Semi-Conducter Detector (Radiation Detectors)

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Introduction to Radiation Detectors and Electronics, 18-Mar-99 Helmuth Spieler

IX.1 Semiconductor Detectors II The Signal LBNL

1

IX. Semiconductor Detectors Part II

1. Fluctuations in the Signal Charge: the Fano FactorIt is experimentally observed that the energy required to form anelectron-hole pair exceeds the bandgap.

C.A. Klein, J. Applied Physics 39 (1968) 2029


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The mean ionization energy exceeds the bandgap for two reasons

1. Conservation of momentum requires excitation of latticevibrations

2. Many modes are available for the energy transfer with anexcitation energy less than the bandgap.

Two types of collisions are possible:

a) Lattice excitation, i.e. phonon production (with no formationof mobile charge).

b) Ionization, i.e. formation of a mobile charge pair.

Assume that in the course of energy deposition

Nx excitations produceNPphonons and

Ni ionization interactions formNQ charge pairs.

On the average, the sum of the energies going into excitation andionization is equal to the energy deposited by the incident radiation

whereEi andEx are the energies required for a single excitation orionization.

Assuming gaussian statistics, the variance in the number ofexcitations

and the variance in the number of ionizations

xx N=

ii N=

xxii NENEE +=0


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For a single event, the energyE0 deposited in the detector is fixed(although this may vary from one event to the next).

If the energy required for excitationEx is much smaller than requiredfor ionizationEi, sufficient degrees of freedom will exist for somecombination of ionization and excitation processes to dissipateprecisely the total energy. Hence, for a given energy deposited in thesample a fluctuation in excitation must be balanced by an equivalentfluctuation in ionization.

If for a given event more energy goes into charge formation, lessenergy will be available for excitation. Averaging over many eventsthis means that the variances in the energy allocated to the two typesof processes must be equal

From the total energy EiNi + ExNx = E0

yielding

0=+ iixx NENE

xxii EE =

x

i

xi N

E

E=

x

iix

E

NEEN

= 0

i

x

i

xi

xi NE

E

E

E

E

E

= 0


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Since each ionization leads to a charge pair that contributes to thesignal

where i is the average energy loss required to produce a charge

pair,

The second factor on the right hand side is called the Fano factorF.

Since i is the variance in signal charge Q and the number of charge

pairs isNQ=E0 /i

In Silicon Ex= 0.037 eV

Ei = Eg= 1.1 eV

i = 3.6 eV

for which the above expression yieldsF= 0.08, in reasonable

agreement with the measured valueF= 0.1.

The variance of the signal charge is smaller than naively

expected

iQi

ENN

0==

ix

i

xi

xi

E

E

E

E

E

E

E

00 =

10

=

i

i

i

x

ii

EE

EE

QQ FN=

QQ N3.0


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A similar treatment can be applied if the degrees of freedom aremuch more limited and Poisson statistics are necessary.

However, when applying Poisson statistics to the situation of a fixedenergy deposition, which imposes an upper bound on the variance,one can not use the usual expression for the variance

Instead, the variance is

as shown by Fano [1] in the original paper.

An accurate calculation of the Fano factor requires a detailedaccounting of the energy dependent cross sections and the density ofstates of the phonon modes. This is discussed by Alkhazov [2] andvan Roosbroeck [3].

References:

1. U. Fano, Phys. Rev. 72 ( 1947) 26

2. G.D. Alkhazov et al., NIM 48 (1967) 1

3. W. van Roosbroeck, Phys. Rev. 139 (1963) A1702

NN =var

NFNN = 2)(


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Intrinsic Resolution of Semiconductor Detectors

Si: i= 3.6 eV F= 0.1

Ge: i= 2.9 eV F= 0.1

Detectors with good efficiency for this energy range have sufficientlysmall capacitance to allow electronic noise of ~100 eV FWHM, sothe variance of the detector signal is a significant contribution.

At energies >100 keV the detector sizes required tend to increase theelectronic noise to dominant levels.

iiQi FE

w

EFFNE === 35.235.235.2

Intrinsic Resolution of Si and Ge Detectors

0

50

100

150

200

250

0 5 10 15 20 25

E [keV]

EFWHM[eV] Si

Ge


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2. Induced Charge

When does the current pulse begin?

a) when the charge reaches the electrode?

orb) when the charge begins to move?


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Although the first answer is quite popular (encouraged by the phrasecharge collection), the second is correct.

When a charge pair is created, both the positive and negativecharges couple to the electrodes and induce mirror charges of equalmagnitude.

