Rivelatori X e Gamma per spettroscopia e imagingAntonio LongoniPolitecnico di Milano, Dipartimento di Elettronica e InformazioneINFN, Sezione di Milanoantonio.longoni@polimi.it

Scuola Nazionale INFN“Rivelatori ed Elettronica per Fisica delle Alte Energie, Astrofisicaed Applicazioni Spaziali“, Laboratori Nazionali di Legnaro4-8 Aprile 2005

Summary

First part•why “low capacitance” for spectroscopic detectors (in other words, why SDDs?): some basic concepts on signal processing

Second part•Introduction to some new SDDs and SDD monolithic arrays•Some applications in material analysis, in biomedicine and in atomic physics

The interaction of the radiation with the detector

The main interactions of X raysin semiconductor detectors

Fast electron

phE hν=0el ph bE E E= −Photon

*0ph bE E=

Photon Auger electron*

0Ae b bAeE E E= −

Photoelectric absorption

Primary electronPrimary electron

e-h pairs

Electron-hole pairs generation

PhotonphE hν= Fast electron

Photon

ϕ

* ( )ph

E f ϕ=

*el ph phE E E= −

Compton scattering

Signal generation in a detector

Amplification circuit

Fast electron

Electrons

HolesSi atom

X ray

N type Si

N+ anode

P+ cathode

Bias

coup ph coupN E ε=

The average number of e-h couples generatedin the detector is:

where Eph is the photonenergy and εcoup is the average energy requiredto generate an e-h couple(εcoup= 3.6 eV in Si).

The generation of electron-hole couples is a statistical process, the electronic noise introduces further statistical fluctuations

Incident photon.monoenergetic

Eph E

dndE

2cN ph cF Eσ ε= ⋅

Statistical spreading of thenumber of couples e-h

Fano factor F, in silicon 0.12F ≈

c

dndN

cN cN

intrσ meas

dndN

measN measN

measσ

Electronic noise.contribution

meas c elect

2 2 2N N Nσ σ σ= +

The concept of Equivalent Noise Charge: ENC

meas

dndN

measN measN

Nmeasσmeas c elect

2 2 2N N Nσ σ σ= +

elect

2 2 2NENC q σ=

Q

peak outV 1SQ ENCN

= ⇔ =

A lower ENC means a better resolution of the detection system

Signal and noisein the detection system

Block diagram of a typical detection system for X-ray spectroscopy

AmplifierFilter

ADandPC

Detector

PhotonE=hν

Our goal: to reach S/N as good as possible

Block diagram of a typical detection system for X-ray spectroscopy: signal and noise

SA Pk Str ADCto PC

CD

-

+PA

Si

Sv

CG

CF

011010

DetectorCD comprises also the parasitic capacitancesseen by the amplifier input

For what concerns S/N, charge-preamplifier and voltage preamplifier are equivalent

SA Pk Str ADCto PC

CD

-

+PA

Si

Sv

CG

CF

IS

PA SA Pk Str ADCto PC

CD CG

Sv

Si

CF

We will use voltage preamplifierbecause it is simpler to analyze

Block diagram: relevant elements

From now on CD comprises also the feedback capacitance CF:CD=Cdet+Cparasitic+Cfeedback

CG is the input capacitanceof the FET

T(s)

CGCD

IS

ID=gmVG

AIVID

Detector Preamplifier Shaping amplifier

VGVPA VSA

Si

Sv

White series noise of the FET

Thermal noise of the FET channel

CG

Svw=α2kT/gmG G*

S

D

S

gmVG*S

Note: VG*S !!

