Rivelatori X e Gamma per spettroscopia e imaging

Transcription

1 Rivelaori X e Gamma per speroscopia e imaging Anonio Longoni Poliecnico di Milano, Diparimeno di Eleronica e Informazione INFN, Sezione di Milano anonio.longoni@polimi.i Scuola Nazionale INFN Rivelaori ed Eleronica per Fisica delle Ale Energie, Asrofisica ed Applicazioni Spaziali, Laboraori Nazionali di Legnaro 4-8 Aprile 2005

2 Summary Firs par why low capaciance for specroscopic deecors (in oher words, why SDDs?): some basic conceps on signal processing Second par Inroducion o some new SDDs and SDD monolihic arrays Some applicaions in maerial analysis, in biomedicine and in aomic physics

3 The ineracion of he radiaion wih he deecor

4 The main ineracions of X rays in semiconducor deecors Phoon E ph = hν Fas elecron E = E E el ph b0 Phoon E * = E ph b0 Auger elecron * E = E Ae b0 EbAe Phooelecric absorpion Phoon Eph = hν Fas elecron ϕ E Compon scaering Phoon * ph E = E E * el ph ph = f ( ϕ) e-h pairs Primary elecron Elecron-hole pairs generaion

5 Signal generaion in a deecor N + anode N ype Si X ray Si aom P + cahode Bias Elecrons Holes Fas elecron Amplificaion circui The average number of e-h couples generaed in he deecor is: Ncoup = Eph εcoup where E ph is he phoon energy and ε coup is he average energy required o generae an e-h couple (ε coup =3.6 ev in Si).

6 The generaion of elecron-hole couples is a saisical process, he elecronic noise inroduces furher saisical flucuaions dn de dn dn c σinr dn dn meas σ meas E ph E Nc Nc Nmeas Nmeas Inciden phoon. monoenergeic Saisical spreading of he number of couples e-h Elecronic noise. conribuion σ 2 N ph c c = F E Fano facor F, in silicon F 0.2 ε σ = σ + σ N N N meas c elec

7 The concep of Equivalen Noise Charge: ENC dn dn meas σ Nmeas σ = σ + σ N N N meas c elec ENC = q σ N elec Nmeas Nmeas Q V peak ou Q= ENC = S N A lower ENC means a beer resoluion of he deecion sysem

8 Signal and noise in he deecion sysem

9 Block diagram of a ypical deecion sysem for X-ray specroscopy Phoon E=hν Deecor Amplifier Filer AD and PC Our goal: o reach S/N as good as possible

10 Block diagram of a ypical deecion sysem for X-ray specroscopy: signal and noise S i S v - + C F PA SA Pk Sr ADC o PC C D C G Deecor C D comprises also he parasiic capaciances seen by he amplifier inpu

11 For wha concerns S/N, charge-preamplifier and volage preamplifier are equivalen C F S i S v - + PA SA Pk Sr ADC o PC C D C G S v I S PA SA Pk Sr ADC S i o PC C D C F C G We will use volage preamplifier because i is simpler o analyze

12 Block diagram: relevan elemens I S V G I D =g m V G V PA V SA T(s) S v C D S i C G A IV I D Deecor Preamplifier Shaping amplifier From now on C D comprises also he feedback capaciance C F : C D =C de +C parasiic +C feedback C G is he inpu capaciance of he FET

13 Whie series noise of he FET Thermal noise of he FET channel S S vw =α2kt/g m G G* C G Noe: V G*S!! D g m V G*S S S vw α 23 2kT 2kT = α = α g C ω m G T Noe: bilaeral noise specra

14 Whie parallel noise of Deecor and FET Deecor sho noise S id = qi D FET gae sho noise S ig = qi G G D V GS C D C G g m V GS S id S S ig S S = S + S = q( I + I ) = qi iw id ig D G L