As the positive charge moves toward the negative electrode, itcouples more strongly to it and less to the positive electrode.

Conversely, the negative charge couples more to the positiveelectrode and less to the negative electrode.

The net effect is a negative current at the positive electrode and apositive current at the negative electrode, due to both the positive andnegative charges.


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Magnitude of the Induced Charge

(S. Ramo, Proc. IRE 27 (1939) 584)

Consider a mobile charge in the presence of any number of groundedelectrodes.

Surround the charge q with a small equipotential sphere. Then, ifVisthe potential of the electrostatic field, in the region betweenconductors

CallVq

the potential of the small sphere and note thatV=

0 on the

conductors. Applying Gauss law yields

Next, consider the charge removed and one conductor A raised to

unit potential.

Call the potential V1, so that

in the space between the conductors, including the site where the

charge was situated. Call the new potential at this point Vq1.

Greens theorem states that

02 = V

=

surfacessphere'

4 qdsn

V

012 = V

=

surfacesboundary

11

boundariesbetweenvolume

122

1 )( dsn

VV

n

VVdvVVVV


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Choose the volume to be bounded by the conductors and the tinysphere.

Then the left hand side is 0 and the right hand side may be dividedinto three integrals:

1. Over the surfaces of all conductors except A. This integral is 0

since on these surfaces V= V1= 0.

2. Over the surface of A. As V1= 1 and V= 0 this reduces to

3. Over the surface of the sphere.

The second integral is 0 by Gauss law, since in this case thecharge is removed.

Combining these three integrals yields

or

Asurface dsnV

+

surfacessphere'

surfacessphere'

11 ds

n

VVds

n

VV qq

1

surfacessphere'

1

Asurface

440 qAq qVQdsn

VVds

n

V =

=

1qA qVQ =


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If the charge q moves in directionx, the current on electrode A is

Since the velocity of motion

the induced current on electrode A is

where Vq1 is the weighting potential that describes the coupling of acharge at any position to electrode A.

The weighting potential is for a specific electrode is obtained bysetting the potential of the electrode to 1 and setting all otherelectrodes to potential 0.

If a charge q moves along any paths from position 1 to position 2,the net induced charge on electrode kis

The instantaneous current can be expressed in terms of aweighting field

The weighting field is determined by applying unit potential to themeasurement electrode and 0 to all others.

=== dtdxxV

qdt

dVq

dtdQi qqAA 11

xvdt

dx=

dx

Vvqi

q

xA

1 =

( ))1()2())1()2(( 11 kkqqk qVVqQ =

kk Fvqi =


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Note that the electric field and the weighting field are distinctlydifferent.

The electric field determines the charge trajectory and velocity

The weighting field depends only on geometry and determineshow charge motion couples to a specific electrode.

Only in 2-electrode configurations are the electric field and theweighting field of the same form.


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Example 1: Parallel plate geometry, disregarding space charge(semiconductor detector with very large overbias)

Assume a voltage Vb applied to the detector. The distance

between the two parallel electrodes is d.

The electric field that determines the motion of charge in thedetector is

so the velocity of the charge

The weighting field is obtained by applying unit potential to thecollection electrode and grounding the other.

so the induced current

since both the electric field and the weighting field are uniformthroughout the detector, the current is constant until the chargereaches its terminal electrode.

d

VE b=

d

VEv b ==

dEQ

1=

2

1

d

Vq

dd

VqqvEi bbQ ===


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Assume that the charge is created at the opposite electrode and

traverses the detector thickness d.

The required collection time, i.e. the time required to traverse thedetector thickness d

The induced charge

Next, assume an electron-hole pair formed at coordinatex fromthe positive electrode.

The collection time for the electron

and the collection time for the hole

Since electrons and holes move in opposite directions, they inducecurrent of the same sign at a given electrode, despite their

opposite charge.

bbc

V

d

d

V

d

v

dt

2

===

qV

d

d

VqitQ

b

bc ===

2

2

bee

ce

V

xd

v

xt

==

bhh

chV

dxd

v

xdt

)( =

=


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The induced charge due to the motion of the electron

whereas the hole contributes

Assume thatx= d/2. After the collection time for the electron

it has induced a charge qe/2.

At this time the hole, due to its lower mobility he /3, has

induced qe/6, yielding a cumulative induced charge of 2qe /3.

After the additional time for the hole collection, the remaining

charge qe /3 is induced, yielding the total charge qe .

