Simplified modelling of an ion linac used for ion implantation

Nuclear Instruments and Methods in Physics Research B 217 (2004) 143–157

www.elsevier.com/locate/nimb

Simplified modelling of an ion linac used for ion implantation

John Gordon *,1

Applied Materials, Foundry Lane, Horsham RH13 5PX, UK

Received 25 March 2003; received in revised form 2 September 2003

Abstract

A spreadsheet-based simulation of an ion-implanter linac section has been developed for use by implant process

engineers and other users, who are not accelerator physics specialists. The simulation covers longitudinal motion

through a two-gap pre-buncher and several three-gap accelerating stages. The effect of a combined electrostatic and

magnetic dipole filter on the resulting ion energy spectrum is included. Several simplifications are included in the

calculation to allow rapid recalculation. Control parameters equivalent to those on a specific implanter linac allow the

model to be tuned in real time much as the physical hardware is tuned, and the result is displayed as an energy spectrum

which can be compared directly with spectra taken on the implanter. Calculated results from the model are compared

with experimental data. Comparisons are also made with simplified linear approximations, and with numerical inte-

gration of gap crossings.

� 2003 Elsevier B.V. All rights reserved.

PACS: 29.17.+w; 41.47.AK; 41.85.Ja

Keywords: Ion implanter; Linear accelerator

1. Introduction

The requirement to introduce deep dopant layers

into silicon has been met by the addition of RFlinear accelerator stages to ion implanters [1–5]. The

linac extends the maximum ion energy maximum

from 300 to 800 keV range typical of conventional

DC systems up into the 1–3 MeV range. In this

* Tel.: +44-1403-222345/1273-495-685; fax: +44-1403-

222288/1273-493-590.

E-mail address: [email protected] (J. Gordon).1 Present address: Pyramid Technical Consultants, Inc., 259

Bishops Forest Drive, Waltham, MA 02154, USA. Tel.: +1-

781-891-1169.

0168-583X/$ - see front matter � 2003 Elsevier B.V. All rights reser

doi:10.1016/j.nimb.2003.09.026

application, the accelerator is simply one part of a

routine wafer processing tool, and as such must be

operated by users who are not expert in the physics

of its operation. The complexity is of course gen-erally hidden behind a controls and software system

that permits either the user, or more commonly, the

factory automation host software, to interact at the

level appropriate to the task. However, because ion

implanters are capable of running a wide range of

different processes, there is usually an ongoing need

to develop and qualify new process recipes from

time to time. This task is typically entrusted to aprocess engineer with a more detailed knowledge of

the implanter. The introduction of a linac compli-

cates the work of this engineer.

ved.

mail to: [email protected]

144 J. Gordon / Nucl. Instr. and Meth. in Phys. Res. B 217 (2004) 143–157

Ion beam energy, U , along with ion species,

total ion dose and incidence angle, is a primary

implantation process parameter. On a conven-

tional DC system with grounded target wafer, it isknown immediately from

U ¼Xi

Viq;

where Vi are the various voltage drops along thebeamline and q ¼ �ne where n is the positive

ionisation state and e the electron charge. In a RF

accelerator, by contrast, the relationship between

the electrode voltages and energy gain is not so

direct, and also the electrode voltages may not be

known exactly. The process engineer attempting to

develop a new implant recipe needs to know how

to set the RF control parameters such as ampli-tude and phase for each separate section, but the

values are not immediately obvious. Moreover, if

incorrect values are used when the beam is being

accelerated by multiple linac sections, then the

transmitted beam current can be very low when

the acceptance of one section is not matched to the

time and velocity structure imposed on the beam

by the previous section. It is very time consumingto search for the optimum parameters on the im-

planter without some initial starting values close to

the actual values.

The objective of the work reported here was to

provide a simple to use and widely available

modelling tool to allow process engineers and

similarly experienced users to understand the

relationships between linac settings and energygain, and make reasonably accurate predictions of

the settings needed for a given recipe. Extensive

simplifications were introduced for computational

efficiency, such that a simple spreadsheet without

iteration was sufficient to contain the model. No

attempt was made to find a unique solution for a

given energy gain requirement; rather the model

was presented with �controls� equivalent to thoseof the implanter. The user adjusts the controls as

he or she would on the implanter and observes

the resulting energy spectrum directly. When the

spectrum appears satisfactory, the settings are

transferred to the implanter as the starting point

for final beam optimisation.

2. The Swift xE linear accelerator

The simplified linac modelling method has beenapplied to the Applied Materials Swift xE ion

implanter, although it would be applicable to any

multi-section ion linac using conventional accel-

erating gaps. The Swift xE implanter includes an

RF booster linac section following a 100 kV DC

injector. The booster is designed to accelerate ions

with a mass to charge ratio (m=n) of 5 to 15 amu/

charge, corresponding to the phosphorus dopantspecies at 2+ and 3+ charge state, and the boron

dopant at 1+ and 2+ charge state. The relevant

portions of the system are shown schematically in

Fig. 1. The booster linac comprises a two-gap pre-

buncher followed by three, three-gap accelerating

sections. Magnetic quadrupoles along the booster

contain the beam radially. A combination filter

employs magnetic, electrostatic or a combinationof both types of field to define the output energy

and remove off-energy ions. Fig. 2 is a photograph

of the linac booster section of the implanter.

