Ultram~croscopy 47 (1992) 35-54 North-Holland

Idzl~'nln~-t~r~a,~n

Emission microscopy and related techniques: resolution in photoelectron microscopy, low energy electron microscopy and mirror electron microscopy Gertrude F Rempfer Department of Phystcs, Portland State Umt erstty, Portland, OR 97207, USA

and

O H a y e s Grlfflth Instttute of Molecular Btology and Department of Chemtstry, Umterstty of Oregon, Eugene, OR 97403, USA Recewed 17 October 1991, at Editorial Office 13 May 1992

A umfied treatment of the resolution of three closely related techmques ~s presented em~sston electron m~croscopy (parhcularly photoelectron m~eroscopy, PEM), low energy electron mLcroscopy (LEEM), and mirror electron m~croscopy (MEM) The resolution calculation is based on the intensity distribution m the image plane for an ob}ect of fimte size rather than for a point source The calculations take into account the spherical and chromatic aberrations of the accelerating field and of the objectwe lens Intens|ty d~str|but~ons for a range of energies m the electron beam are obtained by adding the single-energy d~stnbuhons weighted according to the energy dmtnbut~on function The d~ffractlon error Ls taken into account separately A working resolution ~s calculated that mcludes the practical reqmrement for a fimte exposure t~me, and hence a fimte non-zero current m the ~mage The expressions for the aberration coefficients are the same m PEM and LEEM The calculated aberrahons m MEM are somewhat smaller than for PEM and LEEM The resolution of PEM is calculated to be about 50 ,~, assuming conventional UV excitation sources, which prowde current densities at the specimen of 5 × 10 - s A / c m 2 and emmslon energies ranging up to 0 5 e V A resolutmn of about 70 ,~ has been demonstrated experimentally The emission current density at the specimen ~s higher m LEEM and MEM because an electron gun ~s used m place of a UV source For a current density of 5 × 10 4oA/cm2 and the same electron optical parameters as for PEM, the resoluhon ts calculated to be 27 A for LEEM and 21 A for MEM

1. Introduction PEM (photoelectron microscopy), L E E M (low energy reflection electron microscopy), and MEM (m~rror electron m~croscopy) are imaging teehtuques m which electrons interact w~th the spec~men at very low energies and are then accelerated by an electric field between the specimen and an anode and focused to an image of the specimen by an electron lens system PEM, also called p h o t o e m i s s l o n electron m i c r o s c o p y (PEEM), is a form of emission electron ml-

croscopy [1-3] In PEM the tmagmg electrons are photoelectrons released from the specimen surface by ultraviolet hght In L E E M [4] and MEM [5,6] the image Is formed by electrons which are reflected from the specimen The specimen potential Vs ts offset from the potential Vc of the electron gun cathode by a small adjustable bias voltage Vb,as (corrected for the contact potential due to the difference in cathode and specimen work functions) The electrons are directed toward the specimen and decelerated to low energies by the electric field between the specimen

0304-3991/92/$05 00 © 1992 - Elsevier Science Pubhshers B V All rights reserved

36

G F Rempfer, 0 H Gnfftth / Emission microscopy and related techniques PEM

LEEM-MEM-PEM

~z~-----

SPECIMEN

r

l

~

--

OBJECTIVE - -

h

LENS

t

/ - -

MAGNET

INTEGMEDIATE - IMAGE

PBOJECTION LENS ~-

FIt~A~L

~ i ....

O EMFALGEG NCES TIN

INMMEOEIAT E

C ONLOEENNE EH

.......... / - 7

I

...... \ < : ~ >

~" /a)

I

i

~I

IA~

/ELECTRON

~oo~EE

(b)

