shockley ramo

Nuclear Instruments and Methods in Physics Research A 463 (2001) 250–267

Review of the Shockley–Ramo theorem and its application insemiconductor gamma-ray detectors

Zhong He*

Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USA

Received 10 July 2000; received in revised form 7 November 2000; accepted 14 November 2000

Abstract

The Shockley–Ramo theorem is reviewed based on the conservation of energy. This review shows how the energy istransferred from the bias supplies to the moving charge within a device. In addition, the discussion extends the originaltheorem to include cases in which a constant magnetic field is present, as well as when the device medium is

heterogeneous. The rapid development of single polarity charge sensing techniques implemented in recent years onsemiconductor g-ray detectors are summarized, and a fundamental interpretation of these techniques based on theShockley–Ramo theorem is presented. # 2001 Elsevier Science B.V. All rights reserved.

PACS: 07.85.Nc; 07.85.�m

Keywords: Shockley–Ramo theorem; Radiation detectors; Single polarity charge sensing; X- and g-ray spectrometers; Wide band-gap

semiconductors; Position sensitive detectors

1. Review of the Shockley–Ramo theorem

1.1. Introduction

The common principle of a wide range ofradiation detection techniques can be describedas follows: the incoming radiation generates free-moving charge q within the detection apparatus,and then the charge Q induced on an electrode bythe movement of q is amplified and converted tothe output signal. Because of the unique relation-ship between the energy deposited by the radiation

to the moving charge q generated in the device,and between the induced charge Q on the electrodeand the output signal, the deposited energy can beobtained from the output pulse amplitude. Thegeneral relationship between these quantities isillustrated below:

Deposited energy ! Moving charge q

! Induced charge Q ! Output signal:

We can understand this process by considering anexample of a g-ray spectrometer using semicon-ductors. The incident g-ray interacts with thesemiconductor and generates a large number ofelectron–hole pairs proportional to the depositedenergy. The movement of these electrons and holesdue to the electric field within the device, which

*Tel.: +1-734-763-7130; fax: +1-734-763-4540.

E-mail address: [email protected] (Z. He).

0168-9002/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 0 2 2 3 - 6

correspond to the q discussed above, causevariation of induced charge Q on an electrode.The change of Q is converted to a voltage pulseusing a charge sensing amplifier and the amplitudeof the output voltage signal ideally is proportionalto the deposited energy.

The time dependent output signal of a chargesensing device can be predicted if the inducedcharge Q on the readout electrode can becalculated as a function of instantaneous positionof the moving charge q within the device. Priorto the Shockley–Ramo Theorem, one had tocalculate the instantaneous electric field Ewhen the moving charge q is at each pointof its trajectory, and then calculate the inducedcharge Q by integrating the normal componentof E over the surface S surrounding theelectrode:

Q ¼IS

eE � dS

where e is the dielectric constant of the medium.This calculation process is very tedious since largenumbers of E, which correspond to differentlocations of q along its trajectory, needed to becalculated to obtain good precision. Shockley [1]and Ramo [2] independently found a simplermethod of calculating the induced charge on anyelectrode of a vacuum tube. Later, it was proven[3,4] that the Shockley–Ramo theorem can beapplied not only to vacuum tubes, which can beconsidered as having no space charge within theapparatus, but is also valid in the presence ofstationary space charge. This generalization leadsto wider application of the theorem in predictingoutput signals from many types of charge sensingdevices.

1.2. The Shockley–Ramo theorem

The Shockley–Ramo theorem states: The chargeQ and current i on an electrode induced by amoving point charge q are given by:

Q ¼ �qj0ðxÞ

i ¼ q* � E0ðxÞ

where * is the instantaneous velocity of charge q.j0ðxÞ and E0ðxÞ are the electric potential and fieldthat would exist at q’s instantaneous position xunder the following circumstances: the selectedelectrode at unit potential, all other electrodes atzero potential and all charges removed.

j0 and E0 are called the weighting potential andthe weighting field, respectively. While the trajec-tory of the charge q is determined by the actualoperating electric field, the induced charge Q canbe calculated much easier with the help of theweighting field. Because there is only one field thatmust be calculated, which is independent of themoving charge q, and the space charge is notinvolved. The original Shockley–Ramo theoremmakes the usual assumptions that the magneticeffects are negligible and the electric field propa-gates instantaneously. Under these assumptions,the problem can be treated as electrostatics at eachmoment of charge movement.

1.3. An important corollary: the charge induced byq is independent of applied potentials on electrodesand the space charge

An actual device is illustrated in Fig. 1(a). It hasa number of electrodes kept at constant voltagesVi ði ¼ 1; 2; . . . ;KÞ, a moving charge q and sta-tionary space charge rðxÞ which is represented bythe darker background within the device. Thesurface S of the volume t is surrounded completelyby surfaces Si ði ¼ 1; 2; . . . ;KÞ of electrodes. Themost outside boundary can be either the surface ofa conductor at potential VK or at infinity with zeropotential (VK ¼ 0 V) corresponding to an openapparatus. The electric potential jðxÞ within thedevice satisfies Poisson’s equation with Dirichletboundary conditions, and the electric field EðxÞ ¼�rjðxÞ is uniquely determined:

r2jðxÞ ¼ �½rðxÞ þ qdðx� xqÞ=e

jjSi¼ Vi; i ¼ 1; 2; . . . ;K ð1Þ

where xq is the location of the moving charge qand e is the dielectric constant of the detectormedium. From the linear superposition principle,

Z. He / Nuclear Instruments and Methods in Physics Research A 463 (2001) 250–267 251

let j ¼ j0 þ js þ jq, where

r2j0ðxÞ ¼ 0; j0jSi¼ Vi; i ¼ 1; 2; . . . ;K

r2jsðxÞ ¼ �rðxÞ=e; jsjSi¼ 0; i ¼ 1; 2; . . . ;K

r2jqðxÞ ¼ �q � dðx� xqÞ=e; jqjSi¼ 0;

i ¼ 1; 2; . . . ;K

where j0 would be the potential within the devicewhen the electrodes are still kept at the actualvoltages but with no charge within the device, js

would be the potential generated only by thestationary space charge rðxÞ while all electrodesare grounded, and jq would be the field only fromthe presence of the moving charge q while keepingall electrodes at zero voltage. Similarly, the electricfield E consists of three components E ¼ E0þEs þ Eq, where E0 ¼ �rj0ðxÞ, Es ¼ �rjsðxÞ andEq ¼ �rjqðxÞ. The three electric field compo-nents correspond to those due to applied voltages,the space charge and the moving charge, respec-tively.