2 2 1vw

m G T

kT kTSg C

α αω

= =

2 3α ≈

Note: bilateral noise spectra

White parallel noise of Detector and FET

iD DS qI=Detector shot noise

iG GS qI=FET gate shot noise

CG

G

S

D

S

gmVGS

VGS

CD

SiD SiG

( )iw iD iG D G LS S S q I I qI= + = + =

1/f series noise – “flicker noise”

Mainly due to charge trapping and de-trappingin the FET channel

( ) 11 2 12

f fv f

G T

A A kTSf C

π ωω αω ω ω

= = =CG

G G*

S

D

S

gmVG*S

VG*S

( ) 12v f fS A fω =

ωω 1

SV

1 constant for a given technology 2 2

f Gf

T

A CH

kT kTπω π

ω α α= = ≈

T(s)

CGCD

IDET

ID=gmVG

AIVID

Detector Preamplifier Shaping amplifier

VG

VPA VSA

Evaluation of the S/N: a) the output signal

( )DETI s Q= ( ) 1DRAIN m

T

QI s gC s

=

( ) 1PAout m IV

T

QV s g AC s

=( ) 1 ( )m IVSAout

T

g AV s Q T sC s

=

( ) 1G

T

QV sC s

=

2 2

2SAout peak

no

VSN v

⎛ ⎞ =⎜ ⎟⎝ ⎠T D GC C C= +

Evaluation of the S/N: b) the output noise

T(s)

CGCD

ID=gmVG

AIVID

Detector Preamplifier Shaping amplifier

VGVPA VSA

Si

Sv

( ) 22 22 2( ) ( ) iw

no vw vf m IVT

SS S S g A T jC

ω ω ωω

⎛ ⎞= + +⎜ ⎟⎝ ⎠

( )2 12no nov S dω ωπ

+∞

−∞

= ∫2 2

2peak out

no

VSN v

⎛ ⎞ =⎜ ⎟⎝ ⎠

Evaluation of the S/N

( )

22 2 2 1

2 22

2 2

22 22 2

1 ( )

1 ( )2

IV mpeak out T

noiw

vw vf m IVT

QA g Max L T sV C sSN v SS S g A T j d

Cω ω ω

π ω

−

+∞

−∞

⎧ ⎫⎡ ⎤⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭= =

⎛ ⎞+ +⎜ ⎟

⎝ ⎠

⌠⎮⌡

2 1 1 ( ) 1Max L T ss

−⎛ ⎞⎡ ⎤ =⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠The ENC does not depend on the gain of the shaper

The Equivalent Noise Charge

Evaluation of the ENC

1SQ ENCN

= ⇔ =

( ) 22 22 2

1 ( )2

iwT vw vf

T

SENC C S S T j dC

ω ω ωπ ω

+∞

−∞

⎛ ⎞= + +⎜ ⎟

⎝ ⎠

⌠⎮⌡

( ) ( ) ( )2 2 22 2 22

1 1 1 1 12 2 2T vw T f iwENC C S T j d C A T j d S T j dω ω π ω ω ω ωπ π ω π ω

+∞ +∞+∞

−∞ −∞−∞

= + +⌠ ⌠⎮⎮ ⌡⌡∫

Parallel noisecontribution

1/f noisecontribution

Series noisecontribution

The “shape factors”

The angular frequency ω can be normalised to a characteristic frequency ωc=1/τ, where τ is a characteristic time which represents the width of the output pulse (for instance the peaking time, or the time width at half height, or a characteristic time constant of the filter). The characteristic time τ is also called ‘shaping time’of the filter.

c

x ω ωτω

= =

( ) ( ) ( )2 2 22 2 22

1 1 1 1 12 2 2

1T vw T f iwENC C S T jx dx C A T jx dx S T jx dx

xπ

π π π ωτ τ+∞ +∞+∞

−∞ −∞−∞

= + +⌠ ⌠⎮⎮ ⌡⌡∫

Parallel noisecontribution

1/f noisecontribution

Series noisecontribution

21

22

23 2

1 ( )2

1 1 ( )2

1 1 ( )2

A T jx dx

A T jx dxx

A T jx dxx

π

π

π

+∞

−∞

+∞

−∞

+∞

−∞

=

=

=

∫

∫

∫

The three factors A1, A2 and A3 depend on the SHAPE of the output pulse and on the choice of the characteristic time τ of the output pulse used to normalise the angular frequency ω. The factors A1, A2 and A3 do not depend on the particular value of the shaping time τ.