Fig I Schematic diagrams of the electron optical systems of (a) PEM, and (b) LEEM-MEM-PEM Both systems utilize the accelerating field and long focal length oblectlve lens The PEM optical system is linear and this is also the arrangement of other types of emission microscopes (e g thermlonic and ion emission) In LEEM or MEM a means of separating the illuminating beam heading towards the specimen and the imaging beam leaving the specimen is required In (b) the separating system is shown inserted between the intermediate and projection stages Deflections are produced by magnetic fields and take place at image planes A LEEM electron optical system can also be used for PEM

and the a n o d e b e f o r e b e i n g r e f l e c t e d D e p e n d i n g on the bias voltage, the e l e c t r o n s a r e r e f l e c t e d e i t h e r after striking the s p e c i m e n with low e n e r gles ( L E E M m o d e ) , or just b e f o r e r e a c h i n g the specimen (MEM mode) The same accelerating field c o n f i g u r a t i o n a n d objective lens design a r e used in L E E M a n d M E M as m P E M H o w e v e r , m L E E M - M E M i n s t r u m e n t s a m e a n s of s e p a r a t ing the i l l u m i n a t i n g a n d i m a g i n g b e a m s m u s t be i n c o r p o r a t e d at some stage in the e l e c t r o n optical system, w h e r e a s P E M studies can be c a r r i e d out e i t h e r with or w i t h o u t a s e p a r a t i n g system A s c h e m a t i c d i a g r a m o f a P E M optical system with a straight axis is shown in fig l a , a n d of a

LEEM-MEM-PEM optical system, with a separating a r r a n g e m e n t i n t r o d u c e d b e t w e e n the interm e d i a t e a n d p r o j e c t i o n stages, in flg l b T h e r e have b e e n a n u m b e r o f r e s o l u t i o n calculations for emission m i c r o s c o p y [7-19], for L E E M [4,20,21], a n d for M E M [5,6,22,23] (see also references in the historical review [21) H o w e v e r , we are u n a w a r e of any untried t r e a t m e n t o f all t h r e e t e c h n i q u e s for the s a m e e l e c t r o n optical p a r a m e ters T h e p u r p o s e o f the p r e s e n t work is to present a c o m m o n analysis for p h o t o e l e c t r o n mic r o s c o p y ( P E M ) , L E E M , a n d M E M that invites a c o m p a r i s o n of the r e s o l u t i o n o f these t e c h n i q u e s F i r s t - o r d e r imaging in the a c c e l e r a t i n g a n d objective stages t a k e s p l a c e in the s a m e way in the t h r e e imaging m o d e s A b e r r a t i o n s In the acc e l e r a t i n g field, however, are d i f f e r e n t for the m i r r o r m o d e from those for the P E M and L E E M m o d e s In the p r e s e n t calculations the a b e r r a tions of the a c c e l e r a t i n g field a r e e x p r e s s e d separately in t e r m s of s p h e r i c a l a n d c h r o m a t i c a b e r r a ttons a n d a r e c o m b i n e d with the c o r r e s p o n d i n g a b e r r a t i o n s o f the objective lens to yield the overall a b e r r a t i o n s o f the imaging system T h e r e s o l u t i o n c a l c u l a t i o n ts b a s e d on the intensity d i s t r i b u t i o n in the i m a g e a n d t a k e s into account the effect o f the object size n e e d e d for a r e q u t r e d c u r r e n t in the i m a g e T h e starting p o i n t is a g e o m e t r i c a l optics a p p r o a c h to resolution, develo p e d for e l e c t r o n optics in g e n e r a l [24] a n d discussed in a n o t h e r p a p e r o f this v o l u m e [25] This a p p r o a c h avoids the simplifying a s s u m p t i o n of a p o i n t source, a n d the a t t e n d i n g lnfinttles in the intensity d i s t r i b u t i o n It also r e p l a c e s reliance on the p l a n e of least confusion as the o p t i m a l focuslng p l a n e In a r e c e n t p a p e r [19] we have a p p l i e d this a p p r o a c h to P E M H e r e we s u m m a r i z e the e a r h e r c a l c u l a t i o n a n d e x t e n d it to include L E E M a n d M E M R e s o l u t i o n m l c r o g r a p h s t a k e n with a P E M a r e i n c l u d e d for c o m p a r i s o n with t h e o r e t i cal results

2 First-order imaging in PEM, LEEM, and MEM T h e a r r a n g e m e n t for a c c e l e r a t i n g the e l e c t r o n b e a m leaving the s p e c i m e n in P E M , L E E M , a n d M E M ( a n d for d e c e l e r a t i n g the i l l u m i n a t i n g b e a m