The induced charge on electrode i can then bewritten as:

Qi ¼ISi

eE � dS i

¼ISi

eE0 � dS i þISi

eEs � dS i þISi

eEq � dS i:

The first term shows the induced charge due to thecoefficients of capacitance between electrodes,which depends on the actual potentials. Thesecond term shows the contribution of space

charge. It is evident that only the last term, whichcorresponds to jqðxÞ, relates to the moving chargeq. Therefore, the induced charge produced by q onelectrode i, which is

HSieEq � dS i, cannot depend

on the applied potentials Vi on each electrode, noron the stationary space charge rðxÞ. In otherwords, the induced charge on any electrodeproduced by q depends only on the location ofthe moving charge and the configuration of thedevice, and is independent of the actual biasvoltages and space charge distribution.

1.4. Proof of the Shockley–Ramo theorem

We now prove the Shockley–Ramo theoremfrom the conservation of energy. Because of thelinear superposition principle, the case in Fig. 1(a)can be divided into two cases shown in Fig. 1(b)and (c). Case (b) keeps the actual potentials, but allcharges ðrðxÞ þ qÞ are removed. Case (c) keeps thecharges, but all electrodes grounded. The electricpotential j1ðxÞ of case (c) is equivalent to jsðxÞþjqðxÞmentioned in the previous section. In a linearand isotropic medium, the electric displacement isD ¼ e � E where the dielectric constant e is a scalar.The energy density w of the electric field is givenby:

w ¼1

2E �D ¼

1

2eE2:

The total energy of the electric field is theintegral of energy density over the entire volume oft. From the conservation of energy, this totalenergy of the field can only be changed by energy

Fig. 1. Application of the linear superposition principle.

Z. He / Nuclear Instruments and Methods in Physics Research A 463 (2001) 250–267252

exchanges between the field and the moving chargeq (in the form of kinetic energy), and between thefield and power supplies.

Imagine a device having the same configuration,space charge distribution and boundary conditionsas shown in Fig. 1(c). When the charge q movesfrom xi to xf within such a device, the work doneby the electric field is

Z xf

xi

qE01 � dx

where E01 is E1 excluding q’s own field. While the

charge q moves from xi to xf , the induced chargeson electrodes redistribute accordingly at the samepotential of zero voltage since all conductors aregrounded. Therefore, no work is done on themoving induced charges by power supplies (noenergy exchange between the detector systemand the power supplies). From the conservationof energy, the work done on charge q must comefrom the energy stored in the field. Therefore:

Z xf

xi

qE01 � dx ¼

1

2

ZteðE2

1i � E21f Þ dt ð2Þ

where E1i and E1f are the electric fields in Fig. 1(c)when the point charge q is at its initial ðxiÞ andfinal ðxf Þ positions, respectively.

In the actual case shown in Fig. 1(a), sinceE ¼ E0 þ E1, the work done on charge q by theelectric field is

Z xf

xi

qðE0 þ E01Þ dx:

In this case, when the induced charge DQi on eachelectrode moves between that electrode and theground due to the movement of q, the total workdone on the induced charges by the power suppliesis

XKi¼1

ViDQi:

Applying the conservation of energy in this case,i.e., the work done by the power supplies minusthe energy absorbed by the moving charge qshould be equal to the increase of energy stored in

the field, we have

XKi¼1

ViDQi �Z xf

xi

qðE0 þ E 01Þ dx

¼1

2

Zte½ðE0 þ E1f Þ

2 � ðE0 þ E1iÞ2 dt: ð3Þ

Apply Green’s first identity ([5], p. 41)ZV

ðj1r2j0 þrj0 � rj1Þ dV ¼

IS

j1rj0 � dS

where j0 and j1 can be any arbitrary functions.Notice j1jS ¼ 0 (all electrodes are grounded inFig. 1(c)) andr2j0 ¼ 0 within the device volume t(there is no space charge in Fig. 1(b)), we have:Z

tE0 � E1 dt ¼

Ztrj0 � rj1 � dt

¼IS

j1rj0 � dS �Ztj1 � r

2j0 � dt ¼ 0: ð4Þ

ThereforeZte½ðE0 þ E1f Þ

2 � ðE0 þ E1iÞ2 dt

¼ZteðE2

1f � E21iÞ dt: ð5Þ

Eq. (5) means that when charge qmoves from xi toxf , the change of energy of the electric field inFig. 1(a) is the same as that in Fig. 1(c). This canbe interpreted as stating that the energy stored inthe actual field E equals the sum of that in E0 andE1, and the energy in E0 is kept constant.Substituting Eqs. (2) and (5) into Eq. (3), we get

XKi¼1

ViDQi ¼Z xf

xi

qE0 � dx ¼ �q½j0ðxf Þ � j0ðxiÞ:

ð6Þ

Eq. (6) can be understood as follows: It can beseen from Eq. (3) that the work done on charge qconsists of two parts. The first part

R xfxi

qE0 � dx isfrom the interaction between q and the actual biaspotentials. The second part

R xfxi

qE01 � dx is from the

interactions between q, the induced charges by qon electrodes and the stationary space chargewithin the device. This second part comes from thechange in energy stored in the field indicated byEqs. (2) and (5). Therefore, the work done by thepower supplies is converted completely to the


kinetic energy of the moving charge q as if qmoved from xi to xf in the field E0 generated bythe actual bias voltages, with all charges removedfrom the device (in the presence of the samemedium). Notice that in a real (non-vacuum)medium, the kinetic energy gained by q from theelectric field can be lost in collisions with themedium.

But recalling our earlier result, the inducedcharge by q is independent of the actual potentialsVi applied on each electrode. Therefore, in orderto single out the variation of the induced charge onthe electrode of interest while q moves from itsinitial to final positions, we can choose thepotential to be 1 V on this electrode (say electrodeL) and zero on all other electrodes. The inducedcharge by q on electrode L in this simplified settingshould be the same as that with actual biases.Considering the dimensions of VL and j0 arecanceled, Eq. (6) gives:

DQL ¼Z xf

xi

qE0 � dx ¼ �q½j0ðxf Þ � j0ðxiÞ: ð7Þ

The E0 and j0 in Eq. (7) correspond to the electricfield and potential when electrode L is biased atunit potential (dimensionless), all other electrodesgrounded and all charges removed. Therefore, 04j041 and has no dimension. In fact, anarbitrary bias voltage VL can be assigned toelectrode L in computational modeling. Theweighting potential is the calculated electricpotential divided by VL. Because j0 is propor-tional to VL, it is evident from Eq. (6) that thesame DQL will be obtained when charge q movesfrom xi to xf . However, letting VL ¼ 1 makes iteasier to see how much the induced charge changesas a fraction of the moving charge q. This is themeaning of weighting field.