The RF system operates at a 20.34 MHz fixed

frequency, with an independent control processor

and amplifier for each section. The pre-buncher

section uses a single-ended coil in atmosphere to

make up a parallel resonant circuit. Typical opti-mum RF powers are around 40–100 W, which

with a circuit loaded Q of around 460, gives peak

voltage amplitudes on the driven electrode of

around 4–7 kV. The three accelerator sections are

identical aside from the length of the driven elec-

trodes, which increases to match the increasing ion

velocity. The driven electrodes are connected to

the ends of a pair of strongly mutually coupledwater-cooled coils in vacuum. At the 20.34 MHz

frequency, these ends oscillate in anti-phase, such

that the peak voltage across the central gap is

twice the individual voltage amplitudes to ground.

Effective loaded Q values of around 1700 and

maximum RF power of 6 kW gives maximum

RF voltage amplitudes in the 75–80 kV range, or

150–160 kV between the driven electrodes.Fig. 3 shows the equivalent schematic of the

accelerating section circuits. RF power is coupled

into the circuit via an input coupling loop which

acts as the primary of a transformer. The high

shunt impedance of the resonant secondary circuit

Fig. 1. Schematic diagram of the Swift xE linac booster. A beam extracted from the ion source is mass-analysed and injected from the

left. It leaves at the right after being pre-bunched, accelerated and then filtered by the booster section. Each RF resonator has a power

input coupler, pickup loop for feedback, and tuner for maintaining resonance. A magnetic scanning and collimation system (not

shown) follows, to present a uniform scanned ribbon beam to the target wafer.

Fig. 2. Swift xE linac booster showing the beamline with pre-

buncher, quadrupole lenses, accelerating sections with resona-

tor bodies below, and energy filter. The length of the booster

linac section is 1.74 m excluding the filter.

50R

E1

E2

Input coupler Pickup loop

Z = 50R

Gce Gep

Gcp

Fig. 3. The parallel resonant equivalent circuit for the accel-

erating sections, with an input coupling transformer which

provides impedance matching to 50 X, and a very weakly

coupled pickup loop which samples the resonator voltage. The

transmission factor Gcp can be measured directly at low power

with a network analyser. Gep is required in order to convert

measured pickup voltage (Vp, also known as ‘‘AT’’) to electrode

voltage; its value is constrained by measurable and calculated

parameters of the resonant circuit.

J. Gordon / Nucl. Instr. and Meth. in Phys. Res. B 217 (2004) 143–157 145

is transformed down to 50 X. The RF voltage andphase in the resonant circuit is sampled to provide

feedback for closed-loop servo control using a very

weakly coupled loop pickup. The RF amplitude

target control parameter (‘‘AT’’) entered by the

user is the voltage amplitude in the line from this

pickup.

An unusual feature of the control system is that

the individual accelerating sections operate with a

fixed RF phase relationship [6]. Correct beam

transfer between the sections is achieved byadjustment of only the RF amplitudes. The max-

imum number of RF parameters that must be set is

therefore three RF amplitudes, plus the amplitude

and phase of the pre-buncher.


3. Energy spectrum calculation

All calculations and data presentation are per-formed with the Microsoft Excel spreadsheet

software, using only native functions. This ensures

wide availability and user familiarity.

The first major simplification is to deal only

with the axial longitudinal motion. Experience has

shown that the RF parameters can be optimised

for efficient energy gain and beam transfer between

the accelerator sections independently of lateralmotion considerations. However this does mean

that the simple model gives no guidance on how to

tune the quadrupole magnets, which must be done

once the initial RF settings have been found. The

second simplification is to ignore relativistic ef-

fects. These are negligible for the ion velocities of

interest (<0:02c).The next simplification is to ignore the actual

electric field distribution in the RF gaps, and as-

sume instead that an ion of charge q with energy

Uj will exit gap k with energy Uk given by

Uk ¼ Uj þ Vgðcosðum þ usÞÞTq;

where for the linac sections in the Swift implanter,

Vg is the peak effective gap voltage derived from

the RF voltage amplitude Ve for the relevant sec-

tion as follows:

Pre-buncher section: Vg ¼ Ve for the first gap,

Vg ¼ �Ve for the second gap.

Accelerator sections: Vg ¼ Ve for the first gap,

Vg ¼ �2Ve for the central gap, Vg ¼ Ve for the

third gap.

um is the particle phase xt, where x is the angularfrequency of the RF field in gap k and t is the time

at which the particle is at the centre of gap k. All

phases are relative to a master clock. At gap k the

phase um is incremented from the value at prior

gap j by the amount

xdj=vj ¼ xdj

ffiffiffiffiffiffiffiffi2Uj

m

r,;

where dj is the drift distance from gap j to gap k, vjis the drift velocity corresponding to Uj and m the

ion mass.

us is any imposed phase shift relative to the

master clock. On the Swift implanter this is a user

adjustable parameter for the pre-buncher, but

fixed for the accelerating sections.T is a transit time factor to compensate for the

fact that the time the ion takes to cross the gap is

not negligible compared to the RF period [7].