( ) ( )( ) ( )

( ) ( )

1 1

2 2

3 3

" '

" '1" '

A k A

A A

A Ak

τ τ

τ τ

τ τ

=

=

=

" 'kτ τ=

12 2

22

31

T vw T f iwE A ANC C S C A ASπτ τ= + +

SHAPE FACTORSA 1 A 2 A 3

Infinite cusp τ 1.00 0.64 1.00 1.00 1.00Triangular Tbase/2 2.00 0.88 0.67 1.16 1.73Gaussian σ 0.89 1.00 1.77 1.26 0.71CR-RC RC 1.85 1.18 1.85 1.85 1.00CR-RC4 RC 0.51 1.04 3.58 1.35 0.38CR-RC4

τpeak 3.06 1.04 0.60 1.35 2.26Semigaussian 7 poles σ 0.92 1.03 1.83 1.30 0.71Semigaussian 7 poles τpeak 2.70 1.03 0.62 1.30 2.08Trapezoidal (Tf =0.5xTr) Tr 2.00 1.18 1.16 1.52 1.31Trapezoidal (Tf =Tr) Tr 2.00 1.38 1.67 1.83 1.09Trapezoidal (Tf =2xTr) Tr 2.00 1.64 2.67 2.31 0.87

1 3A A 1 3A A

The dependence on τ of the 3 contributions to the ENC

2 2 21 2 3

1T vw T f iwENC C S A C A A S Aπτ τ= + +

2 2 2 21/series f parallelENC ENC ENC ENC= + +

Series

1/f

Parallel

τ

ENC 2

The dependence on transistor and detector parameters of the ENC

2 2 21 2 3

1T vw T f iwENC C S A C A A S Aπτ τ= + +

( ) 1

2 2 1

( )

2 1

vwm

iw D Req L

fv f

G T

G

G T

kT kTSg

S q I I qIC

I

CA kTS

ωα α

πω α

ωωω ω

⎧= =⎪

⎪⎪ = + + =⎨⎪⎪ = =⎪⎩

2 2

21 2 1 3

2 1 2G GD DD D T

G D T G D T

kT kTENC A AC CC CC C IC C

AC

qC

ωω ω

α α ττ

⎛ ⎞ ⎛ ⎞= + + + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Series noise 1/f noise Parallelnoise

How to optimise the ENC

How to optimise the ENC (1)

Choose the optimum shaping time

2 2series parallelENC ENC=

1/fτ

ENC 2

ENC 2opt

τopt

( ) 1 1

3 3

2 1Vw GDD G D

Iw G D T Lopt

S CA C kT AC C CS A C C qI A

τ αω

⎛ ⎞= + = +⎜ ⎟⎜ ⎟

⎝ ⎠

How to optimise the ENC (2)

Match the Detector capacitance with the Gate capacitance

2

2 GD

G D

CCC C

M⎛ ⎞

= +⎜ ⎟⎜ ⎟⎝ ⎠

Series white and series 1/f noise contributions are minimised

0.1 1 101

10

M2 /(M

mat

ched

)2CD/CG

* For constant FET current density

2 2

21 2 1 3

2 1 2G GD D

G D G DD D T

T T

kT kTENC AC A C AC CC CC C C C

qIα α ω τω τ ω

⎛ ⎞ ⎛ ⎞= + + + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

How to optimise the ENC (3)

10-9 10-8 10-7 10-6 10-5101

102

103E

NC

(el r

ms)

Tm (s)

Cd=1.5pF

Cd=0.15pF

Reduce the detector capacitance (in matched conditions)

leakopt

T

DIEN CCω

≈

optleak T

D

ICτω

=

* With 1/f noise negligible

ENCopt = 13 e- rms

τopt= 0.4 µs

ENCopt = 23 e- rms

τopt= 1.3 µs

Series white and series 1/f noise contributions are minimised

How to optimise the ENC (4)