15 /f series noise flicker noise Mainly due o charge rapping and de-rapping in he FET channel Svf ( ω ) = Af f G 2 G* D C G V G*S g m V G*S S vf A π A 2kT ω 2 f ω C ω ω f f ( ω) = = = α G T S S S V ω π AC f G π = = H f ω α2kt α2kt T consan for a given echnology ω ω

16 Evaluaion of he S/N: a) he oupu signal I DET I D =g m V G V PA V SA V G T(s) C D C G A IV I D Deecor Preamplifier Shaping amplifier DET ( ) = VG ( s) = DRAIN ( ) I s Q Q C T s Q I s = g C T m s gmaiv VSAou ( s) = Q T () s C s CT = CD + CG T S V = N 2 2 SAou peak v 2 no Q V s = g A ( ) PAou m IV CT s

17 Evaluaion of he S/N: b) he oupu noise V G I D =g m V G V PA V SA T(s) S v C D S i C G A IV I D Deecor Preamplifier Shaping amplifier S iw 2 2 Sno( ω) = Svw + Svf ( ω) + g 2 2 m AIV T jω ω CT ( ) 2 S V = N v 2 2 peak ou 2 no 2 no + ( ω) v = Sno dω 2π

18 Evaluaion of he S/N S N Q A g Max L T () s IV 2 m V peak ou C T s = = vno S iw 2 2 vw vf ( ) 2 2 m IV 2π + + ω CT ( ) S S ω g A T jω dω 2 The ENC does no depend on he gain of he shaper 2 Max L T () s = s

19 The Equivalen Noise Charge

20 Evaluaion of he ENC Q S = ENC = N Siw ENC C ( ) ( ) 2 T Svw Svf T j d ω = π + + ω C ω ω T = T vw ( ) + T f ( ) + iw 2 ( ) 2π 2π ω 2π ω ENC C S T jω dω C πa T jω dω S T jω dω Series noise conribuion /f noise conribuion Parallel noise conribuion

21 The shape facors The angular frequency ω can be normalised o a characerisic frequency ω c =/τ, where τ is a characerisic ime which represens he widh of he oupu pulse (for insance he peaking ime, or he ime widh a half heigh, or a characerisic ime consan of he filer). The characerisic ime τ is also called shaping ime of he filer. ω x = = ωτ ω c = T vw ( ) + Tπ f ( ) + iwτ 2 ( ) τ 2π 2π x 2π ω ENC C S T jx dx C A T jx dx S T jx dx Series noise conribuion /f noise conribuion Parallel noise conribuion

22 + 2 A = T( jx) dx 2π + 2 A2 = T( jx) dx 2π x + 2 A = T( jx) dx 3 2 2π x The hree facors A, A 2 and A 3 depend on he SHAPE of he oupu pulse and on he choice of he characerisic ime τ of he oupu pulse used o normalise he angular frequency ω. The facors A, A 2 and A 3 do no depend on he paricular value of he shaping ime τ. τ " = kτ ' A A A ( τ ") = ka ( τ ') ( τ" ) = A ( τ ') 2 2 k ( τ ") = A ( τ ') 3 3

23 ENC = C S A + C π A A + S A T vw T f 2 iw τ τ 3 SHAPE FACTORS A A 2 A 3 AA 3 A A3 Infinie cusp τ Triangular T base / Gaussian σ CR-RC RC CR-RC 4 RC CR-RC 4 τpeak Semigaussian 7 poles σ Semigaussian 7 poles τpeak Trapezoidal (T f =0.5xT r ) T r Trapezoidal (T f =T r ) T r Trapezoidal (T f =2xT r ) T r

24 The dependence on τ of he 3 conribuions o he ENC ENC = C S A + C π A A + S A T vw T f 2 iw 3 τ ENC = ENC + ENC + ENC τ series / f parallel ENC 2 Series Parallel /f τ