T ¼ sinðh=2Þðh=2Þ ;

with

h ¼ xgeff=vj;

geff is an effective inter-electrode gap width. To

compensate for field penetration into the elec-

trodes, which is particularly significant for the pre-

buncher, an arbitrary correction is made whereby

the known physical gap, has added to it a pro-

portion, b, of the width of the electrode aperture

slots, w.

geff ¼ gactual þ bw:

Comparison with the results of numerical inte-

gration of ion gap crossings, described later,

showed that a value of b of 1.0 was appropriate for

the pre-buncher and 0.25 for the accelerating sec-

tions. It would be more justifiable to use thevelocity at the centre of the gap rather than the

velocity before the gap, vj, in the calculation of

transit angle, h. However, this would require iter-

ation to converge on the correct transit time fac-

tor, so the above simplification is used in the

interests of calculation efficiency.

A given ion is injected with an initial velocity

due to the DC ion source voltage, and initialarbitrary RF phase relative to the master clock.

The calculations described above are then per-

formed for each of the 11 RF gaps to arrive at

the final energy. If the velocity reverses during the

calculation, then the ion is assumed to be lost. The

process is repeated for 360 ions, typically with one

degree increments of starting phase, so as to sim-

ulate the effect of the linac on a continuous in-jected beam. Finally an energy spectrum is

displayed by binning the table of final energies and

plotting bin counts against energy. On a modest

PC (300 MHz Pentium processor), a complete

recalculation of the spreadsheet completes in

Mass 31 amu Source potential 90 kV 270.0 keV Filter HV 0 V Mode #N/ACharge 3 Filter magnet I 0.00 A Energy 0.0 keV

ATB (buncher ampl) 2.30 V 5.0 kVRF frequency 20.34 MHz PTB (buncher ph targ) -25 degr 58 W Slit (n/m/w) w Slit 25.0 mm

AT1 (RF1 ampl) 3.30 V 62.88 kV Filter on/off (1=on) 0 dU/U +/- 0.0 %Phase step 1 degr RF1 phase (fixed) 0 degr 3.78 kWFirst ion phase 0 degr AT2 (RF2 ampl) 3.70 V 70.50 kV Filter: off

RF2 phase (fixed) 140 degr 4.75 kW Transmission 80.3%Energy bin size 50 keV AT3 (RF3 ampl) 3.80 V 72.41 kVSpectrum start 0 keV RF3 phase (fixed) -60 degr 5.01 kW

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 500 1000 1500 2000 2500 3000Energy (keV)

Rel

ativ

e cu

rren

t

Fig. 4. Sample energy spectrum calculation (unfiltered) with the parameters tuned to boost Pþþþ from 270 keV to 2400 keV. The

parameters ATn are the pickup voltage amplitude targets, Vp. The derived electrode voltages Ve and input coupler powers Pc are also

displayed in the central data boxes.

Mass 31 amu Source potential 90 kV 270.0 keV Filter HV 0 V Mode #N/ACharge 3 Filter magnet I 0.00 A Energy 0.0 keV

ATB (buncher ampl) 2.30 V 5.0 kVRF frequency 20.34 MHz PTB (buncher ph targ) -25 degr 58 W Slit (n/m/w) w Slit 25.0 mm

AT1 (RF1 ampl) 3.15 V 60.02 kV Filter on/off (1=on) 0 dU/U +/- 0.0 %Phase step 1 degr RF1 phase (fixed) 0 degr 3.44 kWFirst ion phase 0 degr AT2 (RF2 ampl) 3.70 V 70.50 kV Filter: off


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 500 1000 1500 2000 2500 3000Energy (keV)

Rel

ativ

e cu

rren

t

Fig. 5. The same parameters as Fig. 4, except for a small change to the RF amplitude in the first accelerating section (parameter AT1)

which causes a substantial detuning, as the ‘‘handover’’ of the beam from first to second accelerating section is spoilt.


about 2 s, making it convenient to ‘‘tune’’ themodel dynamically to achieve a spectral peak at

the desired energy.

Fig. 4 shows a typical calculated spectrum for

Pþþþ ions accelerated from 270 keV to 2.4 MeV.

The input parameters shown were tuned to achievea well-defined peak at the high energy end. Fig. 5

shows identical conditions except that the first

accelerating section amplitude is detuned slightly

so that the beam is not transferred efficiently to the


second accelerating section. These sharp tuning

effects are seen with real beams, and are one reason

why the correct tuning solution can be missed on

the implanter without the guidance of a modelledsolution.

