Electrostatic ion optics and beam transport for ion implantation

Nuclear Instruments and Methods 189 (1981) 71---91 North-Holland Publishing Company

71

ELECTROSTATIC ION OPTICS AND BEAM TRANSPORT FOR ION IMPLANTATION J.D. LARSON 10011 E. 35th St. Terr., Independence, MO 64052, U.S.A.

Ion optical properties of electric and magnetic elements used for beam transport are reviewed. Devices are categorized according to predominantly transverse fields (found in quadrupole lenses, deflectors, velocity filters, etc.) and predominantly longitudinal fields (found in accelerators, axially symmetric lenses, etc.). Comparisons are made in the general context of systematic beam transport analysis using matrix methods. Emphasis is placed on perceiving the entire system as a coherent entity. Brief consideration is given to nonlinear effects or aberrations that adversely modify the beam.

1. Introduction

Ion optics derives its name and many analytic techniques from the parent study of light optics. The closest analogy remains in the field of electron microscopy which laid the foundation for optical imaging of charged particles using primarily axially symmetric or "round" electric and magnetic lenses [ 1 - 3 ] . Particle accelerator technology has broadened ion optics to encompass the concept of beam transport [4,5] wherein the emphasis shifts away from reproducing faithful images toward constraining beams of charged particles to remain within specified boundaries. In either case, the properties of ion optical systems are ascertained by studying the transit of individual particles or rays through regions containing electromagnetic fields. To lowest order, particle trajectories vary linearly with respect to initial position and angle. Equations of linear motion may be processed conveniently by hand or by computer using standard matrix methods reviewed in the following section. Analytic solutions to the equations of motion exist for a few idealized devices, including dipole deflectors and quadrupole lenses, that apply fields transverse to the path of the beam [6]. Very little analytic information is available to describe devices such as electrostatic gap lenses and einzel lenses that impose longitudinal fields although detailed calculations for many specific geometries have been published [3,7,8]. Requirements for ion implantation cover a relatively modest range of energies but a much broader range of currents and a generous spectrum of ion species. Electrostatic accelerating devices are required to produce the requisite beam energy and to vary the energy. Electric devices also are preferred for focusing and deflecting the beam because they operate in proportion to beam energy and are insensitive to particle mass. Compared to equivalent magnetic devices, electric lenses and deflectors are lighter in weight, lower in power consumption, and responsive to higher frequencies. At high beam currents, however, the advantages of electrostatic devices are overshadowed by the need to maintain space charge neutralization which electric fields tend to disrupt. Consideration then should be given to combining the functions of electric elements in order to minimize exposure of the beam to disruptive fields.

2. Matrix methods First-order beam transport utilizes the linear transport equations t

x = a l l X o +a12xo , X t

= a21xo

+

(1)

s

a22Xo .

where x represents perpendicular displacement and x' = tan a represents angular divergence of a particle or ray at position x having angle a with respect to the beam axis z. Ordinarily, ~ is small (part of the usual paraxial ray 0029-554X/81/0000-0000/$02.50 © 1981 North-Holland

II. ION SOURCES AND BEAMOPTICS

72

.I.D. Larson / Electrostatic ion optics

assumption) so that tan a ~ a. (Throu~out this discussion "divergence", "divergence angle", and "angle" all serve to identify x'.) Similar expressions describe beam displacement y and divergence y' in the plane perpendicular to x; motion in y is presumed to be independent of motion in x permitting trajectories in x and y to be studied separately. The 2 ×2 matrix equivalent to eqs. (1) is

(xx /

t

,a:l

(2)

a:2/\Xo/

Missing from eq. (2) are rows and columns often added to incorporate constant displacement, momentum dispersion, path length, transit time, or other quantities that may be expressed in linear form. Such additions are easily comprehended once the 2 × 2 matrix is mastered. The advantage of matrix representation is the ease with which an overall matrix representing the total beam transport system may be built up by multiplying together separate matrices describing individual components of the system. While studying beam transport matrices it often is helpful to recall the following matrices for a field-free drift space of length l and a thin lens of focal length f drift

lens

Drift spaces preserve angle but change position x from Xo to Xo + lx'o. Thin lenses preserve ray position but change divergence x' from x~ to Xo + xo/f. Note that positive values of f correspond to diverging lenses while negative values of f correspond to converging lenses. Negative values of I are useful artifices for projecting rays backward to a virtual origin but negative distance (analogous to negative passage of time) has no direct physical meaning. Qualitatively, the 2 X 2 matrix relates input to output as follows

I

SIZE-TO-SIZE (magnification)

ANGLE-TO-SIZE (length-like)

SIZE-TO-ANGLE L(focus-like)

"~

f

(4)

ANGLE-TO-ANGLE ] (concentration) .~J

Descriptions in parentheses provide guidance but, as the following paragraph demonstrates, must not be taken literally. Any one element of the matrix may become momentarily zero; for example, matrices for rays passing through the focal point of a thin converging lens are lens

drift

drift

point-to-parallel

lens

parallel-to-point

The resulting matrix conditions for parallelness correspond to rays emitted from a point emerging parallel to each other (a22 = 0) and parallel incident rays converging to a point (al t = 0). Another familiar lens relationship is the general point-to-point focus condition (a12 = 0) drift

(10

lens

drift

I(0

point-to-point

,

provided 1/p + 1/q = l / f . Image inverstion (a11 < 0) and proportional magnification (lall I = q/p) for the thin

£D. Larson/Electrostaticion optics

73

lens are clearly revealed. (Elsewhere in this discussion the term "ibcus" is used more loosely to imply lens adjustment or an effort to minimize a beam spot on target.) The 2 × 2 matrix is adequate to encompass all the standard asymptotic focal properties of a lens including object and image focal lengths (fl and f2), midfocal lengths or focal points (Fx and/72) and derived principal planes in the form [9]

( F2/f2 -l/f~

(FiF2- f, f2)/f2 ),

A=-fl/f2

,

(8)

-r~/f2

where quantities measured toward the object (typically fl and F I ) are negative. The lens matrix in eq. (8) occupies zero length and produces an impulse at the reference plane. Reciprocity observed between eqs. (5) and (6) is not accidental. The expression for a 2 × 2 beam transport matrix used in the reverse direction is [9] (b,l

b21

~x222' =1(a22

ax2), ) A~,a21 a11:

(9)

where matrix A is the forward directed matrix. Reverse matrix B is constructed from the inverse of A but with signs of off-diagonal elements reversed to account for reversal in the sense of divergence angles for rays travelling in the opposite direction. For beam transport components that preserve phase space area (normalized by the square root of energy) the determinant A = ( a l 1 a 2 2 - a 12a21) = ( V l / U 2 ) 1 / 2 ~ ,),2 where qUl and q U2 are beam energies preceding and following the matrix, respectively. Each particle in a beam has position and divergence coordinates (x, x') in particle phase space that differ from coordinates of other particles. Measurements reveal that well-compacted beams occupy some non-vanishing area in phase space and that contours of equal intensity often approximate ellipses. The shape and orientation of these phase space ellipses change as the beam moves along the z axis but the energy-normalized area (in the absence of aberration) remains constant (Liouville's theorem). Because of this natural propensity of real beams and because linear beam transport transforms one ellipse into another, whole beams frequently are represented as ellipses in phase space. Although a variety of ellipse representations are possible, the description of an ellipse using two vectors in phase space is particularly easy to use, simple to analyze, and compatible with other linear ray tracing procedures. Two rays, properly chosen, describe an ellipse as follows [10]: ray 1

{e,~ Xt (~))]

e2x

ray 2

beam ellipse

e, l(c°s01 = e : 2 / \ sin ~ I

(10) \sin ~ /

Ray 1 consists of a column vector having displacement el 1 and divergence e21 ; ray 2 consists of column vector e12 and e22. As parameter q5 varies from 0 to 27r, coordinates x(qS) and x'(O) trace out an ellipse in phase space. Properties of the ellipse are contained in the ellipse matrix E which consists of two ordinary phase space vectors that may be transported simultaneously using the matrix product E = A × Eo. If el2 = e21 = 0, the beam ellipse has semi-axes ell and e22 aligned parallel to the coordinate axes. (The center of the ellipse may be offset from the origin of coordinates by adding a displacement column to the beam transport matrix [ 10--12] ; displacements will be considered later in conjunction with deflectors.) Such an upright ellipse corresponds to a minimum in cross section or a beam waist. At a waist, phase space area may be.determined by taking n/4 times the product of minimum beam diameter by maximum included cone angle. The convenient transformation [10]

e~ = 0, e;2 = (e~l + e~2) 1/2 , e'11 = (elle22 - elze21)/(e~l + e~2) w2 ,

(11)

e12 = (eale21 + e12e22)/(e~t +e~2) 1/2 , t

shifts the phase of q~ ensuring e;~ = 0 but does not alter the ellipse. If the ellipse is upright then e'12 = 0; otherII. ION SOURCES AND BEAM OPTICS