Reduce the parallel noise sources

•Reduce the detector leakage current by•cooling the detector•reducing the detector active volume•improving the detector technology

•Increase the value of the resistors connectedto the FET input (bias or feedback resistors)

Parallel white noise contribution is minimised2 2

21 2 1 3

2 1 2G GD D

G D G DD D T

T T

kT kTENC AC A C AC CC CC C C C

qIα α ω τω τ ω

⎛ ⎞ ⎛ ⎞= + + + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

1E-8 1E-7 1E-6 1E-5 1E-410

100

1000

IL eq=10pA

IL eq=1pA

CD=1.25pF

Gaussian shapingmatched conditions

EN

C [e

- rms]

☯s

1E-8 1E-7 1E-6 1E-5 1E-410

100

1000

IL eq=10pA

CD=1.25pF

CD=125fF

Gaussian shapingmatched conditions

E

NC

[e- rm

s]

☯s

CD and IL

How to optimise the ENC (5)

Choose the best transistor

2 2

21 3

12

2 1 2 ( )D DG G

GTD

DT

D D

G G D

C CC CC C C C

kTENC AC A C kT A I Iqατ ω

τωω

α⎛ ⎞ ⎛ ⎞

= + + + + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Bandwidth 1/f noiseGate

Leakage

1

2 2f G

fT

A CH

kT kTπω π

ω α α= =

DEVICE Hf [J]

JFET, n-channel, discrete 2x10-26

JFET, n-channel, in CMOS process

10-25

MOSFET, p channel, in CMOS process

6x10-25

MOSFET, n channel, in CMOS process

2.5x10-23

MESFET, GaAs, discrete 10-23

How to optimise the ENC (6)

Perform the optimum signal processing

If only the white noise sources (series and parallel) are present,the best ENC can be obtained by using an ideal filtering amplifierwhich gives at its output an ‘infinite cusp’- shaped pulse

( ) expsout

tv t

τ⎛ ⎞

= −⎜ ⎟⎝ ⎠

with a shaping time τ set equal to the ‘noise corner’ time constant τc.

( ) Vwc D G

Iw

SC CS

τ = +

Ideal filter (for white noise sources)

Optimum signal processing

-4 -2 0 2 40.0

0.2

0.4

0.6

0.8

1.0Infinite cusp

v s ou

t(t)

t/τc

1 2 321 0.64 1A A Aπ

= = ≈ =

( )2 2 D G Vw IwENC C C S S∞ = +

Ideal filter (In presence of white and 1/f noise)

Optimum signal processing

( ) fD G

Vw Iw

AK C C

S S= +

Practical filters

12 2

22

31

T vw T f iwE A ANC C S C A ASπτ τ= + +

SHAPE FACTORSA 1 A 2 A 3

Infinite cusp τ 1.00 0.64 1.00 1.00 1.00Triangular Tbase/2 2.00 0.88 0.67 1.16 1.73Gaussian σ 0.89 1.00 1.77 1.26 0.71CR-RC RC 1.85 1.18 1.85 1.85 1.00CR-RC4 RC 0.51 1.04 3.58 1.35 0.38CR-RC4

τpeak 3.06 1.04 0.60 1.35 2.26Semigaussian 7 poles σ 0.92 1.03 1.83 1.30 0.71Semigaussian 7 poles τpeak 2.70 1.03 0.62 1.30 2.08Trapezoidal (Tf =0.5xTr) Tr 2.00 1.18 1.16 1.52 1.31Trapezoidal (Tf =Tr) Tr 2.00 1.38 1.67 1.83 1.09Trapezoidal (Tf =2xTr) Tr 2.00 1.64 2.67 2.31 0.87

1 3A A 1 3A A

Appendix:Noise modelling

Noise modellingA noise waveform can be represented as a random sequence of pulses:

( )∑ −=k

kk ttfatx )( The distribution of tk is Poissonian*

The bilateral spectrum Sx(ω) is given by:

( ) ( ) 2xS F jω λ ω= Carson’s theorem

λ = mean rate of pulses

( ) =ωjF Fourier transform of the pulse shape f(t)

* Processes whose probability of occurrence is small and constant: exp( )!x

xmP mx

= −

White noise modelling

White noise with spectrum So can be modelled by a random sequence of δ pulsesof unit area and average rate λ=So

Log(ω)

Sv(ω)So

vn(t)

t

White noise modellingSmall signal model of the front-end.