4. RF power calculation

Although the control parameter for the reso-

nators is the pickup voltage amplitude, which is a

linear function of electrode voltage, an importantconstraint on the feasibility of any proposed set-

tings is the RF power requirement, ultimately

limited by the RF amplifier capability. Making the

realistic assumption that the resonator is matched

to 50 X at the power input coupler, and on reso-

nance, it is straightforward to derive the minimum

power requirement to achieve a given pickup

voltage. The attenuation of a 20.34 MHz signalfrom input coupler to pickup loop, Gcp, can be

measured directly at low power with a RF network

analyser. Then

Gcp ¼ 20 lgVc

Vp

� �

for Gcp measured in dB and where Vc and Vp are the

peak voltage amplitudes in 50 X in the input

coupler and pickup loop respectively. Then the

power at the input

Pc ¼1ffiffi2

p 10 �Gcp=20ð ÞVp

� �2

50¼ 10�ð0:1Gcpþ2ÞV 2

p :

The actual measured power delivered by the

accelerating section amplifiers always exceeds this

minimum due to cable losses, imperfect resonance

and match, beam loading and reduction in Q athigher temperatures. The relatively low power pre-

buncher shows good agreement when correctly

tuned, however.

5. Electrode voltage

Linac electrode voltage generally is not mea-sured directly due to the high voltages involved,

and the loading effect on the circuit of any direct

probe. A weakly coupled pickup is thus used to

sample the electromagnetic fields in the resonator

for feedback. In order to compare calculatedspectra with measured ones, it is necessary to

know how the sampled voltage relates to the actual

electrode voltage. On the reasonable assumption

that there is no direct coupling from the input

coupler to the pickup loop, then

Gcp ¼ Gce þ Gep;

where Gce and Gep are the voltage transformerratios from input coupler to electrodes, and from

electrodes to pickup loop, in dB. Gcp is measured

directly during setup of a resonator, which leaves

Gce say, as a free parameter. In practice this can be

adjusted to give good agreement between calcula-

tion and measurement. However Gce is constrained

within a narrow range by other measurements.

The resonator loaded Q can be measured directlyby looking at the )3 dB bandwidth in a network

analyser transmission measurement. The shunt

impedance of the circuit can be measured directly

by probe methods. The inductance of the resonant

circuit was calculated to about 10% confidence

during the initial design work. These known values

can be compared to values calculated from Gce and

Gep. The three gap accelerating sections have shuntimpedance

Rs ¼ð2Ve=

ffiffiffi2

pÞ2V 2

p

P¼

2ð10�Gep=20Þ2V 2p

P;

where the power, P , equals Pc at perfect

match and resonance, and the factor 2 appears

because the voltage across the whole coil in this

case is twice the electrode voltage. Substituting

for Pc

Rs ¼ 2 � 10ð0:1ðGcp�GepÞþ2Þ:

Then, given ZL ¼ xL, Qunloaded is given by

Rs=ZL, and the loaded Q at perfect match, forcomparison with measurement, by Qloaded ¼Qunloaded=2.


6. Energy filtering

Ion implantation requires a relatively mono-energetic beam so as to control the depth of the

implant. The energy filter serves both to define the

beam energy, and filter out off-energy beam such as

the extended low energy tail visible in Fig. 4. The

Swift filter is a combination magnetic dipole and

electrostatic sector analyser, shown schematically

in Fig. 6. Various operating modes are possible [8]:

B mode. Magnetic dipole only. The filter selectsaccording to mv=q. Suitable for higher current

lower energy beams where the linac is not in use.

E mode. Electrostatic sector only. The filter se-

lects according to mv2=q. Improved dispersion over

B mode and direct absolute kinetic energy analysis.

Suitable for higher energy lower current beams,

such as those obtained from the linac.

E � ð�BÞ mode. The magnet polarity is reversedto buck the electrostatic field, which has to be in-

creased to compensate. This provides further dis-

persion improvement over E mode.

ð�EÞ � B mode. The electrostatic field polarity is

reversed to buck the magnetic field. This gives re-

duced dispersion compared to B mode, and allows

qvxB

(a)

qvxB

qE

+ve

-ve

(c)

Fig. 6. Energy filter modes. (a) B mode, which gives good transmis

absolute ion energy analysis and improved dispersion. (c) E � ð�BÞbucking B field, with consequent further improvement in dispersion.

doubly charged ions when transmitting triply charged ions, but at th

double energy solutions in the general case. How-

ever it has the interesting property that settings can

be found where there is no solution at all for

potentially contaminating doubly charged ionswhen the filter is set to transmit triply charged ions.

The beam tuning process on the implanter for

linac accelerated beams involves setting the filter to

provide a window at the required energy (or equiv-

alent momentum), then tuning the linac such that an

energy spectrum peak coincides with this window.

The model calculates the filter energy setting for any

magnet current and high voltage settings, and sim-ulates the filtering effect on the spectrum, but with

the advantage for theoretical work that the filter can

be turned off such that all energies pass.

In the general case, the energy setting of the

filter, U , for a particle of mass m, charge q, given

electric field strength E and magnetic field strength

B is given by [9]

ffiffiffiffiU

p¼ k �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 þ 4U0

p

2;

where

k ¼ qBRffiffiffiffiffiffi2m

p

qE

+ve

-ve

(b)

qE

qvxB

-ve

+ve

(d)

sion of space–charge limited beams. (b) E mode, which gives

mode, where the E field has to be increased to compensate a

(d) ð�EÞ � B mode, which can be arranged to completely reject

e cost of reduced dispersion.


and

U0 ¼qER

2:

Mass 31 amu Source potential 90 kV 270.Charge 3

ATB (buncher ampl) 2.30 V 5.RF frequency 20.34 MHz PTB (buncher ph targ) -25 degr 5

AT1 (RF1 ampl) 3.30 V 62.8Phase step 1 degr RF1 phase (fixed) 0 degr 3.7First ion phase 0 degr AT2 (RF2 ampl) 3.70 V 70.5

RF2 phase (fixed) 140 degr 4.7Energy bin size 50 keV AT3 (RF3 ampl) 3.80 V 72.4Spectrum start 0 keV RF3 phase (fixed) -60 degr 5.0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 500 1000 150Energy

Rel

ativ

e cu

rren

t

Fig. 7. The spectrum of Fig. 4 with E mode filtering imposed. This is

low-energy tail in the linac spectrum is eliminated. Any doubly charg

through the filter.