G F Rempfer, 0 H Grtfftth / Emission microscopy and related techniques SPECIMEN Vs

EMITTING OR REFLECTING POINT

ANODE Va

I

i

(a)

VIRTUAL

[/SPECIMEN

Fa~.,~ ~'~va

I

(b)

1 I [

I

)'~

I"

I'

APERTURE LENS TO

[ OBJECTIVE ~.~D~" ~ LENS

OBJECT FOR OBJECTIVE LENS

--

I ~ 1

2[

I"~

4_

3~"

411

Fig 2 (a) Trajectories of electrons emitted or reflected from a specimen point on the axis showing the curved paths in the

accelerating region and the dwerging action of the aperture lens (radial distances have been exaggerated, the beam is much narrower than shown here) (b) Detail of the accelerating region showing a trajectory and a tangent ray defining the position of the virtual specimen at a distance l* from the anode The components of the emission or reflection velocity i,~ are re and z~, and the components of the velocity La after acceleration are r, and za The lmtla| and final tangents make angles a~ and c%, respectively, with the axis (c) Electron optical equivalent for the case of a uniform accelerating held combined with the diverging aperture lens The focal length of the aperture lens for low emission energies is fA = --41 The virtual specimen is at a distance of 21 from the aperture lens The aperture lens forms a virtual linage of the virtual specimen at a distance of ~l and magnification ma = ~ The ray angle after divergence by the aperture lens is ~1 = ~,a~ Adapted from ref [191

in L E E M a n d M E M ) is shown schematically in fig 2a T h e d i a g r a m r e p r e s e n t s the region n e a r the axis A p l a n a r - t y p e s p e c i m e n at p o t e n t m l Vs

37

serves as the c a t h o d e of the accelerating field T h e a n o d e at p o t e n t i a l VA also is planar, with a c e n t r a l o p e n i n g for passage of electrons R a d i a l distances have b e e n exaggerated T h e d i a m e t e r of the a n o d e o p e n i n g is actually considerably smaller t h a n the accelerating gap l T h e electric field is essentially u n i f o r m except n e a r the a n o d e a p e r t u r e T h e effect of the a p e r t u r e ~s to create a thin diverging lens ( a p e r t u r e lens) T h e accelerating field is r e g a r d e d as a u n i f o r m electric field t e r m i n a t e d by the a p e r t u r e lens Fig 2a dlustrates the paths of electrons e m i t t e d or reflected from a specimen p o i n t on the axis I n the u n i f o r m field the electron trajectories are parabolic arcs A t the a n o d e the trajectories are diverged by the a p e r t u r e lens, a n d b e c o m e object rays for the objective lens I m a g e f o r m a t i o n m the accelerating process can be t h o u g h t of as occurring in two steps I n the u m f o r m field the t a n g e n t s to the p a r a b o h c trajectories after acceleration a p p e a r to come from a virtual s p e c i m e n havmg unit lateral magmficatlon, a n d located (to first order) at a distance l* = 21 (twice the s p e c i m e n distance) from the a n o d e I n hght-optlcal terms the c h a n g i n g index of refraction in the axial d~rect~on (proportional to the electron velocity) causes the rays to b e n d so that they a p p e a r to come from a virtual s p e c i m e n In fig 2b a trajectory from the specim e n p o i n t on the axis ts shown, with the t a n g e n t extrapolated backward intersecting the ax~s at a distance l* from the a n o d e T h e subscript " e " on the angle a , velocity r, and velocity c o m p o n e n t s r a n d z stands for the emission m o d e T h e diagram is the same for the reflection m o d e with the subscript " e " replaced by " r " T h e quant~tles after acceleration are subscripted with " a " T h e virtual s p e c i m e n created by the u n i f o r m accelerating field is the object for the a p e r t u r e lens T h e focal length of the a p e r t u r e lens for electrons e m i t t e d or reflected from the s p e c i m e n with very low energies is fA = --41 T h e a p e r t u r e lens forms a virtual ~mage of the virtual s p e c i m e n at a distance of 41 from the a n o d e a n d wlth a 2 lateral m a g n i f i c a t i o n of x This d e m a g n l f i e d image is the object for the objective lens, as dlustrated in fig 2c T h e angle a 1 of the object ray is ~-O/a