74

J.D. Larson I Electrostatic ion optics

wise, L = -e'12/e'22

(12)

,

is the drift distance required to turn the ellipse upright. The maximum projections of the ellipse onto the coordinate axes are Xmax = (e211 + e~2) 1/2 ,

(13)

t

Xmax = (e~l + e~2) 1/2 ,

(14)

where Xmax corresponds to the outer envelope of the beam. Computer programs may be implemented to seek a variety of different beam conditions including focus (a12 = 0), parallelness (air = 0 or a22 = 0), waist (e'12 = 0), minimum size [minimize eq. (13)], minimum anne [minimize eq. (1 4)], specified size [eq. (1 3)], specified angle [eq. (14)], etc., all conveniently accessible using the 2 X2 transport matrix and two rays. (The parameter ¢ is suppressed during calculations and invoked only as needed to actually draw an ellipse.) Analyses based on transport o f a single ray extend in an obvious way to the whole beam. Each component along the beam line restricts the region of available phase space acceptance to an area [12J (15)

A12 = 7rrl r2/lal 21 ,

where r~ and r2 are radii of entrance and exit apertures. Element a~2 comes from the 2 × 2 transport matrix between apertures. The complementary emittance is AA 12, where A is the determinant. For example, a drift space [eq. (3)] bounded by limiting apertures forms a collimator having a12 = L and A 12 = .,rrl r2/L. Note that a collimator cannot transport beams outside its acceptance area no matter how carefully focused. Multiple apertures within a system require separate analyses to determine which pair are most restrictive: usually only a limited number o f apertures participate in limiting phase space area. Increases in phase space area of the beam by aberration, e.g. caused by lens distortions or space charge, represent not only local enlargement of the beam envelope but portend deterioration in quality that prevents the abcrrated beam from passing through the same limiting collimator as before aberration. 3. Transverse field devices Using suitable electric and/or magnetic pole configurations it is possible to generate electromagnetic forces that vary linearly with position transverse to the beam. In the usual non-relativistic approximation, particle motions in such fields vary as = ~ +_ co2x ,

(16)

where x is perpendicular to beam direction z. Constant transverse acceleration (8) steers or deflects the beam as a whole. Linearly changing acceleration acts as a lens causing either convergent (-co2x) or divergent (+co2x) focusing of the beam. Solutions of eq. (16) for convergent focus t ~ e the form x =Xo cos cot

+

t

•

X o ( ~ / w ) sin w t

--

(8/co2)(cos c o t - 1)

,

x' = -Xo(co/k) sin cot ÷ Xo cos cot + (~/co2)(co/2) sin cot ;

(17)

and for divergent focus x = Xo cosh cot + Xo(~/co) sinh cot + (~/co2)(cosh cot - 1), t

t

x = Xo(co/~) sinh cot + Xo cosh cot + (~/co2)(co/:?) sinh c o t ,

(18)

where t = time, k = longitudinal velocity, Xo = initial transverse position of particle, Xo = xo/zo = initial divergence of particle. 3.1. Quadrupole lenses

The idealized quadrupole singlet lens element consists of tbur opposed, alternately polarized hyperbolic pole tips aligned parallel to the beam axis. The quadrupole lens converges (C) in one plane and simultaneously diverges

J.D. Larson / Electrostatic ion optics

75

(D) in the orthogonal plane. Particle motions in a pure quadrupole field are described by eqs. (17) and (18) with = 0, or by the matrices cos0

-(O/L) sin 0

(L/O) sinO) cos 0

c '

/cosh 0 ~(O/L) sinh 0

(L/O)sinhO) cosh 0

(19) D '

where L = i t = effective length of lens, 0 = cot = coL/z = coL/(2qU/m) 1/2 , m = mass, qU = kinetic energy (KE). The effective length includes typically about one bore radius added to the physical pole length; exact values depend on pole geometry and neighboring structures [13]. Closer modeling becomes necessary only for very precise beam control (e.g. in spectrometers or microprobes). The dimensionless parameter 0 provides a geometric description of beam trajectories through quadrupole elements. Finding values of 0 that satisfy conditions of beam magnification, waist, focus, etc. is one of the principal tasks performed by beam transport computer programs. Once a particular geometric solution is obtained, it may be applied to particles of different energy, charge, and mass and to both electric and magnetic lenses. For the electric quadrupole lens co2x = (q/m)E(x); it is convenient to let

E(x) = (m/q) co2x = (2Va/a 2) x ,

(20)

so that Va = foaE(x) dx = potential at distance a; consequently,

0 E = coL/~ = L [(q/m) (2 Va/a 2)/(2q U/m)] 1/2,

(2 l)

= (L/a)(Va/U) 1/2 . Note that E(a) = 2 Va/a; that is, the electric field at distance a (e.g. at a pole tip) is twice the average field Va/a. Also, for a fixed gradient (e.g. fixed focal length), 2Va/a 2 remains constant; consequently, the electrode potential Va increases as a 2 . In electric fields the particle kinetic energy depends on position within the potential distribution. The quadrupole electric field increases linearly from the axis [eq. (20)] so the potential changes quadratically causing a nonlinear change o f particle speed with position and contributing to aberration in electric quadrupole lenses. For the magnetic quadrupole lens co2x = -(q/m)z, By(x); therefore, By cx x, Bx c~y and magnetic gradients in x and y must be held constant. Then

co2 = (q/m ) ~,G ,

(22)

and

OB = coL/~, = L [(q/m) G/(ZqU/m) 1/2 ] 1/2 ,

(23)

= L[ G/(2mU/q)I/2] 1/2 , where G = (dBx/dy) = - ( d B y / d x ) = constant magnetic gradient and (2mU/q) x/2 is the magnetic rigidity Bp (containing the "mass-energy-product" expression mqU/q 2). As already noted [eq. (19)], the quadrupole singlet lens always diverges in one plane. Suppose, however, that two sin~et lenses are placed sequentially along the beam path to form a doublet lens structure. If polarities are opposite in the two lenses then in one plane convergence is followed by divergence (CD) and in the orthogonal plane divergence is followed by convergence (DC). Matrices for the quadrupole doublet are [from eqs. (19)] CD

={cosh 0 ~(O/L) sinh 0

(L/O)sinhO)(10 cosh 0

l'~[cosO 1]\-(O/L) sin 0

(L/O) sinO) cos 0

(24) '

where l represents some (usually small) separation between lenses and (for purposes o f illustration only) quadrupole singlet elements here have equal lengths L and equal strengths 0. Multiplying out eq. (24) and invoking eq. (9) for the DC plane yields the following matrix elements for the doublet: cdl I = dc22 = cos 0 cosh 0 - sin 0 sinh 0 - O(l/L) sin 0 cosh 0 , cda2 = dc12 = (L/O)[cos 0 sinh 0 + sin 0 cosh 0] + l cos 0 cosh 0 ,

(25)

Cd=l = dc2a = (O/L)[cos 0 sinh 0 - sin 0 cosh 0 - O(I/L) sin 0 silth O] , cd== = dclx = cos 0 cosh 0 + sin 0 sinh 0 + O(I/L) cos 0 sinh 0 . II. ION SOURCES AND BEAM OPTICS

J.D. I_arson/ Electrostatic ion optics

76

Although 0 is not necessarily small in practice (nor equal in both sin~ets as assumed above) it is instructive to expand these expressions in powers of 0 to obtain the following approximate matrices for the doublet l-V02(1 +l/L) 2L +l )

-(04/[,)(2/3 + l/L)

1 -+ 02(1 + l/L)

'

(26)

where the upper signs correspond to CD and lower signs to DC polarities, respectively. Several properties of quadrupole doublet lenses are now evident. First, length-like terms cdl2 and dc12 are equal and, for small 0, close to the overall physical length 2L +/. Second, when both elements have the same length and strength the focus-like terms cd21 = dc=l ~ l / f are the same in both planes. (For optimum lens adjustment 0 usually differs by small but significant amounts for each element.) Third, for reasonable values of 0 the focal length is negative [eq. (26)]; consequently, convergence in both planes is possible using alternating gradient quadrupole fields. Fourth, increasing the separation length l in cd=l makes the 1/f term more negative and thus shortens the focal length. Beware, however, that increasing singlet separation in order to shorten focal length alters the ratio of magnifications in x and y and tends to increase beam size in the downstream element; careful analysis of the consequence 'always is advisable. Fifth, magnificatioMike terms cdl~ and dcla differ in x and y for the doublet lens so that initially circular beams become elliptic in cross section. A quadrupole triplet combination comprising three singlet lenses of alternating polarities ameliorates some of the asymmetry caused by a doublet. The popular symmetric triplet lens consists of two doublets placed back-toback. In one plane the triplet matrix [for constant 0, see eqs. (25)] is CDDC

( c d , i c d : 2 + cd,2cd2, = \ 2 edited21

2 cd22cd,:

)

cdllcd2; + cdl2cd21

(27) '

and in the orthogonal plane DCCD = (cd,,cd22 + cd,2cd2a \2 cdz2cd21

2 cdlacdt2

) .