Mass 31 amu Source potential 90 kV 270.Charge 3

ATB (buncher ampl) 2.30 V 5.RF frequency 20.34 MHz PTB (buncher ph targ) -25 degr 5

AT1 (RF1 ampl) 3.30 V 62.8Phase step 1 degr RF1 phase (fixed) 0 degr 3.7First ion phase 0 degr AT2 (RF2 ampl) 3.70 V 70.5

RF2 phase (fixed) 140 degr 4.7Energy bin size 50 keV AT3 (RF3 ampl) 3.80 V 72.4Spectrum start 0 keV RF3 phase (fixed) -60 degr 5.0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 500 1000 150Energy

Rel

ativ

e cu

rren

t

Fig. 8. The spectrum of Fig. 4 with B mode filtering imposed and a

visible as some of the low-energy tail starts to be transmitted. Any dou

also pass through the filter.

The electric field is generated by DC voltages þVand �V applied to plates with centre geometric

radius R, and constant spacing d, so E ¼ 2V =d. Themagnetic dipole field is generated by a magnet of

0 keV Filter HV 37855 V Mode EFilter magnet I 0.00 A Energy 2400.0 keV

0 kV8 W Slit (n/m/w) m Slit 10.0 mm8 kV Filter on/off (1=on) 1 dU/U +/- 2.4 %8 kW0 kV Filter: on5 kW Transmission 34.6%1 kV1 kW

0 2000 2500 3000(keV)

the normal operating mode for linac boosted beams. The long

ed ions present at 2/3 of the triply charged energy will also pass

0 keV Filter HV 0 V Mode BFilter magnet I 79.41 A Energy 2400.0 keV

0 kV8 W Slit (n/m/w) w Slit 25.0 mm8 kV Filter on/off (1=on) 1 dU/U +/- 11.6 %8 kW0 kV Filter: on5 kW Transmission 39.5%1 kV1 kW

0 2000 2500 3000(keV)

wider resolving slit. The reduced effective energy resolution is

bly charged ions present at 4/9 of the triply charged energy will

Mass 31 amu Source potential 90 kV 270.0 keV Filter HV -15000 V Mode (-E)xBCharge 3 Filter magnet I 110.88 A Energy 2400.0 keV

ATB (buncher ampl) 2.30 V 5.0 kV 376.8 keVRF frequency 20.34 MHz PTB (buncher ph targ) -25 degr 58 W Slit (n/m/w) w Slit 25.0 mm

AT1 (RF1 ampl) 3.30 V 62.88 kV Filter on/off (1=on) 1 dU/U +/- 19.4 %Phase step 1 degr RF1 phase (fixed) 0 degr 3.78 kWFirst ion phase 0 degr AT2 (RF2 ampl) 3.70 V 70.50 kV Filter: on


0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 500 1000 1500 2000 2500 3000Energy (keV)

Rel

ativ

e cu

rren

t

Fig. 9. The spectrum of Fig. 4 with the filter operating in ð�EÞ � B mode, showing degraded resolution and a second lower energy that

can pass through the filter. The latter is of no practical consequence on the Swift xE beamline because further magnets following the

filter will deflect it completely away from the beam path. There is no solution for potentially contaminating Pþþ ions with these filter

settings.


known good linearity, so B ¼ cI where I is the coil

current and c the measured linear proportionality.

In the case where E is positive (E and E � ð�BÞmodes), there is always one positive root giving thefilter energy setting for a particular charge state. In

the case where E is negative (ð�EÞ � B mode),

there are two real roots provided that k2 > 4jU0j,otherwise there are no real solutions and the par-

ticle cannot pass through the filter.

A qualitative sense of how the filter modifies the

spectrum is provided by assuming simply that there

is a soft-edged passband of �DU around the setenergy U , where DU ¼ Us=Dt. Dt is the velocity

dispersion of the filter and s is the width of the

resolving slit. Dt is given by the sum of the lateral

displacement plus the angular displacement multi-

plied by the drift distance L from filter exit to slit [9]

Dt ¼1

j2

1

r

�� 2

R

�ðcosðkRhÞ � 1Þ

þ L� 1

j1

r

�� 2

R

�sinðkRhÞ

;

where h is the filter deflection angle, r is the radius

that would occur due to the magnetic field com-

ponent alone and

j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

R� 1

r

� �2

þ 1

R2

s:

The calculated linac energy spectrum is filtered

by a Gaussian function centred at the calculated

energy U , and with full width half maxi-

mum¼ 2DU . Figs. 7–9 show the effect of various

filter operating modes on the raw linac spectrumshown in Fig. 4.