G F Rempfer, 0 H Gnffith / Emission microscopy and related techmques

38

3. Image aberrations in PEM, LEEM, and MEM

are displaced from the spectmen point, and the distance to the intersection of the tangents ~s less than 21 Since z e depends on both the angle o~ and the energy eve of emission or reflection from the specimen, both sphertcal aberration and chromatic aberration are present The distance to the virtual specimen, written in terms of a~ and V~, and the accelerating voltage V,~ = VA - Vs, is

3 1 Aberranons of the accelerating fwld m PEM and LEEM The aberrations of the accelerating field arise because the tangents to the parabolas from a specimen point do not meet exactly at a point in the virtual specimen T h e dtstance to a virtual spectmen p o m t d e p e n d s on the ratto of the axial c o m p o n e n t s of velocity before (Ze or z~) and after (z~) acceleration, and the length l~ over which the electron is accelerated, as given by l* = 2 l J ( 1

+Ze/Z~),

l* = 2/(1 - ;,,V~/V, cos a~),

(2)

plus terms of higher order of smallness, and the angle of the tangent to the parabolic trajectory after acceleration is

(1)

where the subscript " e " is meant to represent either " e " or " r " In L E E M as in P E M the acceleration length is just the distance l between the spectmen and the anode The image aberrations of the accelerating field are due to the d e p e n d e n c e of l* on z e The family of parabolic arcs originating at a specimen point, in general, do not have their vertices at the point Only for z~ = 0 ( a e = 90 °) do the parabolic vertices coincide with the specimen pomt, and the tangents meet at 21 As illustrated i n f i g 3, if z~ lS not zero ( a e < 90°), the parabolic verttces

OZ, = v"Ve/Va s i n oz e

(3)

The paraxlal value of the virtual specimen distance is l,* = 2 / ( 1 -

~'V~/V~)

(4)

3 1 1 Spherical aberranon m PEM and LEEM Longitudinal spherical aberration refers to the difference in image distance for off-axis rays and paraxial rays, l e , the difference between eqs (2) ANODE PLANE

VIRTUAL SPECIMEN PLANE

S PpELCAINMEE N

t

t

/ / ~ - "

:-

!P 2~

Fig 3 Spherical a b e r r a t i o n in the i m a g e f o r m e d by a uniform a c c e l e r a t i n g field M o n o e n e r g e t l c e l e c t r o n s are e m i t t e d or reflected with velocity t',~ and a r a n g e of angles o~ from a s p e c i m e n point and are a c c e l e r a t e d t o w a r d the a n o d e p l a n e T h e t a n g e n t s to the trajectories after a c c e l e r a t i o n m a k e angles a~ with the axl~ For a t = 90 ° the parabolic v e m c e s are at the s p e c i m e n point For o t h e r emission or reflection angles the vertices are d i s p l a c e d from the s p e c i m e n point as i n d i c a t e d by the d o t t e d extension of the p a r a b o l i c arcs (the vertices are at the last dot) E x t r a p o l a t e d backward, the t a n g e n t s form a virtual i m a g e of the s p e c i m e n point The i m a g e exhibits u n d e r c o r r e c t e d s p h e r i c a l a b e r r a h o n T a n g e n t rays c o r r e s p o n d i n g to small emission angles intersect n e a r the paraxlal focus at l~* T a n g e n t rays c o r r e s p o n d i n g to the largest emission a n g l e s intersect m the m a r g i n a l i m a g e p l a n e (at a d~stance 2 l for a . = 90 °)

39

G F Rempfer, 0 H Grtfftth /Emtsston mtcroscopy and related techmques

a n d (4) for a g w e n v a l u e o f Ve If t h e i m a g e moves f o r w a r d (in t h e d i r e c t i o n o f t h e b e a m ) as t h e angle of t h e rays ~s i n c r e a s e d , the s p h e r i c a l a b e r r a t i o n Is stud to b e o v e r c o r r e c t e d If the i m a g e m o v e s b a c k w a r d s , as for u n c o r r e c t e d elect r o n lenses, t h e a b e r r a t i o n Is u n d e r c o r r e c t e d In PEM and LEEM the spherical aberration of the a c c e l e r a t i n g field is u n d e r c o r r e c t e d T h e longitudinal s p h e r i c a l a b e r r a n o n zlsz = - ( l * - l~') can b e e x p r e s s e d as