The induced current on electrode L is thenobtained from Eq. (7) as

iL ¼dQL

dt¼ qE0 �

dx

dt¼ q* � E0: ð8Þ

We define the space charge free field E0 associatedwith electrode L at unit potential and all otherelectrodes at zero potential as the ‘‘weightingfield’’, and j0 as the ‘‘weighting potential’’. Themaximum induced charge on electrode L by q is

�q when q is infinitely close to the surface ofelectrode L where j0 ¼ 1. This result can beunderstood from the method of images ([5], p. 54)since the mirror-image charge �q is infinitely closeto q in this case, and equals the induced charge onelectrode L. The minimum induced charge onelectrode L by q is 0, when q is infinitely close tothe surface of another electrode or at infinitywhere j0 ¼ 0. The charge q is neutralized by itsmirror charge �q when it is near the surface ofanother conductor. If we set j0ðxiÞ ¼ 0 in Eq. (7),it is evident that the induced charge on electrode Lby qðx) is

QL ¼ �qj0ðxÞ:

The Shockley–Ramo theorem is thus proven.It should be noted that the Shockley–Ramo

theorem is just a special case of Eq. (6). Forexample, if we want to calculate the total inducedcharge on all electrodes, we can set potentials onall electrodes to be 1. In this case, if the wholevolume t is enclosed by electrodes, j0 ¼ 1 every-where. The total induced charge on all conductorsis �q which is a constant and the individualcharges on each electrode are simply redistributedby the motion of q. The total induced current iszero since E0 ¼ 0. This result reflects the con-tinuity equation for current. Finally, note that ifthe most outside boundary is infinity (an opendevice), then the potential at infinity is always zeroand cannot be set to 1. In this case, the totalinduced charge on all conductors is less than q ðQL ¼ �qj0ðxÞ and j0ðxÞ51 inside t.)

1.5. Further discussion

This proof of the Shockley–Ramo theoremallows the presence of a constant magnetic fieldB, such as an external field. There is no energytransfer between the magnetic field and the movingcharge q because the force acting on q( f ¼ q* B) is always perpendicular to thevelocity *. The work done by the magnetic fieldR xfxi

f � dx ¼ 0. The energy component ðw ¼ 12H � BÞ

stored in the magnetic field is invariant when B iskept constant. This term is canceled out on theright hand side in Eqs. (2) and (3).


It is implied that the dielectric constant e may bea function of position. Therefore, the Shockley–Ramo theorem may also be applied to a hetero-geneous medium. If the device medium is notisotropic, such as when D ¼ e � E and the electricpermittivity e is a tensor, as long as the linearsuperposition principle still allows the decomposi-tion of the actual case shown in Fig. 1(a) into thetwo parts in Figs. 1(b) and (c), the Shockley–Ramo theorem should still be valid (since Eq. (4) isalways valid). However, the calculation of theweighting potential could be very complex in anon-isotropic medium.

Although this proof assumes that there is onlyone moving charge q within the device, the result isalso valid when multiple moving charges arepresent. For example, electrons and holes are bothgenerated when a g-ray deposits its energy in asemiconductor detector. While electrons and holesare moving in opposite directions in the device, thetotal induced charge on an electrode is simply thesum of induced charges by all moving charges. Inthese cases, each E0

1k on the left of Eq. (2)corresponding to charge qk includes also theinteractions between qk and all other movingcharges.

This proof of the Shockley–Ramo theorembased on the conservation of energy shows howthe energy is transferred from external powersupplies to the moving charge q in an actualdevice. It should be kept in mind that the chargeinduced on an electrode by q is independent of theactual bias voltages on each electrode. Forexample, some detectors can be operated withoutexternal bias [6]. The cathode and the anode can bekept at the same potential (both electrodes can begrounded at zero voltage as illustrated inFig. 1(c)), the charges generated by radiation candrift in the internal electric field formed by spacecharge, such as by a p–n junction in a semicon-ductor device ([7], p. 369–371). The chargesinduced on each electrode redistribute while thecharge q moves, and the change of the inducedcharge on a specific electrode can be read out usingan amplifier for radiation detection. The weightingpotential on the electrode of interest is artificiallyset to 1 (zero potential on all other electrodes)which is different from the actual bias voltage. For

example, in a semiconductor device having a p–njunction, electron–hole pairs that are createdwithin the depletion region by the passage ofradiation will be swept out of the depletion regionby the electric field. The initial energy for sweepingout charges comes from the energy stored in theelectric field as shown in Eq. (2), and later, theoriginal electric field (as well as its energy) isrestored by the diffusion of electrons and holes.

In order to use the Shockley–Ramo theoremcorrectly, the implied condition that the fieldpropagates instantaneously during our derivationshould be kept in mind. This means that the transittime of the moving charge needs to be much longerthan the propagation time of the field across thevolume t. This restriction also requires that themovement of charge q is non-relativistic and theapplied bias voltages do not change too fast. Thiscondition is satisfied in almost all practical cases.

The weighting potential j0ðxÞ (04j0ðxÞ41)provides a convenient way for calculating theinduced charge on any electrode of interest. With agiven configuration of a device and the specifiedelectrode, only one weighting potential needs to becalculated from the Poisson equation. After thetrajectory xðtÞ of the moving charge q is deter-mined from the actual operating electric field (andmagnetic field if there is one), the charge inducedon electrode L by q as a function of time can beobtained as QLðtÞ ¼ �qj0ðxðtÞÞ.

2. Single-polarity charge sensing on wide band-gap

semiconductor c-ray detectors

2.1. Conventional devices using planar electrodes

Semiconductors having high atomic numbers Zand wide band gaps are desired for efficient g-raydetection operating at room temperatures. HgI2,CdTe and CdZnTe are materials that haveattracted most attention to date. However, dueto trapping of charges, the output signal of aconventional detector having planar electrodesdepends not only on the deposited energy, butalso on the position of that interaction. Therefore,the deposited energy cannot be obtained uniquelyfrom the amplitude of the output signal.