7. Comparison of calculations with experimental

data

Sweeping the energy filter while monitoring the

beam current passing through the energy resolvingslit generates a spectrum that can be compared

with a calculated spectrum. If the sweep is per-

formed in E mode, then the scale is linear in U=n,

and the result may be considered an energy spec-

trum, albeit convolved with the effects of finite

beam and slit width, and the effects of non-parallel

rays entering the filter. In order to make direct

comparison with the calculated spectra, the mea-sured input coupler to pickup loop sensitivity

0

0.2

0.4

0.6

0.8

1

1.2

1.4

100 200 300 400 500 600 700Energy (keV)

Rel

ativ

e cu

rren

tAT1=1.00 AT1=1.50 AT1=2.00 AT1=2.50 AT1=3.00 AT1=3.50 AT1=4.00

0

0.2

0.4

0.6

0.8

1

1.2

1.4

100 200 300 400 500 600 700Energy (keV)

Rel

ativ

e cu

rren

t

AT1=1.00 AT1=1.50 AT1=2.00 AT1=2.50 AT1=3.00 AT1=3.50 AT1=4.00

Fig. 10. Measured (top) and calculated (bottom) spectra for Pþþ ions injected at 200 keV with various settings of feedback amplitude

target, AT1 (i.e. Vp), for the first accelerating section resonator. There is generally good agreement in the position of the high energy

peak, and the calculation also reproduces many of the overall features of a particular spectrum.

0

0.1

0.2

0.3

0.4

0.5

0.6

800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800Energy (keV)

Rel

ativ

e cu

rren

t

AT1=3.50AT2=1.00

AT1=3.50AT2=1.50

AT1=3.56AT2=2.00

AT1=3.56AT2=2.50

AT1=3.56AT2=3.00

AT1=3.56AT2=3.50

0

0.1

0.2

0.3

0.4

0.5

0.6

800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800Energy (keV)

Rel

ativ

e cu

rren

t

AT1=3.55AT2=1.00

AT1=3.55AT2=1.50

AT1=3.55AT2=2.00

AT1=3.53AT2=2.50

AT1=3.50AT2=3.00

AT1=3.46AT2=3.50

Fig. 11. Measured (top) and calculated (bottom) spectra for Pþþþ ions injected at 285 keV with various settings of feedback amplitude

target, AT2, for the second accelerating section resonator, and with the first accelerating section amplitude target, AT1, tuned for good

beam transfer. There is good agreement in the energy peak positions, although slight differences in the first accelerating section

amplitude, AT1, needed for efficient beam transfer can be seen.



factors for the particular system, Gcp, are entered

into the spreadsheet. Other parameters such as the

step up ratio from input coupler to electrode

voltage, Gce, and electrode width factors, b, areconsidered constant over all Swift implanters, and

this assumption gives adequate agreement.

Figs. 10 and 11 show some sample comparisons

for Pþþ accelerated by the first booster section

only and Pþþþ accelerated by the first two sections.

The voltage amplitude target parameter (AT) was

set in 0.5 V increments and the resulting spectrum

P++ U(AT1)

200

300

400

500

600

700

0.0 1.0 2.0 3.0 4.0 5.0AT1 (V)

U fi

nal (

keV)

P++ U(AT2)

400

500

600

700

800

900

1000

0.0 1.0 2.0 3.0 4.0AT2 (V)

U fi

nal (

keV)

P+++ U

1800

1900

2000

2100

0.0A

U fi

nal (

keV)

Fig. 12. Measured (diamonds) and calculated (dashed line and crosses

of the RF amplitudes of the three accelerating sections. The calculatio

RF settings required for a given energy, to within 5%. Only two data p

and collimation magnets required upgrade to higher bending power

taken for Pþþ up to 1500 keV and Pþþþ up to 2500 keV.

recorded. There is generally good agreement be-

tween the peak position and overall shape of the

spectra. This is despite the calculation taking no

account of lateral motion.The energy position of the peak at the top of the

spectrum is the main practical concern, rather than

accurate prediction of all the spectral features.

Under the fixed phase/variable amplitude opera-

tional method used on the Swift xE linac, the first

part of the linac energy range is covered with the

first accelerating section and pre-buncher only

P+++ U(AT1)

200

400

600

800

1000

1200

0.0 1.0 2.0 3.0 4.0 5.0AT1 (V)

U fi

nal (

keV)

P+++ U(AT2)

1000

1200

1400

1600

1800

0.0 1.0 2.0 3.0 4.0AT2 (V)

U fi

nal (

keV)

(AT3)

1.0 2.0T3 (V)