2, a n d t h e s p h e r i c a l a b e r r a t i o n coefficient can b e a p p r o x i m a t e d by

A . z ~ = - [ 2 1 ~ / - ~ J V . / ( 1 + c o s a ~ ) ] sm2a~,

3 1 2 Chromattc aberratton m PEM and L E E M

Cs. = l V~a/Ve

A c c o r d m g to eq (6) o r ( 6 ' ) Csa b e c o m e s very large as Ve b e c o m e s small H o w e v e r , this is not c a t a s t r o p h i c b e c a u s e from eq (5) the s p h e r i c a l a b e r r a t i o n itself goes to z e r o A l s o the e n e r g y d i s t r i b u t i o n f u n c n o n goes to z e r o at V~ = 0

L o n g i t u d i n a l c h r o m a t i c a b e r r a t i o n refers to the d i f f e r e n c e m ~mage & s t a n c e for e l e c t r o n s having d i f f e r e n t e n e r g i e s It is said to b e u n d e r c o r r e c t e d if t h e i m a g e moves f o r w a r d as t h e e l e c t r o n e n e r g y ~s i n c r e a s e d , a n d o v e r c o r r e c t e d if t h e ~mage moves b a c k w a r d Eq (4) shows t h a t t h e c h r o m a t i c a b e r r a t i o n of the a c c e l e r a t i n g field m P E M a n d L E E M is u n d e r c o r r e c t e d F o r e l e c t r o n s which have an a v e r a g e emission o r r e f l e c t i o n e n e r g y ePe a n d an e n e r g y d l s t n b u t l o n of effectwe width eAV~, t h e c h r o m a t i c a b e r r a t i o n d e r i v e d from eq (4) is

o r in t e r m s o f t h e a n g l e after a c c e l e r a t i o n

AsZa:

- [2l~/(1

2 +cos O/e)]O/a,

(5)

w h e r e a . = ~-ee/VaSm a e E x c e p t for sign t h e coefficient o f a a2 is the s p h e r i c a l a b e r r a t i o n coefficient C,. of the a c c e l e r a t i n g field r e f e r r e d to virtual s p e c i m e n s p a c e at m a g n t f l c a t l o n 1 By c o n v e n t i o n the s p h e r i c a l a b e r r a t i o n coefficient ~s d e n o t e d as p o s l t w e for u n d e r c o r r e c t e d a b e r r a tion F r o m eq (5), Csa = 2 / V ~ f @ ~ / ( 1

+ cos O/e)

(6')

Acza= 2'[ V~/(~I - AVe/2~

(6)

F o r angles such t h a t cos a e Is not very d i f f e r e n t from umty, t h e d e n o m i n a t o r in eq (5) is close to

4-~/l 4- AVe/2Ve )]

AVe//Va

(7)

ANODE PLANE VIRTUAL SPECIMEN PLANE Spsolmeo--

SPECIMEN PLANE |

--> I 5~-~

2£.a

"~

2£.

Fig 4 Trajectories of electrons reflected from the specimen m MEM The electron that travels on path (1) has sufficient energy to reach the specimen before being turned back, whereas the electron which travels along path (2) has less energy associated wtth the axml &rectmn and ~s turned back before reaching the specimen The vertices m both cases are in hne wxth the specimen point The tangents to the parabolas are drawn as dashed hnes

G F Rempfer, 0 H Grtffith / Emission mtcroscopy and related techmques

40

The chromatic aberration coeffmlent Cca In virtual specimen space is the coefficient of A V e / V ~, and from eq (7) is

C~,, = 21~'V,/P¢ "'/( ~1 - A ~ / 2 P ~ + ~1 + A V ~ / 2 < )

(8)

Ordinarily the d e n o m i n a t o r in eq (8) is close to 2, and the chromatic aberration coefficient can be approximated by

Cc,, =

l(V,/Vo

(8')