(28)

cdx tcd22 + cdazcd2t

When expanded in 0 the two triplet matrices differ by order 02 only in one term (

1

-2(04/L)(2/3 +ILL)

2(2L +l)[1-+02(1 1

+I/L)]) '

(29)

where the + sign corresponds to CDDC and the - sign to DCCD polarity. Spatial magnification terms are identical for both planes indicating that triplet lenses tend to preserve circular beams. Ilowever, the angle-to-size terms differ by the ratio (2 cd== cd~ =)/(2 cdx 1 cd 12) = cd22/cdl 1 ; consequently, if angle dominates over size, as may be the case when either object or image distance is very short, then the triplet lens will produce unequal beam envelopes in x andy and will not preserve circular beams. Often the question arises as to whether a doublet or triplet lens should be selected for a given application. The difference lies primarily in magnification. When equal magnifications in x and y are essential then the doublet is excluded and three or more quadrupole elements must be considered. On the other hand, unequal magnifications are not necessarily harmful and often prove quite beneficial either by producing non-circular beana spots when desired or by reducing non-circular beams to more circular shape. A crude estimate of beam asymmetry caused by doublet lenses may be helpful. Let object and image distances each be of order 20L, corresponding to a focal length of 10L. Then from eq. (26), 1/)"~ (2/3) 04/L ~ (1/10L) or 04 ~ 3/20, 02 ~ 3/8, and 0 ~ 5/8. The doublet converts an incident circular beam into an emerging ellipse having axes approximately in proportion [from eqs. (25) and (26)] dc~ 1/cd,, = cd22/cd,1 ~ (1 + 02)/(1 - 02) ~ (1 + 3/8)/(1 - 3/8) ~ 2 .

(30)

This presumes that the beam is small at object and image compared to its dimensions within the lens. The beam from an initially circular object leaves the doublet lens with proportions about 2 : 1. It arrives at the image with proportions about 1 : 2 because beam size leaving the lens determines the maximum angle to the image and the

77

J.D. Larson / Electrostatic ion optics

product of this maximum angle with the minimum spot size on target is constant (from conservation of phase space area). Note that within doublets and triplets the beam expands in the DC plane (e.g. by 40% in the above examplc and occasionally by much more). 3.2. Deflectors

Uniform transverse dipole fields that deflect or steer the beam sideways correspond to (31)

= (q/m)(E - iB),

in eq. (16) with E directed along x and B along y. The familiar solutions for motion in a uniform field for the assumption that co = 0 are + r. + t , X = X o XoZt+½~t2=Xo x°L+½~L2/22 (32) x ' =Xo' + ~t/£ = Xo + ~L/k:

where L = zt is the device length. Particles entering an electrostatic deflector pass through a region of nonuniform fringing fields which decompose into a transverse component that adds effective length to the poles and a longitudinal component that alters the energy of the incident particle as a function of its transverse position. If the fringing field is short compared to L then impulses 6(re.r) and 6(nff) change the incident energy from qUo to qU

=

lr,"/(.72 +

.~2)

:

½m[(ko + 6~) 2 + (ko + 62) 2 ] ~ qUo + m~o 6~,

(33)

provided :~ < < ~. But q U = qUo + qEx, where E x is the potential between deflecting electrodes; therefore, (34)

Zo6Z ~- (q/m) E x .

Intuition suggests 6,~ should be evaluated at z = Zo with x = Xo but if instead 8~ is evaluated with x = Xo + ½Lxo, corresponding to the average transverse coordinate of an undeflected particle, then a symmetric matrix that preserves phase space area is obtained. Therefore, let 1

t

Zo 6~ ~ ( q / m ) E(xo + ~Lxo) ;

(35)

whereupon, kz = ( k o + 5 ~ ) 2 ~ . k g + 2 ( q / m ) E ( x o

+ ~1 L x o, ) ,

1

(36)

t

2 ( q / m ) U[1 + (E/U)(xo + ~-Lxo)] ,

and, from eqs. (31), (32) and (36), (37)

~ 2/~2 = (q/m)(E/k2 _ B/£) L 2 , 1

i

1

t

( q / m ) ( E / U ) L 2/[1 + (E/U)(xo + ~Lxo)] - (2mU/q) - ' / 2 BL 2 /[1 + q:;/U')(Xo + ~Lxo)] x/2,

~ a L 11

1

-

t

( 2 a / L ) ( x o +~Lxo)]

-

13L11

-

(c~/L)(xo +1 Lxo)] ,

( ~ - 2~2) Xo + (}oq~ - a 2 ) L x o + (a - [3)L, where c~ = ~(E/U) L = electric deflection angle,

13= B L / ( 2 m U / q ) 1/2 = magnetic deflection angle ,

(38)

or, in practical units, 13= B (G) L (cm)/{ 144.0x/[M(amu) KE(eV)/Q 2 ] } ,

(39)

where KE is the particle kinetic energy and the mass--energy product under the radical includes Q measured in units of the elementary electron charge. Substitrating ~I~/~ ~ from eq. (37) into eqs. (32) and converting to matrix notation produces the following II. ION SOURCES AND BEAM OPTICS

78

ZD. Larson /Electrostatic ion optics

beam transport matrix equation for a deflector:

IiPl Ii~~a2 1

+goq3

=

(1-~a2+¼043)L

2

- 2a2)/L

½(a-/3)/.

1 - a °" +-1o43

a-j3

0

1

ll~('Xo~ ] "[~ 0•

(40)

The third column in this matrix accommodates displacements of the beam as a whole that arise independently of initial position and divergence. The deflector matrix in eq. (40) may be factored into more easily interpreted components as follows:

I! l, !lli 00 ;Ii 1

If

0

l

a-

0

1

~

1

0

!1 ,

(41)

where f = - L / ( 2 a 2 - a/3) is the thin lens focal length. The drift matrices of half length on either side in eq. (41) convey the beam to and from an impulse matrLx at the center which imparts the deflection angle (a -/3) to the beam as a whole and, when electric fields are present, acts as a (usually week) thin lens. To track the beam inside a deflector it is appropriate to evaluate eqs. (38) and (40) or (41) using intermediate values for L. 3.3. Wien velocity filter

The Wien filter [14] balances crossed electric and magnetic fields transverse to the beam so that particles of one specific velocity are transmitted without deflection. For particles of the same energy the Wien filter has application as a mass separator, l e t opposing electric and magnetic forces cancel in the x plane for some mass me such that [using eqs. (38)] a - ;3 = ~ ( F . I U ) L - B L / ( 2 m o U/q) 1,2

= O.

(42)

When m ~ me the forces no longer balance; therefore, 3 = B L / ( 2 m U/q) 1/2 = B L [ [(2mo U/q) 1/2 (m/mo)a/2 ] , =

a ( m o l m ) '`~ ~ a(l - ½zSan/mo)

(43)

;

whereupon [for use in eq. (41)] a - [3 = a - a ( m o / m ) 1/2 ~ ½ a ( ~ m / m o ) ,

(44)

f = - L / ( 2 a = - o43) = - L I [2a 2 - a2 ( m o / m ) 1/= ] ,

(45)

- ( L l a 2 ) [ 1 - ½(ZDn/mo)] .

When strong fields are present in the Wien filter the lens effect [eq. (45)] is not necessarily small. This is a cylinder lens that converges in one plane to produce a line focus. Quadrupole lenses, one-dimensional einzel lenses, and dipole magnets are among the components capable of correcting the astigmatism of a Wien filter. Approximations used above are invalid if the Wien Filter is long compared to its focal length [eq. (45)], that is, when bfl = L / a : ~ L or a = ½L(E/U) ~> 1 or L >>.2 U / E . For example, if U = 20 kV and E = 4 kV/cm then 2 U / E = 10 cm constitutes a "long" filter. To estimate focal properties of the long Wien filter, recourse must be made to eqs. (17) with co2 4: 0. Rewriting eq. (31) with co 4= 0 and using eq. (36) for arbitrary values o f x (with x' ~ 0) provides the approximation - w~x = (q/m)(E - 2B),

or

(46)

( q / m ) { E - £0B[l + ½ ( E I U ) x ] } ,

~ (q/m)(E -ioB) ~ O, co2 .~ ~ ( q l m ) ( E / U ) B£o ~ ~ ( q / m ) ( E 2 / U ) .