) energy peak positions for Pþþ and Pþþþ for a range of settings

n generally predicts energy for given RF settings or, conversely,

oints were taken for the three section case because the scanning

at the time of the measurements. Subsequently, data has been


operating, to give a response curve UðAT1Þ, where

AT1 is the amplitude target, Vp, for the first sec-

tion. The next part of the energy range is then

covered by increasing the amplitude target for thesecond section, AT2, with AT1 essentially con-

stant at a value that gives good transfer of the

beam, to give response curve UðAT2; ½AT1�Þ. Fi-

nally the highest energies are reached with all three

sections operating, AT1 and AT2 essentially fixed,

to give UðAT3; ½AT2;AT1�). Fig. 12 summarises

the peak positions, as read from spectra such as

those shown in Figs. 10 and 11, as a function ofRF amplitude target for Pþþ and Pþþþ for one,

two and three (limited data) accelerating sections

P+++, U injection = 270keV

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10 12Sum of AT settings

U fi

nal (

keV)

B+, U injection = 90keV

0

200

400

600

800

1000


U fi

nal (

keV)

Fig. 13. Approximations to the calculated final energies as a function

Bþþ. Triangular points are with the first accelerating section only oper

with all three sections operating. The dotted line behind the symbols is

is the product of the charge state and four times the sum of the electrod

and the calculated values shows the extent to which transit time and io

use of the available total RF amplitude because of the constraint that

operating. Agreement is again generally good,

with the main difference of note being that the

calculations are slightly optimistic for the energy

gain for Pþþ acceleration. The worst case errors inpredicted energy are around 5%.

8. Simple linear approximation for final energy

The prediction of final energy for a given

parameter set, or the converse, can be a complex

problem, especially for a multi-section indepen-dently phased linac [10,11]. The adoption of fixed

phase operation allows simple linear approxima-

P++, U injection = 200keV

0

500

1000

1500

2000


U fi

nal (

keV)

B++, U injection = 160keV

0

500

1000

1500

2000


U fi

nal (

keV)

of the RF amplitude targets for the four species Pþþþ, Pþþ, Bþ,

ating, crosses with the first and second operating, and diamonds

a two parameter linear fit to the calculated values. The solid line

e voltages, plus the injection energy. The difference between this

n phase affect the calculated energy gain. Bþþ does not make full

the accelerating sections operate with a fixed relative RF phase.


tions to be exploited. It is clear from Fig. 12 that

the UðATÞ responses are nearly linear, except for

the lowest Pþþ energies. Moreover, the gradients

are similar for the three accelerating sections, and,if divided by the charge state, are also similar for

the four species of interest. Thus, to a fairly good

approximation for all four species, the final energy

is given by

Ufinal ¼ c1qX3

i¼1

ATi

!þ c0Uinj;

where Uinj is the injection energy. The Swift xE

linac has the following best fit values of these

parameters, for nominal setup of Gcp:

At this level of approximation, and with a fixed-

phase linac, the estimation of final beam energy

has become almost as straightforward as the DC

accelerator case. The choice of fixed phase betweenthe accelerating sections does not affect this linear

relationship. Fig. 13 shows the comparison be-

tween the linear approximations and calculated

values for the species and charge states of interest.

The breakpoints where the number of sections

operating change are shown in the calculated val-

ues. Also shown is the result of the na€ıve calcula-

tion that the final energy is the injection energyplus four times the sum of the electrode voltages

times the charge state. The difference between this

and the calculated values is an indication of the

combined influence of the transit time factor, and

phase slip.

c1 c0

Pþþ 67 0.38

Pþþþ 67 0.75

Bþ 66 0.76

Bþþ 64 0.78

9. Limitations of the model

Despite the generally good agreement between

the calculated values and measured data, the lim-

itations introduced by the numerous simplifica-

tions do become apparent under more extreme

conditions. Cases in point are the acceleration of

lower velocity ions such as Asþþþ ðm=n ¼ 25Þ or

Pþ ðm=n ¼ 31Þ where the longitudinal electric field

will generally change direction during gap cross-

ings. Here the transit time factor approximation

becomes unreliable, and the calculation showsgreater energy gain than is measured. This limita-

tion was explored by comparing the simplified

calculations with a numerical integration of ions

across the gaps. The gap fields for each section of

the linac were calculated in three dimensions using

Simion 3D [12,13]. Although Simion provides

good integration of particle trajectories, including

calculation of RF varying fields, the integrationwas performed once again using an Excel spread-

sheet. This allowed a direct comparison with the

simplified model, and allowed the same generally

accessible format to be used. The longitudinal gap

field was tabulated, and a macrofunction was

written to carry out a fourth order Runge–Kutta

integration using interpolated field values and a

5 ns fixed step size.Fig. 14 compares the simple transit time factor-

based calculation with the numerical integration

for the cases of Pþþþ and Asþþþ being accelerated

by the first section only. Except at the lowest en-

ergy end, the Pþþþ spectra are very similar, and the

peak energy at 900 keV is close to the measured

value of 925 keV. The difference between the cal-

culations does become apparent for Asþþþ, how-ever, and the measured maximum energy of 400

keV is best predicted by the numerical integration.

It is instructive to compare the calculation times

under these equivalent conditions, on a 300 MHz

PC. The simplified calculation for Asþþþ generated

the spectrum for 360 ions in 2 s. The numerical

integration for the same number of ions required

540 s. In both cases the calculations covered all 11RF gaps. More efficient calculations could surely

be implemented for both cases, but a large relative

difference would remain.