25 The dependence on ransisor and deecor parameers of he ENC ENC = C S A + C π A A + S A T vw T f 2 iw 3 τ τ 2kT 2kT Svw = α = α gm CG ωt Siw = q( ID + IG + IReq ) = qil π Af 2kT ω Svf ( ω) = = α ω CG ωt ω 2 2 C C α 2kT C C α 2kT τ 2 D G D G ENC = ACD + A2CD A3 qit CG C + ω D ωt τ + + CG C D ωt Series noise /f noise Parallel noise

26 How o opimise he ENC

27 How o opimise he ENC () Choose he opimum shaping ime ENC 2 ENC = ENC 2 2 series parallel ENC 2 op τ op /f τ τ op S A C C 2kT A = + = + α ω Vw D G ( C C ) C D G D SIw A3 CG CD T qil A3

28 How o opimise he ENC (2) Mach he Deecor capaciance wih he Gae capaciance * For consan FET curren densiy 0 2 M 2 CD C G = + C G C D M 2 /(M mached ) C D /C G Series whie and series /f noise conribuions are minimised 2 2 C C 2kT α C C 2kT α ω τ 2 D G D G ENC = AC D + A2C D A3 T CG C + qi D ωt τ + + CG C D ωt

29 How o opimise he ENC (3) Reduce he deecor capaciance (in mached condiions) ENC (el rms) Cd=.5pF Cd=0.5pF ENC τ op op = I I C ω leak C leak D T ω T D Tm (s) * Wih /f noise negligible ENC op = 3 e- rms ENC op = 23 e- rms τ op = 0.4 µs τ op =.3 µs Series whie and series /f noise conribuions are minimised

30 How o opimise he ENC (4) Reduce he parallel noise sources Reduce he deecor leakage curren by cooling he deecor reducing he deecor acive volume improving he deecor echnology Increase he value of he resisors conneced o he FET inpu (bias or feedback resisors) Parallel whie noise conribuion is minimised 2 2 C C 2kT α C C 2kT α ω τ 2 D G D G ENC = AC D + A2C D A3 T CG C + qi D ωt τ + + CG C D ωt

31 ENC [e - rms] Gaussian shaping mached condiions I L eq =0pA C D =.25pF C D and I L C D =25fF 0 E-8 E-7 E-6 E-5 E s ENC [e - rms] 00 Gaussian shaping mached condiions C D =.25pF I L eq =0pA I L eq =pa 0 E-8 E-7 E-6 E-5 E-4 s

32 How o opimise he ENC (5) Choose he bes ransisor 2 2 C C 2kT C C ω 2 ( ) τ 2 D G D G ENC = AC D + 2 D 3 D G CG C α + A C ktα A q I I D ωt τ CG C D ωt Bandwidh /f noise Gae Leakage DEVICE H f [J] ω π AC f G π = = ω α2kt α2kt T H f JFET, n-channel, discree 2x0-26 JFET, n-channel, in CMOS process MOSFET, p channel, in CMOS process MOSFET, n channel, in CMOS process x x0-23 MESFET, GaAs, discree 0-23

33 How o opimise he ENC (6) Perform he opimum signal processing If only he whie noise sources (series and parallel) are presen, he bes ENC can be obained by using an ideal filering amplifier which gives a is oupu an infinie cusp - shaped pulse v sou () = exp τ wih a shaping ime τ se equal o he noise corner ime consan τ c. SVw τ c = ( CD + CG) S Iw

34 Ideal filer (for whie noise sources).0 Opimum signal processing Infinie cusp v s ou () /τ c 2 A= A2 = 0.64 A3 = π 2 ( ) ENC = C + C S S 2 D G Vw Iw

35 Ideal filer (In presence of whie and /f noise) Opimum signal processing ( ) K = C + C D G S A f Vw S Iw

36 Pracical filers ENC = C S A + C π A A + S A T vw T f 2 iw τ τ 3 SHAPE FACTORS A A 2 A 3 AA 3 A A3 Infinie cusp τ Triangular T base / Gaussian σ CR-RC RC CR-RC 4 RC CR-RC 4 τpeak Semigaussian 7 poles σ Semigaussian 7 poles τpeak Trapezoidal (T f =0.5xT r ) T r Trapezoidal (T f =T r ) T r Trapezoidal (T f =2xT r ) T r