(47)

J.D. Larson /Electrostatic ion optics

79

Particle motions [eq. (17)] become approximate sinusoids about curves traced by particles incident on axis with Xo = 0. Over one full period ~ r = 2n, corresponding to a wavelength X = -~r = 2rr~/co ,

(48)

27r(2q U/nO '/= / [~ (q/m ) (E=/ U) I 1: = ~ 4n( U/E) . For a nearly parallel incident beam the first line focus occurs at quarter wavelength or at X/4 ~ n (U/E). Previously, the impulse approximation implied that a long f'dter commenced when L ~ f ~ 2(U/E). That approximation predicts a parallel incident beam will incur a line focus at distance ½L + f ~ 3(U/E) ~ n (U/E), showing that for L <~ 2(U/E) the impulse approximation performs reasonably well. To circumvent intrinsic lens effects it is tempting to operate the Wien filter (or other dipole dispersing component) with a beam waist in the center so that the emerging beam envelope is a mirror image of the incident beam. For example, let the Filter described by eqs. (41), (44) and (45) be followed by an arbitrary, non-dispersing matrix A so that from the center of the filter

I !II lli 21

a22

1

o~-{3 =

0

0

1

2x +a22/f

a22

a22(o~-{3)

0

1

,

(49)

where drift of ½L from eq. (41) has been absorbed in A. Now use a lens of any kind to focus the beam on analyzing slits, causing a12 ~ 0 [see eq. (7)]. Unfortunately, if a12 = 0 then ax2(a -/3) = 0 and the system from filter to slits is dispersionless. The lens collects particles having different velocities fanning out from the center of the filter and (ignoring lens dispersion) directs them all to the center of the analyzing slits. For maximum resolution, the beam within an analyzer should be as large and nearly parallel as possible and lenses should be as close as possible to the virtual pivot from which the beam appears to disperse. 3.4. Space charge lens The space charge lens [15] is a crossed electric and magnetic field device in which a solenoidal magnetic field traps a space charge cloud that (when saturated) produces an electric field everywhere perpendicular to the magnetic field. Prior to saturation, plasma currents produced by beam ionization or by other sources may flow parallel to magnetic field lines to fill tile lens with charge. Under saturated condition coaxial tubes of magnetic flux become electric equipotential surfaces. Control of the space charge distribution within the lens is attained by grading the potential of coaxial, boundary electrodes (preferably located away from the beam in the magnetic fringing field) to conform to the electric potential distribution desired for magnetic field lines that intercept the boundary electrodes. A lens of good quality results when the electric potential is graded such that angular changes imparted to the beam remain proportional to displacement from the axis. Consider an idealized, axially symmetric space charge lens which contains a long central region of uniform solenoidal magnetic field bounded by short fringing fields at entrance and exit that contribute some small amount to the effective length and modulate particle energies to conform to the interior potential distribution but otherwise may be ignored. Equations of motion for the idealized transverse-field lens in both x and y planes are described either by eqs. (18) [matrix D of eq. (19)], if the trapped charge has the same sign as the beam, or by eqs. (17) [matrix C of eq. (19)], if space charge and beam have opposite signs. For the space charge lens a quadratic potential distribution will not be assumed (because this lens geometry does not dictate any particular potential distribution) and instead the ideal electric potential and field will be derived. Using material from eqs. (19), (20) and (36), (co/i) = OIL = const., Er = O~r)/Or = (m/q)w2r, and z = 2(q/m)U[1 + (E/U)r] ; therefore, (w/~) 2 = [ (q/m)(Oq~/Or)(1/r)]/[(2qU/m)(1 + (~/U)] = (O/L) 2 ;

(50)

rearranging and integrating, f ( 1 + O/U) -1 dq~ = .f2U(O/L) 2 r dr ;

(51) II. ION SOURCES AND BEAM OPTICS

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J.D. Larson / Electrostatic ion optics

hence, cp(r) = U[exp(O2r: /L 2) --1] = U(Or/L ) 2 [1 + ½(Or/L) 2 + ...] ,

(52)

Er = 2rU(O/L) ~ exp(O%2/L 2) = (2U/r)(Or/L) 2 [1 + (Or/L) 2 + ...] .

(53)

The radial potential distribution 4~r) scales with U (as required) and is nearly quadratic provided r < < L . The lens strength parameter 0 ~ ( L / f ) t/2 is determined by system geometry and beana focus conditions but does not change for different particles nor for different energies. A long lens graded quadratically is reasonably universal whereas a short lens used with varying 0 conditions requires corrections in grading for each application. Substituting 0 E from eq. (21) (for quadratic potentials) into element c2~ of eq. (19) yields the very approximate focal length (for 0F < < 1) f ~ -L/(O sin 0) ~ - L / O ~ ..... (U/Va)(a2/L ) ,

(54)

where Va = UO~(a/L) 2 is the potential at radius a. Eq. (54), derived previously [15] from an impulse approximation, has rather limited application; the complete matrix [eq. (19)] is preferable because 0 is not likely to be small for space charge lenses. The magnetic field must be strong enough to confhle trapped particles to small orbits [15]. Using eq. (38), let 13>~ 1 for trapped electrons traversing some small distance 6r, then = B 6r/(2meO/e) ~/2 > 1 ,

(55)

or, substituting ~ ~ U(Or/L) 2 from eq. (52) and L ~.f02 from eq. (54), B > (2meU/e)l/2/[(6r/r) Of] ,

B ( G ) >~ 3.4x/[KE(eV)/Q]/[(gr/r) 0f(cm)] ,

(56)

where quantities under the radical refer to the beam kinetic energy in eV and beam charge in elementary units. Typically, 0 ~ 1/3 to 1 and 6r/r might be 10 -~ to 10 -2 depending on engineering considerations and on how precisely the lens potential is graded.

4. L o n g i t u d i n a l field d e v i c e s

Electric or magnetic pole configurations that change polarity along the path of the beam produce fields containing predominantly longitudinal components. Nevertheless, in nearly every application the most important optical effects result from radial field components that arise in the fringing field region where the longitudinal component is changing. Analytic solutions for particle motion in arbitrarily changing fields are not available; therefore, longitudinal field devices are more difficult to analyze than transverse field devices. 4.1. Uniform longitudinal fields

Longitudinal electric fields provide particle acceleration and deceleration. Within a uniform field, particles follow parabolic trajectories according to

(5 7)

: (q/m)E,

where E = [ U ( z ) -

Uo(zo)]/L represents

the z component of uniform electric field. Integrating eq. (57) gives

2 = to + ( q / m ) Et = ( 2 q U o / m ) 1/2 + ( q / m ) E t ,

(58)

= (2qUire) 1/2 ;

therefore, tot = ~

to (U s/2 _. U~/2)/[(q/m)l/2E] ,

= 2 L U ~ / : / ( U ~/2 + U~/:) = 2L/(R + 1),

(59)

J.D. 1,arson / Electrostatic ion optics

81

where R = (U/Uo) 1/2 = Q/~?0) = velocity ratio.

(60)

In the x direction the transverse velocity 97 is not changed; therefore, :? = 20 and t

.

r

X = X o +Sct=Xo + X o Z t = X o + 2 x o L / ( R + I ) , X

~

= x. o / z

.

= x o~ ( C• o ~ z. ) = x ot / R

(61)

,

producing the accelerator matrix [16] (10

2L/(R +

,

A = 1/R .

(62)

The velocity ratio R varies from R <: 1 for decelerating systems through R = 1 for no acceleration to R > 10 for the Van de Graaff type research accelerator. When R > 1 the accelerator resembles a drift space of foreshortened length that contracts an~es by 1/R. 4.2. N o n u n i f o r m longitudhml fields

Quantitative descriptions of longitudinal field lenses have received little generalization. Published data in tabular and graphical form abound for specific lens geometries, e.g. refs. 3,7, but interrelationships among different geometries and different operating conditions are not yet conveniently represented. Nearly "all studies now employ computer-assisted numerical calculations using two basic steps. First, the potential distribution is established for the region of interest and second, rays are traced through the potential using numeric integration to obtain the traditional focal properties of the device. Analytic potential distributions exist only for the most simple geometries [2,3], notably the two-cylinder gap lens that arises from a step change in voltage between abutting cylinders of equal diameter and the aperture lens that arises from a step change in gradient on each side of an aperture penetrating a thin conducting sheet that separates two regions of different gradient. Even in these cases detailed focal properties are obtained by numeric integration although some analytic asymptotic approximations do exist. Where longitudinal fields change in free space, radical components that arise act on the beam like a lens. A power series expansion of the electric potential about an axis of symmetric in free space takes the form, for rectangular and cylindrical coordinates [2,3], U(x, z ) = U ( z ) - ½ U " ( z ) x 2 +~-~4 U""(z) x4 - . . . ,

(63)

U(r, z ) = U ( z ) - 1 U " ( z ) r 2 + ~4 U""(z) r 4 - ...,

where primes represent differentiation with respect to z and odd terms are excluded by reflection symmetry about x = 0 and r = 0. To lowest order, the transverse force is proportional to displacement from the axis; that is, E ( x ) = OU(x, z ) / ~ x = - u " (z ) x + ~ u ' " ' (z ) x 3 - ...,

(64)

E ( r ) = OU(r, z )/3r . . . . ½U" (z ) r + ~ U'"' (z ) ra - ....