The configuration of a conventional detector isillustrated in the top of Fig. 2. When a g-raydeposits energy within a semiconductor detector, alarge number of electron–hole pairs proportionalto the energy deposition are generated. Electronsmove towards the anode and holes move towardsthe cathode in the electric field E formed by thebias voltage. The change of induced charge DQ onone of the electrodes is converted to a voltagesignal by the amplifiers with the pulse amplitudeproportional to DQ. The weighting potential of theanode (assuming the signal is read out from thiselectrode) is calculated using the Shockley–Ramo

theorem by setting the potential on the anode to 1,and assuming the cathode to be grounded. Recallthat the weighting potential is obtained with nospace charge. For a slab geometry, the weightingpotential is simply a linear function of depth Zfrom 0 to 1 (assuming the lateral dimension ismuch larger than the detector thickness) from thesurface of the cathode to that of the anode asshown in the middle of Fig. 2

j0ðzÞ ¼ z; 04z41:

If the loss of charge carriers during the drift timecan be ignored, such as is usually the case in a highpurity germanium detector (HPGe), the totalchange of the induced charge on the anode canbe calculated from Eq. (7) while n holes move fromz ¼ Z to 0 and n electrons move from z ¼ Z to 1:

DQ ¼ �ðne0Þð0� ZÞ þ ðne0Þð1� ZÞ ¼ ne0

where e0 is the electronic charge and Z is theinteraction depth of the g-ray. The first term is thecontribution of holes and the second term is thatof electrons. The normalized electron and holecomponents are shown in the middle of Fig. 2. DQ(also the output amplitude) is always proportionalto the number of electron–hole pairs generated,which is proportional to the deposited energy, andis independent of the interaction depth. For aconstant energy deposition Eg, the amplitude ofthe detector output pulse is also constant as shownin the bottom of Fig. 2, and only fluctuates due tothe noise of the detector system and the statisticalfluctuation in the charge carrier formation. How-ever, when holes can only move very short distancecompared to the detector thickness (Dzh51), theinduced charge on the anode is

DQe � ne0ð1� ZÞ ð9Þ

which is depth dependent. If g-rays interact withdetector material at all depths randomly, theinduced charge would vary from zero to ne0according to Eq. (9). No spectroscopic informa-tion can be obtained under these conditions fromthe pulse amplitude of the detector. The pulseheight spectrum corresponding to this case is alsoshown at the bottom of Fig. 2 for comparison.

Although various pulse processing techniqueshave been applied to improve the spectroscopic

Fig. 2. Illustration of a conventional detector using planar

electrodes. Top: The schematic of a semiconductor g-raydetector using conventional planar electrodes. Middle: The

weighting potential of the anode, and signal components from

the movements of electrons and holes. Bottom: The expected

energy spectra with fixed energy deposition Eg when the g-rayinteraction depth is distributed uniformly between the cathode

and the anode.


performance of room temperature semiconductorg-ray detectors, such as those employed on CdTedevices [8–10], these techniques can lead tosignificant loss of detection efficiency, especiallyon thick detectors. This is because none of thesesignal processing methods can solve the funda-mental problem of charge trapping which causesthe reduction of signal amplitude that depends onthe drift length of charge carriers. With a givennoise level of the detector system, the intrinsicsignal-to-noise ratio of pulses could approach zeroon any detector that has a thickness much largerthan the drift length of holes during the chargecollection time, when the g-ray interaction locationis close to the anode surface. Therefore, it isimpossible to recover the information on g-rayenergy accurately.

In order to overcome the effects of severetrapping of holes in wide band-gap semiconduc-tors, researchers have been investigating techni-ques in which the pulse amplitude is sensitive onlyto one type of charge carriers, normally just theelectrons (and not the holes). These techniques arecalled single polarity charge sensing, which canalleviate the charge trapping problem if the driftlength of just one type of charge (such as electrons)can be long compared with the detector thickness.

2.2. The Frisch grid technique

The first single polarity charge sensing techniquewas implemented in gas detectors by Frisch [11] toovercome the problem of slow drift and loss ofions. The Frisch grid is placed very close ðP51Þ tothe anode as shown in Fig. 3. Electrons passthrough the Frisch grid and are collected by theanode. The weighting potential ðj0Þ of the anode isobtained by applying a unit potential on theanode, and zero potential on both the Frisch gridand the cathode. The weighting potential j0 is zerobetween the cathode and the Frisch grid, and risesto 1 linearly from the grid to the anode as shown inthe bottom of Fig. 3. This configuration meansthat the charge moving between the cathode andthe grid causes no induced charge on the anode,and only those electrons passed through the gridcontribute to the anode signal. Therefore, theoutput pulse amplitude is proportional to the

number of electrons collected, and any inducedsignal from the movement of charges between thecathode and the Frisch grid, including that fromthe slow drift of ions, is completely eliminated.

2.3. Coplanar grid electrodes

Single polarity charge sensing was implementedon semiconductor detectors by Luke based on theuse of coplanar grid electrodes in 1994 [12,13]. Theconcept is illustrated in the top of Fig. 4. Insteadof a single electrode on the anode, parallel stripelectrodes are used and the strips are connected inan alternate manner to give two banks of gridelectrodes (electrodes 2 and 3). A voltage differ-ence between these two banks of electrodes isapplied so that the selected charge carriers (in thiscase electrons) are always collected by oneelectrode, say electrode 2.

The weighting potential j2 of electrode 2 withinthe device is calculated by letting the potential onelectrode 2 be 1 and potentials on electrodes 1 and3 to be zero [14], and solving the Laplace equation(since there is no space charge). Similarly, theweighting potential j3 of electrode 3 is calculated

Fig. 3. Side view of a Frisch grid detector and the weighting

potential of the anode.


by letting the potential on electrode 3 be 1 andpotentials on electrodes 1 and 2 to be zero. Themiddle of Fig. 4 shows the weighting potentials ofeach electrode along a line perpendicular to theelectrode surfaces and intersecting with one stripof the collecting anode (#2) at its center, such asalong the trajectories of electrons and holes shown

in the top of Fig. 4. For simplicity, it is assumedthat the dimensions of the device parallel to theelectrode surfaces are much larger than thedetector thickness (normalized to unity). Notice,j2 and j3 are practically identical for05z51� P, and then j2 approaches 1 and j3

drops back to zero rapidly. P is the period ofcoplanar grid electrodes and is much smaller thanthe thickness of the device (P51). The character-istic shape of these weighting potentials reflectsthe fact that the induced charges on electrodes 2and 3 increase identically due to the symmetrybetween the two coplanar anodes when electronsmove closer (or holes move away) to the anodesurface between 05z51� P. This regime isfollowed by the rapid increase on electrode 2 anddrop on electrode 3 back to zero when electronsapproach electrode 2 and deviate away fromelectrode 3 in z > 1� P driven by the actualelectric field. Single polarity charge sensing isimplemented by reading out the difference signalbetween electrode 2 (collecting anode) and 3 (non-collecting anode). This differential signal corre-sponds to the weighting potential of ðj2 � j3Þ. Ithas zero net value for 05z51� P and rises from0 to 1 when a unit charge travels from z ¼ 1� P to1. This ðj2 � j3Þ is very similar to the weightingpotential of the anode of a gas detector using aFrisch grid shown in the bottom of Fig. 3. If allelectrons are collected by the collecting anode andthe g-ray interaction depth is in the region04Z41� P, the differential pulse amplitude isproportional to DQ2 � DQ3 which can be calcu-lated using Eq. (7):