37 Appendix: Noise modelling

38 Noise modelling A noise waveform can be represened as a random sequence of pulses: x ) ( ) ( = ak f k The disribuion of k is Poissonian* k The bilaeral specrum S x (ω) is given by: x ( ω ) λ ( ω) 2 S F j = Carson s heorem λ = F( jω )= mean rae of pulses Fourier ransform of he pulse shape f() m * Processes whose probabiliy of occurrence is small and consan: Px = exp( m) x! x

39 Whie noise modelling Whie noise wih specrum S o can be modelled by a random sequence of δ pulses of uni area and average rae λ=s o S v (ω) So Log(ω) v n ()

40 Whie noise modelling Small signal model of he fron-end. i D () Signal i D () S I G S V G D C D Deecor S C G Inpu FET g m V G S S i n () v n () Parallel whie noise can be modeled as a random sequence of curren δ-pulses of uni area and average rae λ p = S I Series whie noise can be modeled as a random sequence of volage δ-pulses of uni area and average rae λ s = S V

41 Whie noise modelling In order o beer compare he noise wih he curren signal i is useful o ransform he volage noise generaor in a curren noise generaor which gives he same noise signal a he inpu of he JFET G C T C T G i() C T + () T δ S T v() S v() δ() for T 0 T C T () T δ T

42 Whie noise modelling i D () Signal i D () S I S Ieq g m V G S δ() curren pulse C D C G i n () Parallel noise i neq () Series noise equivalen δ() curren pulses Doubles of δ() curren pulses

43 /f noise modelling The /f noise specrum S vf π Af = ω can be modelled wih a random sequence of pulses ϕ () = u() 2 occurring a a rae r=a f S v/f (ω) V n () Log(ω)

44 V n () u( ) 2 V n () δ () u() Very rough approximaion I n eq () I n eq () +

45 The weighing funcion

46 The concep of he weighing funcion v o () Oupu pulses Weighing funcion w() Weigh Signal pulse posiion p Noise pulse posiion n - n p Signal pulse posiion Noise pulse posiion Curren signal pulses and curren noise pulses produces a he oupu waveforms of he same shape. The weighing funcion w(), which is he ime-reversal of he oupu waveform, gives he weigh wih which an inpu noise pulse conribues o he peak ampliude of he oupu signal, as a funcion of is displacemen from he inpu signal pulse.

47 Noise effecs δ pulses v o () Signal pulse Noise pulse v o () The measuremen of he peak ampliude of he oupu signal pulse is affeced by errors due o he superposiion wih he random oupu noise pulses. The conribuion of a given noise pulse is deermined by he value of he weighing funcion evaluaed a he ime n corresponding o he ime occurrence of he noise pulse wih respec o he signal pulse. m T sh 0 Signal pulse posiion Noise δ pulse posiion

48 Noise effecs doubles of δ pulses v o () Noise double Signal pulse The conribuion o he peak ampliude of he oupu signal of he wo δ of he double is slighly differen because of he slighly differen value of he weighing funcion a he respecive posiion of he wo δ. v o () w() m T sh 0 Signal pulse posiion Noise double posiion

49 How o choose he shaping ime Whie noise - Inuiive consideraions a) Parallel noise (δ curren pulses): he weighing funcion should be as shor as possible in order o collec conribuions from he lowes possible number of δ pulses. b) Series noise (doubles of δ pulses): he weighing funcion should be as long as possible (ails wih small slope) in order o weigh as much as possible equally he wo δ pulses of he doubles. i() i() w() w()

50 How choose he shaping ime /f noise - Inuiive consideraions I n eq () I n eq () + The δ-double would require long shaping ime The δ would require shor shaping ime The opposie requiremens makes he /f conribuion independen of he shaping ime