Leading terms proportional to x and r, respectively, assure lens-like behavior evelb'where along the z axis. Cubic and higher order terms in x and r represent aberrations. Paraxial rays experience nearly ideal lens behavior whereas rays that approach the driving electrodes usually suffer substantial aberration. The I : 2 ratio between rectangular and cylindrical focal lengths evident in eqs. (64) is characteristic of the difference between elongated (slot shaped) apertures and axially symmetric (circular or round) apertures. 4.2.1. A p e r t u r e lenses

An aperture lens is created by opening an aperture through a thin plane electrode separating two regions of uniform electric field directed normal to the plane. Aperture lenses occur at the entrance and exit openings into an otherwise uniformly graded accelerator tube. A simple impulse approximation [17] provides a useful estimate II. ION SOURCES AND BEAM OPTICS

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£D. Larson /Electrostatic ion optics

of the focal length for a circular aperture, (65)

f = 4U/(E~ - E 2 ) ,

and for a slot aperture, f i = 2U/(E1 - E2),

fe = ,,o,

(66)

where U is the aperture electrode potential ( q U = particle energy) and (E 1 -- Ez) is the step change in field along z. Note that the focal length is independent of aperture dimensions. A plane grid mesh covering the opening cancels the macroscopic lens effect but small facet lenses formed by openings in the mesh retain the same focal length and cause aberration. Eqs. (65) and (66) underestimate focal lengths [1] when particle energies change by factors of two or more within distances comparable to aperture dimensions. The converging entrance lens to an accelerator is strong and optically very important whereas the diverging exit lens is relatively weak. For most analyses, eq. (62) combined with eq. (65) (at each cnd) provides an adequate description for the uniformly graded open accelerator tube in the form

A =

1[ f

( 1 -½[R

0

1/R

1/F

(67)

'

- 11

2L/(R + l)

= \ ( 2 R 2 - 2 - [R 2 - 1][3R - 1 ] ) / ( 8 L R 2)

]

(3R - I)/(2R2)/ '

where F(entrance) =--4UIL/(U2 - U1) = - [ 4 L / ( R 2 - 1)] , /(exit) = 4 U 2 L / ( U 2 - U~) = 4 L R 2 / ( R 2 - 1),

(68)

R = ( U 2 / U ~ ) 1/2 ,

where /-/2 - UI is the voltage across the accelerator. Terms within square brackets are contributions from the entrance lens. 4.2.2. Single-gap lenses

A change in potential across a gap between electrodes surrounding tile beam creates the most elementary electrostatic lens. Equipotential surfaces encountered by the beam are convex upstream of the gap and concave downstream creating two "semi-lenses", one converging and the other diver~ng. Whether the change of potential accelerates or decelerates the beam, particles spend more time in the converging part of the lens so that the overall effect is convergence in either case. There are unlimited ways to configure this simplest of lenses. Perhaps the most diligently studied geometry consists of two right circular cylinders of equal diameter for which an exact potential distribution in the form of a series expansion of Bessel functions can be derived provided the gap is infinitesimally small [2,3]. This exact solution is a convenient reference with which to judge the accuracy of various numerical approaches. Focal properties are usually obtained by numerical integration of ray trajectories regardless of how the potential distribution is acquired. Some useful analytic approximations for focal lengths are based on simplified analytic models of the axial potential distribution [2] ; these offer considerable insight but are limited to the fidelity of the model. Perhaps the most interesting new development regarding axially symmetric electrostatic lenses is the observation [9] that over an exceptionally wide range of operating voltages, matrix elements [eq. (8)] for the gap lens are better behaved than traditional focal properties [10]. DiChio et al. [9] offer the following asymptotic approximation for the weak two-cylinder, 0.1D gap lens: (ala a21

a~2t = ( 7 a22/

k l ( 1 - 7)2 D )

-k2(1 - 7)2//)

(69)

7

where = (U,lU~) ''4 ,

k, = 0.12308,

ks = 2.5943,

(70)

J.D. Larson /Electrostatic ion optics

83

for a lens of diameter D. (Curiously, [~k2 - 1] = [~/"2 + l ] k l . ) T h i s approach may be pursued profitably to higher order. For example, the following expressions reproduce the tabulations [9] out to U2/UI = 2, are better (<0.2% error in focal lengths) than many lens approximation formulae out to U2/U~ = 10, and retain some utility (~ 10% errors) out to U2/UI = 40, beyond which errors become prohibitive: axx = 3' - (1 - ,,/)3 _ kl k2(1

-

-r)4/'y +

(kt k2 + ½k2)(1

-

-r)7/7,

a22 = V + (1 - 7) 3 -ffk2(1 - 7)7/7,

a,2/D =

k,(1

- ~,)2 + ½(1 - ,,/)4 + k,(l

(71) - ,¥)s/q,,

a 2 1 D = - k 2 ( l - 7 ) 2 +2(1 - 7 ) 4 +(1 - 7)s/7. Terms of higher order added here to eq. (70) are inexact and speculative at best but the accuracy of the leading terms encourages hope that a unique expression might be discovered. Needed for this purpose are precision calculations of matrix elements based on the analytic potential distribution for the zero-gap lens. A second parameter, in addition to 7, is required to accomodate a finite gap. Other properties of interest, especially cylinders of different diameter, could also be systematically parameterized. The determinant constraint [alia22 - a~2a21 = 72 , partly achieved in eqs. (71)] helps in choosing among possible higher order terms. Although this approach is tedious, endless repetition of some lens calculations could be eliminated.

4.2.3. Multi-gap and einzel lenses The single-gap lens necessarily changes the beam energy and produces a focus only at one ratio of initial to final energies. These limitations may be avoided by using two or more gaps in the lens. Additional gaps also facilitate a reduction in spherical aberration [7,18]. When multiple gaps are close together (less than about 2D), their mutual interaction on the potential distribution must be taken into account [19] and elaborate calculations performed. A few geometries have been studied in detail with attention again lavished on right circular cylinders [3,7]. Some recent studies [20,21] have included fitting of arbitrarily chosen exponential and polynomial expansions to focal properties and aberration coefficients. This method bridges the gap between discrete calculations at different energies and geometries, affords substantial savings in calculational effort, and assists in lens optimization but suffers from lack of any physical meaning in the functions chosen. Often it is desirable, or mandatory, for the beam to enter and leave a lens at the same energy. The resulting equipotential einzel lens ordinarily has one central electrode that may be biased to accelerate or decelerate the incident beam. For a given applied voltage the decelerating mode produces stronger focusing and offers the convenience that voltages on the central element can be designed to be about equal to the source potential. Unfortunately, the decelerating mode also dilates the beam to larger size within the lens causing geometric aberrations that may be an order of magnitude worse than in the accelerating mode [19]. When individual gaps are separated sufficiently (usually by at least two diameters) to be electrostatically independent, the resulting compound lens system may be analyzed as multiple independent single gap lenses [19,21, 22]. Two single gap lenses placed back-to-back and separated by distance l = LID >~ 2 approximate the uniform diameter einzel lens as [using eq. (9)]

b21

b22

't(

1/\a21

az2:

\2a~laz~ +la~

axla22 +a~2a2~ +lazxaz2

1

where atj approximated by eqs. (71) are adequate for most applications. [Eqs. (71) and (72)apply for the accelerating mode; decelerating mode corresponds to interchanging all and a22 on the right side of eq. (72).] Eq. (72) includes a finite insertion length l which may be eliminated, leaving only an impulse matrix at the lens center, by multiplying left and right by drifts of length -½l, as done in eq. (41); however, no obvious simplification occurs in this case.

