I o ) Mβ/α + 4kTB/RM ] S/N is maximum at a M max found as: M max = [(α/β)(4kt/e) /R(I ph +I o )] 1/ for instance, for R=0Ω, I ph +I o =1nA and α/β=1: M max =1 practical limit for M: bandwidth decreasing by Mβ/α at M>α/β

11 APD desirable structure E E m d P P L1 L z two regions desirable, to separate : photon dissipation in L 1 ( 1/α) and the carrier drift (medium E d ) multiplication in L of most efficient ionizing carrier (high E m ) 11

12 APD structures A N E p Si Ge K K A K i n+ n+ n p N N E hole multiplication E drift (a) (b) A Si K K p+ In P A p+ π n+ ν InGaAs p+ π n+ InGaAs n+ InP N N E drift multipl E drift multipl (d) (e) Si A n+ p+ π p+ electron multiplication drift (c) reach-through 1

13 Staircase APD no bias x=1 x=0. p- Ga Al As, variable x x 1-x n+ GaAs inverse bias low field medium field (drift) high field (multiplication) energy n + 1

14 MQW APD drift E g multiplication tunnel 14

15 APD Biasing and Use Requisites GAIN-BANDWIDTH PRODUCT M f (GHz) T=-40 C GAIN M M f Si APD reach-through INVERSE BIAS VOLTAGE (V) 1

16 SPAD +V bb R I v C R 0 I t in quiescent conditions, V bb >V, M= a single charge-pair makes I increase fast up to V ak =V bb -RI < V ; then M returns to a f inite value Max. photon rate: F M <τ rec = R o C; dark rate: F b =J 0 A/q Dynamic range: F M /F b = typ. 16