In this configuration

Electrons and holes contribute equally to the currents on bothelectrodes

The instantaneous current at any time is the same (although ofopposite sign) on both electrodes

The continuity equation (Kirchhoffs law) must be satisfied

Since k=2: i1= -i2

be

ce

V

dt

2

2

=

d

xq

V

xd

d

VqQ e

be

beee ==

2

=

=

d

xq

V

dxd

d

VqQ e

bh

bheh 1

)(2

0=k

ki


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Example 2: Double-Sided Strip Detector

The strip pitch is assumed to be small compared to the thickness.

The electric field is similar to a parallel-plate geometry, except in theimmediate vicinity of the strips.

The signal weighting potential, however is very different.

Weighting potential for a 300 m thick strip detector with strips on apitch of 50 m. Only 50 m of depth are shown.


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Cuts through the weighting potential

Weighting Potential in Strip Detector

Track Centered on Signal Strip

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 50 100 150 200 250 300

Depth in Detector [m]

WeightingPotential

Weighting Potential in Strip Detector

Track Centered on Nearest Neighbor Strip

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0 50 100 150 200 250 300


WeightingPotential


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Consider an electron-hole pairqn, qp originating on a pointx0 on thecenter-line of two opposite strips of a double-sided strip detector. The

motion of the electron towards the n-electrodexn is equivalent to the

motion of a hole in the opposite direction to thep-electrodexp. The

total induced charge on electrode kafter the charges have traversedthe detector is

since the hole charge qp= qe and qn= -qe

If the signal is measured on thep-electrode, collecting the holes,

Qk(xp)= 1,Qk(xn)= 0

and Qk= qe.

If, however, the charge is collected on the neighboring strip k+1, then

Qk+1 (xp)= 0,Qk+1 (xn)= 0

and Qk+1= 0.

In general, if moving charge does not terminate on the measurementelectrode, signal current will be induced, but the current changes signand integrates to zero.

)]()([)]()([ 00 xxqxxqQ QknQknQkpQkpk +=

)]()([)]()([ 00 xxqxxqQ QknQkeQkpQkek =

)]()([ nQkpQkek xxqQ =


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This is illustrated in the following schematic plot of the weighting fieldin a strip detector (from Radeka)


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Cuts through the Weighting Field in a Strip Detector

(d= 300 m,p= 50 m)

Weighting Field of Strip Detector

Track Centered on Strip

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0 50 100 150 200 250 300


WeightingField

Weighting Field in Strip Detector

Track Centered on Nearest Neighbor Strip

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 50 100 150 200 250 300


WeightingField


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Note, however that this charge cancellation on non-collectingelectrodes relies on the motion of both electrons and holes.

Assume, for example, that the holes are stationary, so they don'tinduce a signal. Then the first term of the first equation abovevanishes, which leaves a residual charge

since for any coordinate not on an electrode

Qk(x

0) 0,

although it may be very small.

An important consequence of this analysis is that one cannot simplyderive pulse shapes by analogy with a detector with contiguouselectrodes (i.e. a parallel plate detector of the same overalldimensions as a strip detector). Specifically,

1. the shape of the current pulses can be quite different,

2. the signals seen on opposite strips of a double-sided detector arenot the same (although opposite in sign), and

3. the net induced charge on thep- orn-side is not split evenlybetween electrons and holes.

Because the weighting potential is strongly peaked near thesignal electrode, most of the charge is induced when the

moving charge is near the signal electrode. As a result, most of the signal charge is due to the charge

terminating on the signal electrode.

)]()([ 0 nQkQkek xxqQ =


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Current pulses in strip detectors (track traversing the detector)

The duration of the electron and hole pulses is determined by thetime required to traverse the detector as in a parallel-plate detector,but the shapes are very different.

n-Strip Signal, n-Bulk Strip Detector

Vdep= 60V, Vb= 90V

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30

Time [ns]

SignalCurrent[A]

electrons

holes

total

p-Strip Signal, n-Bulk Strip Detector

Vdep= 60V, Vb= 90V

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30

Time [ns]

SignalCurrent[A]

electrons

holes

total


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Strip Detector Signal Charge Pulses

n-Strip Charge, n-Bulk Strip Detector

Vdep= 60V, Vb= 90V

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 5 10 15 20 25 30

Time [ns]

SignalCharge[fC]

electrons

holes

total

p-Strip Charge, n-Bulk Strip Detector

Vdep= 60V, Vb= 90V

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0 5 10 15 20 25 30

Time [ns]

SignalCharge[fC]

electrons

holes

total


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For comparison:

Current pulses in pad detectors (track traversing the detector)

For the same depletion and bias voltages the pulse durations are thesame as in strip detectors. Overbias decreases the collection time.