4.2.4. Grids on lenses Electrostatic lenses sometimes have one or more plane or curved wire mesh or etched grid structures stretched across the beam path. Grids may be used to shape fields, increase lens strength, isolate lens fields from nearby II. ION SOURCES AND BEAM OPTICS

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J.D. Larson /Electrostatic ion optics

components, and reduce aberrations of one kind at the expense of another [23]. Grids also allow construction of a net diverging lens by elimination of converging semi-lenses normally present. For example, a grid introduced along the equipotential midplane of a symmetric gap lens divides the lens electrically into left and right semilenses, one converging and the other diverging. Either half may be discarded or reversed in polarity to yield a much stronger converging or diverging lens. In some instances a control grid is contoured to conform to an anticipated equipotential surface so that as long as the grid potential remains near the nominal potential for that location very little aberration results. By controlling the field curvature a grid may reduce aberration for the whole lens but the multifaceted grid openings create small aperture lenses that partition the beam and create new forms of aberration. Grids always intercept some part of the beam (typically 5-10%), are subject to erosion especially by heavy ions, and introduce secondary particles that may be troublesome.

5. Beam-dependent processes 5.1. Space charge

The self-interaction of the charge carried by the beam has two distinct consequences. One is a dilation of the beam envelope analogous to the action of a continuous diverging lens whose strength depends on the current, velocity, and radius of the beam. The other is dilation of the phase space area of the beam, an aberration that cannot be reversed by external lenses. Tile diverging lens effect is often crudely analyzed to "first order" by considering radial forces (for a circular beam) acting on a sample particle residing at the surface of an axially symmetric cylinder of charge. The radial electric field is [3] E(r) = I(r)/[27reor(2q U/m)a/2 ] ,

(73)

where fir) is the current flowing inside a cylinder of radius r. The potential difference from center to edge of the beam is approximately r

(74)

A U = / E(p) do = ½rE(r), 0 or

A U / U = 6.47 X 10SP,

where

(75)

z

P = (M/Q) 1/2 I/U 3/2 = ion perveance,

M = mass in a m u ,

Q = charge in elementary units,

(76)

and other units are MKS. [Perveance is essentially a constant of the geometry during space-charge-limited extraction from an ion source plasma but elsewhere in the system P can change. Beams that are largely space-charge neutralized exhibit a much reduced effective perveance that perhaps is better expressed by the measurable ratio &U/U. Intense beams have a more complex potential distribution [24] than assumed for eq. (74).] The assumption usually is made that particles within the beam participate in laminar flow so that the beam remains uniform and the motion of one particle on the surface is representative of the whole beam. Calculations based on this assumption provide useful guidelines but are not quantitatively trustworthy. Various "universal" curves have been derived to describe the envelope of an expanding beam both in free space and while undergoing uniform acceleration [25-27]. An approximation for a beam diverging from a waist in free space is [28] z/ro ~ [2.09(r/ro - 1)1/2]/[2(0.325 × 106p) 1/2] ,

(77)

or

r/ro ~ 1 + 0.46(AU/U)(z/ro) 2 ,

(78)

.I.D. Larson / Electrostatic ion optics

85

(for small r/r0) where ro is the initial minimum radius and z is the drift distance. Perhaps of more general utility is the reverse process o f predicting the minimum spot size given initial beam conditions. An approximation to a universal curve [3,29] that predicts the minimum attainable spot size r s on a target at a distance z from an adjustable lens given the initial beam radius r i and perveance P (neglecting phase space area) is as follows: y = In [(106p) 1/2 z/ri] = l n [ 1 . 2 4 3 ( A U / U ) l / 2 / r '] , x = 0.72)'[1 - (y/8 - - y 2 / 5 ) ( 1 -- tanh[5y - 5 ] ) ] ,

(79)

rs/r i = exp(3.65x - 1.53 - 1.98 In cosh x ) , where r' = ri/z is the half-angle of the beam. Eq. (79) differs from the tabulated curve-2 [3] for rs/r i by about +-10% for 5 × 10 -4 (r/ri) or

(rs/ri)(r') 2 (z/A) > 1 , where angle, curves Q = 1,

r is the minimum beam radius expected in the A = rr' = rri/z is the phase space area divided [3,29]. For example, suppose that the intended and A = rr' = 0.5 mm × 20 mrad; then [from eqs.

(80) absence o f space charge, r' = r.u/z is the beam cone halfb y rr, and rs/r i is obtained from eq. (79) or published beam on target has I = 100 p.A, U = 40 k V , M = 64 amu, (75), (76), (79), (80)]

P = (64/1) 1/2 [10"4/(4 X 104) 3/2 ] = 10 -10 ,

(81)

A U / U = 6.47 X 10SP = 6.47 X I0 -s , AU=6.47×

10 - s × 4 x 1 0 4 = 2 . 6 V ,

y = ln[(106 × 10-1°)1/2/0.020] = - 0 . 6 9 3 , x = 0 . 7 2 ( - 0 . 6 9 3 ) [ I -- (--0.0866 -- 0.0961)(2)] = - 0 . 6 8 1 ,

rs/r i = e x p ( - 2 . 4 9 -- 1.53 -- 0.43) = 0.012 , r s = 0.012(0.020z) = 2.4 × 10-'4z,

(rs/ri)(r') 2 (z/A) = (0.012)(0.020) ~ (z/10 - s ) = 0 . 4 8 z . When the drift distance z = 2 m, r s = 0.5 mm which equals the radius attributable to phase space area; therefore, space charge may be expected to dominate this beam for z >~ 2 m. The potential difference o f 2.6 V within the beam is comparable to ion source noise and probably insignificant relative to 40 kV beam energy; nevertheless, 2.6 V corresponds to a divergence angle o f (AU/U) 1/2 = 8 mrad acquired by a particle moving through the space charge potential from the center to the edge o f the beam and 8 mrad is not insignificant compared to an initial 20 mrad. Space charge effects are not limited to changes in the beam envelope. The author has observed qualitatively from computer simulations [30] that initially Gaussian distributions o f randomly sampled particle positions and angles in a beam reconfigure under the influence o f space charge into relatively square distributions (possibly even peaked at the corners). Square distributions presumably arise as each particle attempts to separate itself as far as possible from nearest neighbors, although the calculation used only an axially symmetric central force. Concurrent with redistribution was a steady increase in the widths o f both distributions. Generally, changes in angular distribution occurred at rates comparable to changes in spatial distribution suggesting (qualitatively) that near a waist the phase space area may grow at about the same fractional rates in all directions. Beam-dependent m a t r i x methods that describe gas and foil scattering [ 11 ] probably could be extended to accomodate beam dilation and aberration caused by space charge.

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£D. Larson / Electrostatic ion optics

5.2. Aberration

Aberration in the context of beam transport refers primarily to nonlinear processes that effectively enlarge the phase space area of the beam. Aberration originates from properties of hardware (geometric aberration), from force fluctuations (chromatic aberration), and from microscopic scattering processes (gas/foil scattering and space charge aberration). Microscopic processes are essentially irreversible and can be reduced by minimizing exposure of the beam to the aberrating medium, by making the beam very small in short gas or foil scatterers (because scattering increases angles by fractionally smaller amounts when beam angles have been made large), and by making the beam very large in the presence of space charge. Macroscopic aberrations of geometric or chromatic origin may be minimized (theoretically eliminated in some instances) by meticulous design. A review of aberration is beyond the scope of this discussion and only a few comments will be offered. 5.2.1. Geometric aberration

Geometric aberration in beam transport usually reduces to studies of parasitic and spherical aberration. Parasitic aberration arises from errors of fabrication and alignment; it becomes more significant in lenses assembled from many separate pieces, especially quadrupole lenses that otherwise exhibit intrinsically low aberration. Spherical aberration arises from failure of rays of different angles leaving an object point on the axis to converge to a single point at the image. Displacement errors in the image plane may be expressed in the form (82)

Ar = MCsOta ,

or related expressions [31 ], where M is the lens magnification, c~is the initial ray angle, and C s is the (third-order) spherical aberration coefficient. For a point object a "circle of least confusion" having minimum radius &r/4 occurs ahead of the image plane. Coefficients Cs cannot be deduced from first-order properties and instead are obtained by tracing non-paraxial rays through the potential distribution. Aberration coefficient defined this way change for different object and image distances [31]. Geometric aberration cannot be eliminated from axially symmetric, round lenses [2,3] but combinations of such lenses may achieve substantially lower aberration than any one element used separately [21]. Calculations support the intuitive expectation that complex lens systems having several adjustable parameters provide more opportunity to reduce aberration [7,21 ]. Quadrupole lenses (and other mttltipole configurations) are capable of functioning as "corrector" lenses to cancel third order and possibly higher order aberrations inherent to round lenses [2,32]. Progress in applying corrector lenses is limited mostly by practical difficulties of precisely aligning many sensitively interacting lens components. 5.2.2. Chromatic aberration

Electromagnetic lenses are roughly an order of magnitude more sensitive to variations in beam energy than are glass lenses to variations in photon energy (color). Fortunately, for many applications the beam is effectively monoenergetic making chromatic aberration negligible. Often it is easier to reduce ripple in accelerator and lens power supplies than to reduce chromatic dispersion within these components by comparable amounts. Some residual energy spread in the beam will always persist because of intrinsic energy noise within the ion source. Chromatic aberration may be estimated by calculating changes in particle position 8x and divergence fix' caused by a change in beam energy; that is,

~x'

~Xo/

\fXo] '

where A is the linear beam transport matrix and 8A is the differential matrix for a change in energy. If the initial beam is unperturbed (by including any source perturbations in 8A) then 8xo = 8x~ = 0; eq. (83) then emulates first order beam transport provided the differential product rule 8A = 8(AtA2) = (6A1) A2 + A I ( 6 A 2 ) ,

is observed when compounding matrix A from separate contributing matrices.