DQcoplanar ¼DQ2 � DQ3

¼ n0e0f½j2ðz ¼ 1Þ � j2ðz ¼ ZÞ

� ½j3ðz ¼ 1Þ � j3ðz ¼ ZÞg

since j2ðz ¼ 1Þ ¼ 1, j3ðz ¼ 1Þ ¼ 0 andj2ðz ¼ ZÞ ¼ j3ðz ¼ ZÞ, we have

DQcoplanar ¼ n0e0 ð10Þ

where n0 is the number of electrons arriving at thesurface of coplanar anodes. The induced charge byholes is eliminated since holes move in the linearregion of the weighting potentials (04z41� P).Since electrons are collected only by the collectinganode (#2) due to the voltage difference between

Fig. 4. Illustration of the coplanar grid technique. Top: The

side view of a semiconductor g-ray detector having coplanar

grid electrodes. Middle: Illustration of weighting potentials of

each electrode. Bottom: The weighting potential of

j2 � G j3. The illustration of electron trapping correction

using the relative gain compensation technique.


the coplanar anodes, j2 is always equal to 1 andj3 equal to zero at z ¼ 1 along the actualtrajectories of electrons.

If the trapping of electrons is negligible, theamplitude of the differential signal is independentof the depth of the g-ray interaction and isproportional to the energy deposition of the g-rays. It should be noted that the differential pulseamplitude has the same pulse height ðn0e0Þ as thatfrom a conventional detector using planar electro-des when charge trapping can be neglected. Theelimination of the sensitivity of the output pulse tohole motion does not result in any loss of signalamplitude.

In practice, not only holes are severely trappedin wide band-gap semiconductors, but the loss ofelectrons also cannot be ignored. From ourmeasurements, about 4–10% of electrons can betrapped across 1 cm of currently available CdZnTeat a bias voltage of 1 kV between the cathode andthe anode. Therefore, a good g-ray spectrometerusing wide band-gap semiconductors must notonly be able to overcome the hole trappingproblem, such as by employing a single polaritycharge sensing technique, but must also be able tocorrect for electron trapping. In Luke’s differentialcircuitry, a gain ðGÞ of less than 1 was applied onthe non-collecting anode signal before subtraction[12] to compensate for electron trapping. Thistechnique can be understood by observing that theoutput signal in this case corresponds to theweighting potential of j2 � G j3 which is shownin the bottom of Fig. 4. Applying a gain less than 1on the non-collecting anode signal effectivelyreduces the pulse amplitude in a linear relationshipas the g-ray interaction depth becomes closer tothe anode surface. Since the loss of electrons is lesswhen the drift length is shorter, this techniquecompensates the electron trapping if its effects arelinear with depth.

It was demonstrated that the coplanar gridtechnique can significantly improve the energyresolution of a CdZnTe detector [13], with arelatively simple electronic circuitry. Some char-acteristics of Luke’s technique should be noticed:(1) If the number of drift electrons as a function ofdrift length is an exponential function, the trap-ping of electrons can be perfectly compensated in

most regions of 04z41� P, except in the vicinityof the cathode where holes can be collected [15].This technique works better if the drift length ofholes is shorter (more trapping). (2) The weightingpotential j2 � G j3 is no longer zero in theregion of 04z41� P. Therefore, the variation inhole transport can contribute to the fluctuation inthe output signal. (3) It can be a challenge toimplement coplanar grid electrodes on a largesurface. This is rooted from some competingfactors. For example, a small pitch of strips isdesired to improve the symmetry between coplanarelectrodes, to make charge collection more uni-form and to reduce the dead region (a layer ofthickness P near the coplanar grid surface).However, a small strip pitch increases the capaci-tance and leakage current between the electrodes,and in turn increases the noise. Smaller gaprequires better surface processing to prevent noisebreakdown at required bias voltage betweencoplanar electrodes.

2.4. Other single polarity charge sensing techniques

The earliest single polarity charge sensingtechnique implemented on semiconductor g-raydetectors was the development of hemisphericaldevices [16]. The original strategy was to create amore uniform charge collection by having elec-trons move in the low field region and holes(having shorter drift length) move in the high fieldregion. On those devices, a small dot anode wasplaced in the focus of the hemispheric cathodeelectrode and the signal was read out from thesmall-area anode. The weighting potential of thissmall anode can be obtained by applying a unitpotential on the anode and grounding the cathode.Because the small scale of the anode relative to thecathode, the weighting potential is very low withinmost volume of the detector, and rises rapidly to 1near the anode electrode. Therefore, the inducedcharge on the anode is dominated by the move-ment of charge carriers near the anode. For themajority of g-ray interactions in the device,the pulse amplitude from the small anode isdominated by the number of electrons collected.Although significant improvement in g-ray energyresolution was demonstrated, the optimum


compensation of the variation of pulse amplitudewas not investigated. The most important limitingfactor is that the operating electric field is veryweak near the cathode where electrons move veryslowly. This causes severe charge trapping near thecathode, and limits the effective volume of thistype of devices.