The calculations reported here deal only with

longitudinal motion. This has been found to be

sufficient for predicting energy spectra and gener-

ating RF parameter settings, but of course it gives

no guidance on how to set the quadrupole focusingelements, which are wholly concerned with lateral

motion. In practice the quadrupole settings have

been found to be less predictable, with multiple

tuning solutions available, and there are strong

0.00

0.02

0.04

0.06

0.08

0.10

0 100 200 300 400 500 600 700 800 900 1000Energy (keV)

Rel

ativ

e cu

rrent

0.00

0.02

0.04

0.06

0.08

0.10

0 50 100 150 200 250 300 350 400 450 500Energy (keV)

Rel

ativ

e cu

rrent

Fig. 14. Comparison of the simplified energy spectrum calculation (crosses and dotted line) with a numerical integration of the ion

motion in the gap fields (circles and solid line) for Pþþþ (top) and Asþþþ (bottom) accelerating in the first linac section only. The

experimentally measured energies for the peaks at the top of the spectra were 925 keV for Pþþþ and 400 keV for Asþþþ. Ions like 270

keV Asþþþ with gap transit times similar to or longer than the RF half period reveal the limitations of the simplified calculation.


interactions with ion source settings, ion source

condition, and with the required beam character-

istics at the target wafer. A simple scaling of the

quadrupole magnet coil currents with the beam

rigidity,ffiffiffiffiffiffiffiffiffiffi2Um

p=q, is reasonably effective to pro-

vide starting points for optimisation.

A full three dimensional ray-tracing of the fulllinac would be computationally intensive, al-

though well-optimised codes like Turtle are avail-

able [14]. More efficient beam envelope and

centroid tracing using transfer matrix techniques is

available in a code such as Trace 3-D [15], and

further work is proposed to see how well such

codes can predict the parameters for the Swift xE

linac. However the intended users of Turtle, Trace3-D and similar codes come from a different

community, use different computer hardware, and

have a different level of experience in the design

and control of RF accelerators than can be as-

sumed for semiconductor process engineers and

technicians. The need would remain for a simple-

to-use, widely available modelling tool for ion

implanters incorporating RF linacs.

10. Conclusions

A simple model of an ion linac for an ion im-

planter can provide non-specialist users with a

widely available and simple to use means to appre-

ciate the interactions of the RF control parameters.

The addition of a physically measurable parameter,

namely the transmission from RF input coupler to

feedback pickup loop, allows the linac electrode

voltages to be estimated, and it is then possible togenerate parameter sets that are a good basis for

final tuning of a specific real system. The combina-

tion of rapid recalculation, with control parameters

and an output display that mimic the real system,

allows experience gained on the model to be trans-


ferred easily to the system. Use of the model has to

be restricted to m=n and injection energy in the

nominal design range of the linac. Numerical inte-

gration of gap crossings overcomes this constraint,but is much less efficient for computation.

The Swift xE implanter linac has fixed RF

phase relationships between the accelerating sec-

tions. This constraint allows the simple linear

relationship between RF amplitude parameters

and final beam energy to be exploited if required,

for example in the automatic generation of new

parameter sets by the implanter control system.The challenge of providing a model that also

gives reliable prediction of quadrupole settings,

but with the same computational efficiency and

ease of use, remains.

References

[1] H.F. Glavish, Nucl. Instr. and Meth. B 21 (1987) 218.

[2] H.F. Glavish, D. Bernhardt, P. Boisseau, B. Libby, G.

Simcox, A.S. Denholm, Nucl. Instr. and Meth. B 21 (1987)

264.

[3] H.F. Glavish, Nucl. Instr. and Meth. B 24–25 (1987) 771.

[4] P. Boisseau, A.S. Denholm, H.F. Glavish, G. Simcox,

Nucl. Instr. and Meth. B 21 (1989) 223.

[5] C.K.C. Jen, J. Gordon, Semicond. Fabtech 13 (2001)

279.

[6] H.F. Glavish, J.S. Gordon, US Patent 6,423,976, July

2002.

[7] W.K.H. Panofsky, Linear Accelerator Beam Dynamics,

University of California Radiation Laboratory Report

UCRL-1216, February 1951.

[8] H.F. Glavish, C.K.C. Jen, International Patent Applica-

tion WO 02/33764 A2, April 2002.

[9] H.F. Glavish, personal communication.

[10] D. Sing, M. Rendon, in: Proceedings of the 13th Interna-

tional Conference on Ion Implantation, 2000, p. 400.

[11] M. Tsukihara, H. Kariya, in: Proceedings of the 12th

International Conference on Ion Implantation, 1998, p.

330.

[12] D.A. Dahl, Simion 3D Version 7.0, Idaho National

Engineering and Environmental Laboratory, 2000.

[13] C. Christou, J. Gordon, Nucl. Instr. and Meth. B 188

(2002) 251.

[14] D.C. Carey, K.L. Brown, C. Iselin, TURTLE with MAD

input and DECAY TURTLE, FERMILAB-99/232, Sep-

tember 1999.

[15] K.R. Crandall, D.P. Rusthoi, LA-UR-97-886, Trace 3-D

Documentation, Los Alamos National Laboratory Report,

May 1997.

Simplified modelling of an ion linac used for ion implantation

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