(84)

J.D. l~rson / Electrostatic ion optics

87

The strength of electric and magnetic quadrupole lenses varies as [differentiating eqs. (21) and (23)]

oz/u,

aoB/au=-¼oB/u ;

(85)

fiOE = -½0E(fU/U),

50B = -¼0B(fiV/V).

(86)

aoE/au=therefore,

Becausc their chromatic dispersions differ by a factor 2 it is possible to combine electric and magnetic fields working in opposition to cancel chromatic effects [32,33]. For example, let 0 = 0E + 0S, where 0E and 0 B represent contributions from perfectly superposed, crossed electric and magnetic quadrupole fields; then 60 = fi0E + 60B = --½(0E + ~0B) f i g / u ,

(87)

and if (symbolically) 0 E = -~10 B, then 0 = ~0 B and 50 = 0. (Here 0 E < 0 symbolizes electric forces everywhere being directed opposite to magnetic forces.) Lenses exhibiting zero chromatic aberration and negative chromatic aberration (useful for compensating positive aberration from other components) have been constructed and verified [33]. Chromatic properties of an accelerator tube may be studied by differentiating eq. (67) to obtain

[ [QR] 5A = \ - ( [ 3R2 + 3R - 2] - 4)/(8LR 2)

- 2 L R / ( R + 1)2 ] 6R/R -(3R -- 2)/(2R 2) ]

(88)

where aR/n = ½(aU2/U~ - aU1/U1) ,

(89)

and, as before, terms within square brackets come from the entrance lens. An accelerator becomes achromatic when fiUl = fiU2/R 2. If fiU2 is of sufficiently low frequently and measurable then 6U1 could be adjusted continuously using feedback to substantially reduce chromatic aberration from accelerator voltage ripple. The static application of this principle is sometimes called "constant-Q" beam injection, where Q = R 2 .

6. Combined functions 6.1. Combined focus and deflection Superposition of focus and deflection fields helps optimize beam transport, shortens system length, and (for electric components) reduces exposure to space charge perturbation. The easiest method, dipole excitation of conventional quadrupole lenses, cannot be recommended because the resulting deflection field is grossly distorted [34]. Mixed electric and magnetic combinations include electric deflector plates placed within a magnetic lens or (less likely) magnetic deflector poles placed outside an electric lens. For such applications the electric and magnetic effects often may be analyzed independently using eqs. (19) and (40). All-electric or all-magnetic systems of high quality commingle pure dipole with pure quadrupole fields insuring that eqs. (17) and (18) are obeyed. Superpcsed magnetic dipole and quadrupole fields can be created inside a "box" magnet [34] consisting of a rectangular yoke surrounding the beam driven by uniform sheets of current flowing parallel to the beam along the inner surfaces. Dipole fields are produced by currents flowing in opposite directions on opposite sides of the beam. Quadrupole fields are produced by currents flowing in the same direction on opposite sides of the beam but in opposite directions on adjacent sides of the box. One box provides one quadrupole singlet lens element plus two independent, orthogonal (x and y) dipole deflectors. Ferrite yoke materials facilitate high-frequency beam scanning. The electric analog of the box magnet consists of rod-shaped electrodes arrayed parallel to and surrounding the beam. Each electrode is operated at a potential corresponding to superposed transverse dipole and quadrupole potentials. The locus of the array may conform to any convenient potential distribution. Dipole fields are easily II. ION SOURCES AND BEAMOPTICS

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J.D. Larson / Electrostatic ion optics

produced using evenly spaced plane arrays forming the sides of a box. For quadrupole fields, electrodes linearly arrayed in space must be graded quadratically whereas quadrupole equipotentials correspond to hyperbolic shapes. One practical compromise consists of hyperbolic poles composed of several independent rod electrodes that may be operated in unison to produce quadrupole fields or graded linearly in space to produce dipole fields. Two methods appear practical for driving an assembly of electrodes to different potentials. The most flexible method is to drive each electrode from a separate programmable power supply. An alternative method expands the GCA [35] system of transformer-driven scanner plates to include multiple transformer taps connected to corresponding electrode segments so that ac signals produce linear dipole fields while dc bias applied to hyperbolic electrode arrays sustains an underlying quadrupole field.

6.2. Combined accelerator and transverse-field lens Electric or magnetic quadrupole fields may be superposed on a uniform field accelerator. Converging and diverging matrices for this combination are obtained by substituting into eq. (16) the conditions that ~ = 0, ~ot = 2L/(R + 1) [from eq. (59)], ~ = ~oR [from eq. (60)], and 0 = cot to yield cos0 )((L/O)[2/(R+l)]sinO)c -(O/L)[(R + 1)/(2R)] sin 0 (l/R) cos 0 cosh 0

(O/L)[(R + 1)/(2R)] sinh 0

(L/O)[2/(R + 1)l sinh O) (l/R) cosh 0 D"

(90)

(91)

Eqs. (90) and (91) provide a possible cosmetic correction which smoothly joins together incoming and outgoing ray trajectories that otherwise change discontinuously at an impulse matrix. For example, the impulse matrix [eqs. (8) or (71)] and some short drift space (~D) on each side of a gap lens could be replaced by a numerically equivalent product of eqs. (90) and (9 l) to simulate semi-lenses on each side of the gap.

7. Systems Ion implantation systems employ a variety of beam handling components to convey selected particles from ion source to target. The ion optics for most of these components has been reviewed in this and companion papers. Equations of particle motion for some components are formally well developed while for others little more than numerical tabulations exist. The matrix method of beam transport combines linear and quasi-linear contributions from individual components into a cohesive first-order description of the system as a whole. Matrix methods add nothing new to the physics but their comprehensiveness sustains a systematic rigor that may escape piecemeal analysis. Linear matrix methods permit changes in the beam or in the hardware at one location to be easily evaluated throughout the rest of the system. For example, changes in the beam at the ion source may be projected forward to the target; conversely, changes observed or desired at the target may be projected back to the source or to any intermediate location in between. One efficacious procedure is to project an initial beam forward and final beam backward until a location is found where a lens (or other component) properly joins the two. Transport mismatches and other weaknesses are revealed when, for example, the beam is too large within some component or the optimum lens position shifts rapidly for different beam conditions. A single lens between two intended waists in a beam, e.g. between source and mass analysis slits or between analysis slits and target, contributes only one adjustable parameter (lens strength) and that adjustment already is committed to producing a beam minimum or waist. The resulting system is restricted to fixed magnification between waists [e.g. eq. (7)]. If space charge or some other influence, such as changing accelerator voltage, changes beam properties then the minimum attainable beam spot on target may grow dramatically because of firstorder effects and not necessarily because of aberration as might mistakenly be inferred. Two lenses (sometimes called compound or zoom lenses) are required between waists to control both magnification and focus. Even with two lenses the range of useful magnification control may not encompass the desired operating range of the sys-

J.D. Larson/Electrostatic ion optics

89

tem. Care must be taken to place lenses where they are most effective. The accelerator section of an ion implantation system poses special problems. Following mass analysis slits, many systems employ a graded accelerator tube, lens, deflectors, and target. A simplified matrix for this region (ignoring deflectors and all fine details) is drift to thin lens target (10

q

1

1 )(--1/f

accelerator drift from [eq. (67)] slits

0](all 1/\a21

a12](1 a22/\0

p 1)

(-(q/R)/(allp+a12) =\-(l'/R)/(axlp +ax2)-alx/q

0 -(a,xp +a12)/p)'

(92)

where the ai/ are supplied by eq. (67) and a focus condition [see eq. (7)] has been imposed between analysis slits and target. (A focus is convenient here for illustration because it does not depend on the beam but, as about to be seen, a focus is not at all equivalent to the beam minimum actually desired.) Of particular interest in eq. (92) is the magnification term

-(q/R )/(a~ ip + a12) = -(1/R )(q/L )/[2/(R + 1 ) - ½ ( P / L ) ( R -

3)] ,

(93)

which becomes singular when R = 2(1

+L/p) ~/2 + 1 .