SI SViD(t)

gmVG’SCGCD

Detector Input FET

G’ D

SS

G

t

iD(t) Signal

vn(t)

t

in(t)

tSeries white noise can be modeledas a random sequence of voltage δ-pulses of unit area and average rate λs= SV

Parallel white noise can be modeledas a random sequence of current δ-pulses of unit area and average rate λp= SI

White noise modelling

In order to better compare the noise with the current signal it is useful to transformthe voltage noise generator in a current noise generator which gives the samenoise signal at the input of the JFET

S

G’

CT CT

G’

S

v(t) → δ(t) for ∆T→0

∆T

v(t)

t

1T∆

∆T

i(t)

t

( )TC tTδ+

∆

( )TC tTδ−

∆

White noise modelling

SI SIeqiD(t)

gmVG’SCGCD

t

iD(t) Signal

δ(t) current pulse

in(t)

t

Parallel noise

δ(t) current pulses

ineq(t)

t

Series noise equivalent

Doublets of δ(t) current pulses

1/f noise modelling

The 1/f noise spectrum fv f

AS

πω

=

1 2

1( ) ( )t u tt

ϕ =can be modelled with a random sequence of pulses

occurring at a rate r=Af

Log(ω)

Sv1/f (ω)

t

Vn(t)

Vn(t)

t

)(tδ

)(tu

t

Vn(t))(1

21 tut

+t

In eq(t)

t

In eq(t)

Very rough approximation

The weighting function

The concept of the ‘weighting function’

vo(t)

tSignal pulse position tp

Noise pulse position tn

Output pulses Weighting function

t

1

tp

w(t)

Signal pulse positionNoise pulse position

Weight

-tn

The ‘weighting function’ w(t),which is the ‘time-reversal’ of the output waveform,gives the weight with which an input noise pulse contributes to the peak amplitude of the output signal,as a function of its displacement from the input signal pulse.

Current signal pulsesand current noise pulsesproduces at the output waveforms of the same shape.

Noise effects – δ pulsesThe measurement of the peak amplitudeof the output signal pulseis affected by errorsdue to the superposition with the random output noise pulses.The contribution of a given noise pulse is determined by the value of the weighting functionevaluated at the time –tn correspondingto the time occurrence of the noise pulse with respect to the signal pulse.

t

vo(t) Signal pulse

Noise pulse

t

vo(t)

tm

Tsh

t

1

0

Signal pulse positionNoise δ pulse position

Noise effects – doublets of δ pulses

vo(t)

t

Signal pulse

Noise doublet

t

vo(t)

tm

Tsh

The contribution to the peak amplitudeof the output signal of the two δ of the doubletis slightly different because of the slightly different value of the weighting function at the respective position of the two δ.

w(t)

t

1

0

Signal pulse positionNoise doublet position

How to choose the shaping timeWhite noise - Intuitive considerations

a) Parallel noise (δ current pulses):the weighting function should be as short as possible in order to collectcontributions from the lowest possible number of δ pulses.

b) Series noise (doublets of δ pulses):the weighting function should be as long as possible (tails with small slope) in order to weight as much as possible equally the two δ pulses of the doublets.

i (t)

t

t

w(t)

i(t)

t

t

w(t)

How choose the shaping time 1/f noise - Intuitive considerations

t

In eq(t)

t

In eq(t)

+

The δ-doublet would requirelong shaping time

The δ would requireshort shaping time

The opposite requirements makesthe 1/f contribution

independent of the shaping time