A variation of coplanar grid technique wasproposed by Amman et al. [17] through which aproper choice of the ratio of strip widths betweenthe collecting and the non-collecting anodes causesthe weighting potential of the collecting anode tobe the same as that shown at the bottom of Fig. 4.Since the weighting potential corresponding to thesum of induced charges on the collecting and thenon-collecting anodes (considering the case of asingle anode on a conventional detector) increaseslinearly from 0 on the cathode surface to 1 on theanode surface, the slow rise from z ¼ 0 to 1� P ofthe weighting potential of the collecting anodeindicates that the induced charge on the collectinganode is much smaller than that on the non-collecting anode. This can be implemented if thetotal area of the collecting anode is much smallerthan that of the non-collecting anode. If thetrapping of electrons matches the change of theweighting potential in the region of 05z51� P,electron trapping can be compensated. Therefore,signals can be read from just one anode electrode,as simple as a conventional detector, in contrast toreading out two signals and performing subtrac-tion on a coplanar grid configuration. Thistechnique can simplify the readout and reduceelectronic noise. However, the electrode config-uration, which determines the weighting potentialof the collecting electrode, must be designed withgiven charge trapping properties which are notknown accurately on a raw semiconductor materi-al. After the fabrication of the device, the amountof charge trapping can be varied by changing thebias voltage between the cathode and the anode.Therefore, there is only one fixed bias voltageunder which the charge trapping is optimallycompensated by the weighting potential. Theoptimum spectroscopic performance of a deviceis determined not only by the compensation ofcharge trapping, but also by other factors, such asthe leakage current and the amount of charge

trapping. Therefore, this technique is not asflexible as a general coplanar device in achievingthe best spectroscopic performance. When electrontrapping is relatively small, as discussed earlier, theratio of the areas of the collecting anode to that ofthe non-collecting anode must also be small. Thereis a region underneath the non-collecting anodewithin which the lateral electric field is very weakdue to the equipotential on the surface of the non-collecting anode [14]. Electrons in this region tendto be trapped under the non-collecting anode, andthey can move out only by thermal diffusion. Thiseffect causes more electron trapping if the energy isdeposited underneath the non-collecting anode[17]. Efforts must be paid to minimize the widthof the non-collecting anode of these devices.

Patt et al. [18] suggested the use of the electrodestructure of a silicon drift detector on HgI2detectors. The electric field formed by a set offocusing electrodes drives the electrons to a smallanode where they are collected. The principle ofthis technique can also be understood easily withthe help of the weighting potential. The weightingpotential of the small anode is calculated byapplying a unit potential on the anode and zeroon all other electrodes. It is evident that thisweighting potential is similar to that shown at thebottom of Fig. 4, with very low values in most ofthe volume of the detector, and sharply rising to 1in the immediate vicinity of the small anode. Thisshape results from the small dimension of thecollecting anode and the closeness of the nearestfocusing electrode to the collecting anode. Becausethe nearest focusing electrode forces the weightingpotential of the collecting anode to be zero veryclose to the collecting anode, the induced chargeon the small anode is dominated by the number ofelectrons collected. This technique was demon-strated on a HgI2 detector having a thickness of2 mm and an area of 2:5 cm2. An energy resolu-tion of 0.9% FWHM at 662 keV g-ray energy wasobtained, which is a significant improvement overa conventional planar detector [18]. However,more investigation is needed to understand thereported results. For example, the effective volumeof the device which contributed to the photopeakevents was not measured. The high continuum inthe 137Cs g-ray energy spectrum between the


photopeak and the Compton edge and at lowerenergies indicated significant electron trappingwithin the detector.

Butler [19] proposed the use of a small-areaanode surrounded by a ‘control electrode’ on thesame surface of the anode. The principle of thisdevice is very similar to that of Patt et al. appliedon HgI2 detectors. The bias voltages on the anode,the control electrode and the cathode form theoperating electric field which drives the electronstowards the small anode. In terms of the weightingpotentials, the function of the control electrode isto set the boundary condition to be zero potentialon the surface covered by this control electrode.This effectively reduces the weighting potential ofthe small anode in the detector volume away fromthe anode and causes the induced charge to bedominated by the movement of electrons near thesmall anode. Similarly, the shape of the weightingpotential of the collecting anode is close to thatshown at the bottom of Fig. 4. However, incontrast to the weighting potential of the coplanargrids, this weighting potential in the region fromz ¼ 0 to 1� P (where P is roughly the relativedimension of the anode electrode to the thicknessof the detector) is not linear and in general variesas a function of lateral position even at the samedepth. Therefore, an accurate compensation ofcharge trapping using the weighting potential ismore difficult, especially on a detector having largedetection volume in which the charge trapping issignificant. Furthermore, this technique has thesame disadvantage of the single electrode readouttechnique proposed by Amman et al. [17] that theoptimum operating voltage is determined by theconfiguration of the electrodes. There is noflexibility in varying the optimum operatingvoltage after the fabrication of the device.

The direct application of the Frisch gridtechnique to room temperature semiconductordevices was demonstrated on CdZnTe detectorsby McGregor et al. [20] and Lee et al. [21]. A pairof Frisch grids were fabricated in parallel onopposite surfaces of thin CdZnTe detectors nearthe anode. The cathode is biased negative and theanode is biased positive relative to the Frisch grids.The signal is read out from the anode and itsamplitude is proportional to the number of

electrons collected. The weighting potential ofthe anode is calculated by applying a potential of 1on the anode and zero on both the Frisch grid andthe cathode. The weighting potential is similar tothat shown in Fig. 3. An energy resolution ofabout 2.7% FWHM at 662 keV g-ray energy wasobtained on a 1 cm3 CdZnTe detector having atrapezoid prism shape which enhances the spectro-scopic performance from the geometric effect [22].The advantage of this technique is simple readout.Since there is no compensation for charge trappingusing the weighting potential, the trapping ofelectrons will degrade energy resolution on thesedevices unless a position sensing technique isemployed. Another limitation of this method isthat the separation distance between the pair ofFrisch grids must be kept small to maintain thegood shielding of induced signal on the anodewhile charge carriers move between the cathodeand the Frisch grids. This limits the detectorvolume of a single device.

2.5. A depth sensing technique using the effect ofcharge trapping

It is sometime desired to know the depth of g-ray interaction, such as for g-ray imaging applica-tions, background rejection, and correction forcharge trapping. He et al. [23,24] proposed to readout signals simultaneously from both the cathodeand the coplanar grid anodes for each g-ray event.The position of g-ray interaction between thecathode and the anode then can be obtained fromthe ratio between these two signals. This is becausethe weighting potentials are different. For exam-ple, the hole mobility is usually much smaller thanthat of electrons in CdZnTe. A shaping time,which is long compared to the electron drift timeand short relative to the hole drift time, can beapplied on the cathode signal to obtain theelectron component of the induced charge shownin the middle of Fig. 2. As discussed before, thecathode signal is described by Eq. (9). Thedeposited energy can be obtained from thecoplanar anode signal which is described byEq. (10). The depth of g-ray interaction is obtainedfrom the ratio of the cathode and the coplanar


anode signals:

DQe

DQcoplanar¼

n

n0ð1� ZÞ � 1� Z; when n � n0:

This technique offers some advantages: (1) Itprovides the depth of g-ray interaction as anindependent parameter, therefore, it can correctfor electron trapping as an arbitrary function ofelectron drift length. (2) It enables the use of equalgain between the collecting and the non-collectinganodes. This makes the differential weightingpotential (see j2 � j3 in Fig. 4) exactly zero withinmost detector volume (04z41� P). Therefore,the induced charge of holes can be completelyeliminated. In contrast, if a relative gain betweencoplanar anodes is used, the variation of holetransport properties and weighting potentials atdifferent lateral positions of the same depth willcontribute to the fluctuation of pulse amplitude.(3) Since this technique allows the use of equalgain between the two coplanar anodes, theamplitude of the coplanar anode signal is onlyproportional to the number of electrons collected,and is independent of the shape of the anodeweighting potentials (as long as they are identicalin 04z41� P). This allows the employment of athird anode (boundary electrode) which can helpto minimize the difference between weightingpotentials of the first two coplanar anodes in theregion 04z41� P [25,26]. The introduction ofthe third anode also allows radial sensing in theregion close to the anode surface. An applicationof this technique is to measure the difference ofweighting potentials between the central twocoplanar grid anodes which causes poor energyresolution [25]. Since such a detector is both aconventional (the cathode signal) and a singlepolarity charge sensing detector, it should workbetter than either type (conventional or singlepolarity charge sensing) alone simply because itprovides more information.

2.6. Strip electrodes

Semiconductor detectors using strip electrodeswere first motivated for two-dimensional positionsensing [27,28], and it was later realized that singlepolarity charge sensing can also be achieved by

reading out signals from individual strip electrodes[14,27,28]. The induced charge on any particularstrip is very low when charge carriers move faraway (when the distance to the strip is much largerthan the pitch of strip electrodes) from theelectrode. This can be understood by noting thatthe weighting potential of the specific stripelectrode is determined by applying a unitpotential on the strip electrode and zero potentialon all other electrodes. The weighting potential isvery low in most of the detector volume exceptvery close to the strip electrode. Therefore, theweighting potential of a strip electrode is verysimilar to that shown at the bottom of Fig. 4 [14].If the linear increase of the weighting potential inthe region 04z41� P matches the loss ofelectrons, the expected pulse amplitude remainsapproximately constant, independent of the driftdistance of electrons. Therefore, the spectroscopicperformance can be significantly improved.Although the implementations of individual stripreadout and the relative gain compensationtechnique on coplanar grid electrodes look verydifferent, their principles of single polarity chargesensing can be unified by the similarity of theirweighting potentials. However, a fundamentaldifference between the two techniques is that thegain G of the relative gain compensation methodcan be easily adjusted according to the actual lossof electrons at any given bias voltage between thecathode and the anode, to achieve the optimumcompensation. In contrast, the change of theweighting potential in the region 04z41� P isfixed by the configuration of strip electrodes. Theoptimum electron trapping compensation can onlybe realized at a particular bias voltage. This is asevere limitation because strip detectors areusually designed not for optimum performance inspectroscopy, but more for imaging applications.Therefore, the pre-determined bias voltage, whichcan best compensate for charge trapping, is notusually employed due to other constraints.

The application of strip detectors has beenmainly for two-dimensional position sensing withfewer readout channels (2N) compared to that ofpixellated anode array ðN2Þ [29]. Orthogonal stripscan be used on the anode and the cathode surfaces,respectively [27,28], where one group of strip


electrodes provides the coordinate in x directionand the other provides the y coordinate. Becausethe weighting potential of one strip electrodeincreases rapidly to 1 only in the immediatevicinity of the strip, the arriving of electrons onone (or a few) anode strip provides the position inone direction, and the arriving of holes on one (ora few) cathode strip gives the position in anotherdirection. As a consequence, the good positionsensitivity depends on the good collection ofelectrons on the anode surface and also on theefficient collection of holes on the cathode surface.Because the drift length of holes is very short (inthe order of � 1 mm) in commercially availableCdZnTe crystals, the strip readout technique hasonly been employed on thin ð� 2 mmÞ CdZnTedetectors. The employment of this method is moreadvantageous if both electron and hole driftlengths are long compared to the detector thick-ness, such as in a high purity Ge detector or ifthe drift length of holes in CdZnTe could beimproved.

2.7. Pixellated anode array

Detectors using pixellated anode arrays werealso motivated for two-dimensional position sen-sitive imaging applications [30,31]. It was foundthat g-ray energy resolution obtained from in-dividual anode pixels on a CdZnTe detector wassignificantly improved compared to that from aconventional detector using planar electrodes. Itwas soon realized that reading out signals fromindividual small anode pixels of an anode array isanother form of single polarity charge sensing [32].The principle is very similar to that of using stripelectrodes. The induced charge on a particularanode pixel from the moving charge q is very smallwhen q is far away (the distance to the pixel ismuch greater than the pixel dimension) due tocharge sharing among many anode pixels, andincreases rapidly when q is in the vicinity of thispixel. The weighting potential of an anode pixelalong a line passing through the center of the pixelis similar to the one shown at the bottom of Fig. 4.It increases slowly in the region 04z4ð1� PÞ andrapidly rises to 1 from z ¼ 1� P to 1, where P isthe relative dimension (to the detector thickness)

of one anode pixel. Because the smaller area of ananode pixel compared to that of an anode strip(assuming the same width), the increase of theweighting potential from z ¼ 0 to ð1� PÞ is muchlower. The smaller the ratio between the dimensionof each anode pixel to the thickness of thedetector, the lower the increase of the weightingpotential from z ¼ 0 to ð1� PÞ. The inducedcharge on each anode pixel is then contributedmostly from the drift of charges in the immediatevicinity near the pixel. This is called the small pixeleffect.

For the same reasons as the strip detectors, mostroom temperature semiconductor g-ray detectorsusing pixellated anode arrays have been designedfor imaging applications, not for optimum g-rayspectroscopy. The pixel dimensions and detectorthickness are usually determined based on therequirements on position resolution and energyrange of the incoming g-rays, respectively. There-fore, the weighting potentials of pixel electrodesare fixed by considerations not for spectroscopy.As discussed previously, there is only one biasvoltage under which the trapping of charges can beoptimally compensated by the slow increase of theweighting potential in the region from z ¼ 0 toð1� PÞ. This is usually not the operating biasvoltage used. However, if the optimum biasvoltage can be carefully selected, the variation ofpulse amplitude due to charge trapping can besignificantly reduced from the compensation effectof the weighting potential, and the spectroscopicperformance can be improved [33]. The bestspectroscopic performance to date obtained atroom temperature on a detector using pixellatedanode array has been reported by Cook et al. [34].