Pad Detector, Vdep= 60V, Vb= 90V

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30

time [ns]

SignalCurrent[A]

electrons

holes

total


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25 30

time [ns]

SignalCurrent[A]

electrons

holes

total


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Operation at or under full depletion leads to long tails from the low-field region.

Note: The steps in the curves are artifacts of the calculation resolution.


0.0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0 10 20 30 40 50

time [ns]

SignalCurrent[A]

electrons

holes

total


0.0

0.1

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0 10 20 30 40 50

time [ns]

SignalCurrent[A]

electronsholes

total


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Application of Induced Charge Concept:Charge Collection in the Presence of Trapping

Practical semiconductor crystals suffer from imperfections introduced

during crystal growth, during device fabrication, or by radiationdamage.

Defects in the crystal

impurity atoms

vacancies

structural irregularities (e.g. dislocations)

introduce states into the crystal that can trap charge.

Charge trapping is characterized by a carrier lifetime , the time a

charge carrier can survive in a crystal before trapping orrecombination with a hole.

Trapping removes mobile charge available for signal formation.

Depending on the nature of the trap, thermal excitation or theexternally applied field can release the carrier from the trap,

leading to delayed charge collection.

Given a lifetime , a packet of charge Q0 will decay

In an electric field the charge will drift. The time required to traverse a

distancex is

after which the remaining charge is

/0)(

teQtQ =

E

x

v

xt

==

LxExeQeQxQ

/0

/0)(

=


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Since the drift length is proportional to the mobility-lifetime product,

is often used as a figure of merit.

Assume a detector with a simple parallel-plate geometry. For a

charge traversing the increment dx of the detector thickness d, theinduced signal charge is

so the total induced charge

In high quality silicon detectors:

10 mse= 1350 V/cm

.s2

E=104 V/cm L 104 cm

In amorphous silicon L 10 m (short lifetime, low mobility)In diamond, however, L 100 200 m (despite high mobility)In CdZnTe at 1 kV/cm,L 3 cm for electrons, 0.1 cm for holes

d

dxxQdQs )(=

==

d

Lx

d

s dxeQddxxQdQ0

/

00

1

)(

1

( )Lds ed

LQQ /0 1

=

d

L

Q

QLd s >>

0

:


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Although there is no magic solution to improving the charge yield ofmaterials with marginal drift lifetimes, it is possible to mitigate thevariations of measured signal.

For example, take a material with a long electron lifetime and a shorthole lifetime used as a gamma ray detector (CdZnTe, for example).The interaction is point-like, so the charge originates from only one

coordinatex0.

In a parallel plate geometry, as shown previously

If the electron drift length xe and the hole drift length xh aresufficient so that

both terms add to yield the full signal charge Qs= Q0.

If, however, the hole lifetime is short, so that holes fail to reach theircollection electrode before being trapped

and Qs< Q0.

Furthermore, the charge yield will depend on the position of theinteraction. If the interaction is at the negative electrode, the signalwill be exclusively due to electrons, so for a long electron lifetime thefull charge will be measured.

If the interaction is at the positive electrode, only holes will contributeand

For intermediate interaction sites the signal will vary between theseextremes.

d

xxQ

d

xQ

d

xQQQQ hehehes

+=

+

=+= 000

dxx he =+

dxx he


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The charge response can be made more uniform by adopting a stripdetector configuration.

Since the induced signal is formed predominantly in the vicinity of thestrips, one can minimize the effect of the trapped charge carrier.

First applied by Paul Luke (IEEE Trans. Nucl. Sci. NS-43(1996)1481)the configuration combines every other strip into two readoutchannels A and B.

Channel A measures the sum of the electron and hole signals qA.

Channel B measures the signal qB from the holes drifting towards

electrode C (cf. discussion on p. 18).

The difference signal is predominantly the electron signal.


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The simulated charge measurement efficiency vs. position for aCdZnTe detector using a simple parallel plate geometry is shownbelow.

In CdZnTe at 1 kV/cm,

L 3 cm for electrons,0.1 cm for holes

The 1 cm thick detectoris operated at 1 kV.

The strip readout provides a substantial improvement in uniformity.

Reducing the gain Gof the non-collecting gridallows compensation ofelectron trapping and yieldsa nearly flat response.

Semi-Conducter Detector (Radiation Detectors)

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