(94)

This singularity is an artifact that arises because a focus is being demanded from the thin lens just as the natural focus of the accelerator coincides with the lens position, making the lens useless as a focusing device. When similar conditions arise in practice, the impotence of a lens becomes quite disconcerting. A more thorough analysis requiring a minimum diameter beam spot on target [instead of a focus, see eq. (13) and related discussion] avoids the sin~larity but will still reveal large variations in minimum beam size. The implication is clear; magnification becomes a sensitive function of accelerator voltage whenever R ~ 2(1 + L/p) a/2 + 1. This problem is avoided in many ion implantation systems by placing the analysis slits very near the accelerator entrance so that p ~ 0 and eq. (93) reduces to -~(1/R)(q/L)(R + 1) which varies smoothly from -q/L to -.½q/L as R varies from 1 to oo. The accelerator section is peculiarly compromized by space charge. Whereas magnetic lenses and deflectors allow space charge neutralization the acceleator must be electric, exposing the beam to full space charge effects [36]. (There exists one possible exception. Heavy particles could be accelerated radially between regions of different potential embedded in magnetic flux within the neutralizing environment of a space charge lens. Formidable engineering may be required to exploit this situation but, in principle, a neutralized "space charge accelerator" appears feasible.) Crossing the beam through a waist at the accelerator entrance stabilizes downstream magnification (see previous paragraph) but promotes increased space charge blowup [e.g. eqs. (73) and (78)]. Reducing the accelerator to one or a few high-gradient gaps may help by minimizing the time of exposure. High fields and small dhnensions increase the risk of geometric aberration but that may be insignificant compared to effects from space charge. Strong over-focusing at the accelerator entrance (e.g. Pierce lens) can be matched to beams of fixed perveance, perhaps obviating the need for additional lenses downstream, but this approach probably is too inflexible for most ion implantation applications. As an alternative, the beam leaving the mass analysis slits might be allowed to expand to larger diameter in a neutral region shielded by a grid across the accelerator entrance. This expanded beam could then be accelerated quickly [use eq. (67) less terms in square brackets] into a region where neutralization is again possible. The resulting larger beam requires larger bore components but, if not seriously aberrated, it can then be focused to a smaller size on target. In many cases a short backbiased region of field at the accelerator exit should prove useful both to keep backstreaming electrons from being accelerated and to assist in re-establishing neutralization downstream. Ion beams of microscopic size are obtained by excluding unwanted particles from a larger beam using defining apertures then (usually) focusing the transmitted portion of the beam to smaller size. Slit apertures ~ 1 /am have been tested [37,38] but as the opening decreases, the amount of beam scattered from the edges eventually exceeds freely transmitted beana, rendering further size reduction counterproductive. Focused ion microbeams of MeV energy and > 2 /am size are available in several laboratories for use as nuclear microprobes [37]. These first-generation systems use ion sources, accelerators, and other hardware designed originally for less demanding II. ION SOURCES AND BEAM OPTICS

90

J.D. Larson / Electrostatic ion optics

service. The final demagnifying lens is often o f special design consisting typically o f magnetic doublet, triplet, or quadruplet elements that, in principle, reduce the object by factors o f 1/5 to 1/25. In fact, only a few beam spots o f 2 - 5 /am size approach the theoretical limits imposed by finite phase space area indicating that many systems are severely limited by aberration. The precision o f fabrication and assembly required for these quadrupole lenses is evident from reported [37] beam spot growth o f some 300/.trn per degree o f rotational misalignment between quadrupole elements. Nevertheless, an order of magnitude improvement ought to be possible from this technology leading to beam spots o f perhaps 0 . 2 - 0 . 5 / a m diameter. A much brighter type o f ion source [39,40] has been combined with a simple 57 kV gap lens accelerator to produce spots ~ 0.1 gm at intensities o f 1.5 A / c m 2 on target [41]. (Beam brightness as used here is beam current divided by the product o f phase space areas in the two transverse planes; alternatively, it is the current density per unit solid angle, Acm -2 sr -l , at a double waist.) Vely brigllt beams appear to emanate from point sources. After collimation, bright beams retain useable currents within very small cone angles whereas to sustain equal currents from dull beams either beam angles must be large or the object must occupy a relatively large area. As a consequence, bright beams confined to small angles suffer lower aberration, provide greater depth o f focus, and produce point-like images without need o f demagnification (with attendant short image distances). Transport o f bright beams becomes relatively easy because the beam often occupies only a tiny fraction o f the phase space acceptance area of the host beam transport system.

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V.K. Zworykin, Electron optics and the electron microscope (Wiley, New York, 1945). P. Grivet, Electron optics, 2nd ed. Transl. P.W. Hawkes (Pergamon, New York, 1972). A.B. E1-Kareh and J.C.J. EI-Kareh, Electron beams, lenses, and optics, vols. 1 and 2 (Academic Press, New York, 1970). A. Septier (ed.), Focusing of charged particles, vols. 1 and 2, (Academic Press, New York, 1967). A.P. Banlbrd, The transport of charged particle beams (Spon, London, 1966). S. Penner, Rev. Sci. Instr. 32 (1961) 150. E. Harting and F.H. Read, Electrostatic lenses (Elsevier, Amsterdam, 1976). D. DiChio, S.V. Natali and C.E. Kuyatt, Rev. Sci. Instr. 45 (1974) 559. D. DiChio et al., Rev. Sci. Instr. 45 (1974) 566. J.D. Larson, IEEE Trans. Nucl. Sci. NS-18, no. 3 (1971) 1088. J.D. Larson, Nucl. Instr. and Meth. 122 (1974) 53. J.D. Larson and C.M. Jones, Nucl. Instr. and Meth. 140 (1977) 489. G.E. Lee-Whiting, Nucl. Instr. and Meth. 76 (1969) 305. W. Wien, Ann. Phys. 8 (1902) 244. R. Booth and H.W. Lefevre, Nucl. Instr. and Meth. 151 (1978) 143. J.G. Cramer, Nucl. Instr. and Meth. 62 (1968) 205. C.J. Davisson and C.J. Calbick, Phys. Rev. 42 (1932) 580. F.H. Read, Inst. Phys. Conf. Ser. No. 38 (1978) 249. A. Adams and F.H. Read, J. Phys. E: Sci. Instr. 5 (1972) 150,156. T. Saito and O.J. Sovers, J. Appl. Phys. 48 (1977) 2306: J. Appl. Phys. 50 (1979) 3050. T. Saito, M. Kikuchi and O.J. Sovers, J. Appl. Phys. 50 (1979) 6123. A. Chutjian, Rev. Sci. Instr. 50 (1979) 347. J.L. Verster, Philips Rcs. Repts. 18 (1963) 465. L.P. Smith and P.L. Hartman, J. Appl. Phys. 11 (1940) 220. P.T. Kirstein, G.S. Kino and W.E. Waters, Space charge flow (McGraw-lliU, New York, 1967). C.D. Moak, Nucl. Instr. and Meth. 8 (1960) 19. T. Morrone, Nucl. Instr. and Mcth. 88 (1970) 323. J.E. Osher, Inst. Phys. Conf. Ser. No. 38 (1978) 201. J.W. Schwartz, RCA Rev. 18 (1957) 3. W.T. Milner and J.D. Larson, unpublished. F.H. Read, A. Adams and J.R. Soto-Montiel, J. Phys. E: Sci. Instr. 4 (1971) 625. P.W. ltawkes, Adv. Electron. Electron Phys. 7 (1970). S.Ya. Yavor, A.D. Dymnikov and L.P. Ovsyannikova, Soy. Phys. Tech. Phys. 9 (1964) 76. J.E. Draper, Rev. Sci. Instr. 37 (1966) 1390. GCA Model 8505 Ion Implanter, GCA Corp., Sunnyvale, CA 94086, USA.

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P.H. Rose and D.F. Downey, Inst. Phys. Conf. Set. No. 38 (1978) 243. J.A. Cookson, Nucl. Instr. and Meth. 165 (1979) 477. R. Nobiling et al., Nucl. Instr. and Meth. 130 (1975) 325; Nucl. Instr. and Meth. 142 (1977) 49. R. Clampitt, et al., J. Vac. Sci. Tech. 12 (1975) 1208. R. Clampitt and D.K. Jefferies, Nucl. Instr. and Meth. 149 (1978) 739. R.L. Seliger et al., Appl. Phys. Lett. 34 (1979) 310.

II. ION SOURCES AND BEAM OPTICS