2.8. Three-dimensional position sensitive detectors

The first fully functional three-dimensionalposition sensitive semiconductor g-ray spectro-meters were demonstrated on two 1 cm3 cubeCdZnTe detectors by He et al. [35]. The config-uration of the pixellated anode array is illustratedin Fig. 5(a). By combining two-dimensional posi-tion sensing using a pixellated anode array yieldinggood energy resolution from the small pixel effect,and the depth sensing technique discussed in


Section 2.5, three-dimensional coordinates of eachg-ray interaction can be obtained together with theenergy deposition. Therefore, the variation ofpulse amplitude due to the variation of chargetrapping, material non-uniformity and gain varia-tion of the readout electronic circuitry can becorrected to the limit of position resolution in 3-D.The challenge is to read out a large number ofindependent channels. Recent development inapplication specific integrated circuitry (ASIC)makes the data acquisition possible. This 3-Dposition sensing method offers some uniqueadvantages: (1) 3-D position sensing could becritical to some g-ray imaging applications [36]. (2)It can correct for material non-uniformity on wideband-gap semiconductors, to the extent of positionresolution [37,38]. (3) It can result in the minimumelectronic noise because the total leakage current isshared among a large number of anode pixels andthe capacitance of each pixel electrode is mini-mized. (4) Since the variation of pulse amplitudecan be corrected in 3-D, it is not necessary tocompensate this variation using the shape of theweighting potential. Therefore, the design of theanode array, which is usually optimized forimaging applications and determines the weightingpotentials of each anode pixel, can be independentof the operating voltage which determines thetrapping of charge carriers.

Other 3-D position sensing techniques havebeen proposed. Hamel et al. [39] suggested aconfiguration that has an array of small anodepixels located along the center lines of parallelnon-collecting anode strips as shown in Fig. 5(b).Anode pixels are interconnected along the ortho-gonal direction to the non-collecting anode stripsand are biased at a higher potential. The energyand x coordinate are obtained from the pulseamplitude and the location of the rows of thecollecting anode pixels, and y position is from theinduced signal on the columns of the non-collecting anode strips. Since each row of collect-ing anode pixels are connected, the weightingpotential of one row of anode pixels is very similarto that of an anode strip in the strip detectorconfiguration discussed in Section 2.6, which isconcentrated in the neighborhood of pixels alongthe row. The difference is that the increase of

Fig. 5. Illustration of 3-D position sensitive semiconductor

devices. (a) Pixellated anode array used by He et al. [35]. (b)

Schematic of anode array suggested by Hamel et al. [39]. (c)

Illustration of a device suggested by Luke et al. [41].


the weighting potential in the region from z ¼ 0 toð1� PÞ ðP is roughly the dimension of one anodepixel here) is in general lower because the presenceof the non-collecting anode strips between anodepixels forces the weighting potential to becomezero between collecting anode pixels. The pulseamplitude on each row of collecting anode pixels isroughly proportional to the number of electronscollected. The variation of pulse amplitude due tocharge trapping can be compensated by the shapeof the weighting potential, note that the optimumcompensation only occurs at a particular voltagebetween the cathode and the anode, or can becorrected by using depth sensing. The authorproposed to obtain the depth of g-ray interactionfrom the negative pulse amplitude derived fromthe non-collecting anode strips. The challenges ofimplementing this technique are: (1) To obtaingood signal-to-noise ratio (SNR) from the non-collecting anode strips. When electrons enter theneighborhood of a non-collecting anode strip, theyinduce a transient signal which intrinsically hasmuch poorer SNR than that on the collectinganode pixels. The poorer SNR causes higherenergy thresholds on the non-collecting anodestrips than on the collecting anode pixels. Thismeans the loss of Y coordinates for low energyevents. (2) Poorer SNR from non-collecting anodestrips could limit depth resolution if signals fromnon-collecting strips were used, especially on thick(detector thickness 4 strip pitch) detectors. If thesignal from only one non-collecting anode were tobe used, the weighting potential of a single anodestrip changes slowly (not sensitively) in the region04z41� P, so that the signal changes onlymildly as a function of depth. If signals frommultiple non-collecting anodes were to be used, thesignal amplitude changes more sensitively as afunction of depth, but the noise would alsoincrease from the required multi-channel readout. Position sensing on this type of detector hasbeen demonstrated using a particles [40]. Measure-ments using low energy g-rays, which will induce amuch lower signal amplitude, have yet to bestudied.

Another method was proposed by Luke et al.[41]. The energy is read out from a single collectinganode as discussed in Section 2.4 [17]. The

difference from the original single-electrode read-out is that the loss of pulse amplitude due toelectron trapping does not have to be compensatedexactly, since it can be corrected from positionsensing. The non-collecting anode is segmented toform two orthogonal arrays of electrodes shown inFig. 5(c). The induced signals by electrons whenthey pass the neighborhood of elements of non-collecting anodes give the x and y coordinates. Thedepth of the g-ray interaction is obtained from theratio of signals of the cathode and the collectinganode as discussed in Section 2.5. Similar toHamel’s method, the challenge of implementingthis technique depends mainly on obtaining goodsignal-to-noise ratio on the non-collecting anodesignals, which is important to identify positionsðx & yÞ. This is due to the intrinsically poorersignal-to-noise ratio of transient signals comparedto that of the collecting anode signal. Thischallenge is more significant for multiple energydepositions, such as a Compton scattering event.In these cases, multiple interaction positions needto be identified and individual energy depositionsat each position need to be separated from thetotal energy signal from the collecting anode. Theadvantage over pixel anode array is a smallernumber of readout channels (� 2N versus N2).However, all the bulk leakage current is collectedon the collecting anode, therefore, the intrinsicnoise from that source is higher than that using ananode array.

2.9. Summary

The Shockley–Ramo theorem is reviewed basedon the conservation of energy. A variety ofrepresentative techniques employed with semicon-ductor g-ray detectors have been summarized anddiscussed based on the Shockley–Ramo theorem.In analyzing many different schemes of readouttechniques, an understanding and correct applica-tion of the Shockley–Ramo theorem are essential.A common feature of all types of single polaritycharge sensing techniques is the rapid increase ofthe weighting potential in the vicinity of thecollecting electrode. Since the weighting potentialonly changes from 0 to 1, it is thus very low in theregion away from the collecting electrode. This


means that the movement of charge carriers awayfrom the collecting electrode contributes very littleto the induced signal. The most importantprogress in recent years on room-temperaturesemiconductor g-ray spectrometers is to makeuse of the differences between the weightingpotentials of different electrodes and the actualoperating potential.

Acknowledgements

The author is very grateful for the suggestionsand comments of Glenn F. Knoll and David K.Wehe.

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