C ~ b e c o m e s large, as does C.., for small values of V~ Again this is not catastrophic because AV~ also becomes small, and the energy distribution goes to zero at ~ = 0

3 2 Aberrattons of the acceleratmg field m M E M

trons ( a d = 0) of velocity v a just reach the specimen surface, non-paraxlal electrons (aa ~ 0) havIng the same velocity do not reach the specimen, but are turned back at a distance from the a n o d e of ld = l cos2a~ T h e distance from the anode to the virtual spemmen for non-paraxlal electrons, 21 cos2a., is less than the distance 2l for paraxlal electrons Thus in M E M , unlike in P E M and L E E M , the image moves forward as the angle a.~ increases, and the spherical aberration of the accelerating field is overcorrected The longitudinal spherical aberration is

k , z d = 2l(1

- cos2a,,)

= 21a 2

T h e coefficient of a ,2 in eq (9), along with the convention of using a negative sign for overcorrectlon, gives the spherical aberration coefficient In v i r t u a l specimen space,

C.., = - 2 I In M E M , as in L E E M , the electron b e a m illuminating the specimen is provided by an electron gun In which the electrons are accelerated from a c a t h o d e at potential Vc to an anode at potential VA In M E M the bias between the gun cathode and the specimen IS adjusted so that electrons are turned back just as, or just before, they contact the specimen Thus z~ = 0, and from eq (1) l* = 2/,~, where l,~ is the distance from the anode at which the electrons are turned back The parabolic trajectories of the electrons which image a given specimen point have a c o m m o n axis in line with the specimen point (fig 4) The tangents corresponding to a vertex at l, intersect m a virtual specimen point at / * = 21d on the parabolic axis For a given bias voltage the distance from the specimen at which the electrons are turned back, and thus the acceleration length l,, depends on the electron energy associated with the axial direction (axial energy) The dependence of l, on the axial energy leads to spherical and chromatic aberrations in M E M

3 2 1 Sphertcal aberratton m M E M T h e axial c o m p o n e n t of the electron velocity at the a n o d e plane IS z, = t'~ cos a , , and the axial I 2 1 ~2 kinetic energy is ~mz~ = ~m~ d cos2a~ If the specimen bias is adjusted so that paraxml elec-

(9)

(10)

The spherical aberration coefficient of the acceleratlng field in M E M is small relative to the aberration coefficients in P E M and L E E M , and being negative, provides a small a m o u n t of correction for the objective lens However, the question of resolution IS complicated by the fact that the farther in front of the specimen the potential surface is at which electrons are reflected, the less sharp is the detail due to microfields at the specimen surface

3 2 2 Chromauc aberrauon m M E M In an electron beam with a range of energies paraxlal electrons of higher energy come closer to the specimen surface than do those of lower energies, and the virtual specimen is farther from the a n o d e Thus the image moves backward, rather than forward, as the electron energy is increased, and the chromatic aberration of the accelerating field, like the spherical aberration, is overcorrected A n electron with an initial energy at the gun cathode equivalent to eAV~ reaches the anode with kinetic energy e ( V A - V c + AVe) If the specimen bias is adjusted so that paraxlal electrons of this energy barely reach the specimen, electrons having no initial kinetic energy do not reach the specimen, but are turned back at

G F Rempfer, 0 H Grtffith / E m t s s t o n mtcroscopy and related techmques

l a = [(Va - A V e ) / V a i l The distance between the vtrtual specimen planes for the two energies, - ( 2 l - 2ld), IS the longitudinal chromatic aberration in virtual specimen space,

Acz ~ = - 2 1 A V e / V a

(11)

The coefficient of A V e / V a IS the chromatic aberration coefficient in virtual specimen space, Cca =

-2l,

41

clents of the accelerating field also are large for PEM and L E E M From eqs (6) and (8), for Ve = A V e = 0 2 5 V, Va = 3 0 kV, and l = 3 mm, C~a = Cc~ = 104 cm These coefficients are of the same sign as those of the objective lens On the other hand the aberration coefficients in MEM are relatively small, C,d = Cca = - 2 1 = - 0 6 cm, and are of opposite sign, so that a small amount of correction occurs

(12)

the same as for spherical aberratton Again, for electrons not reachmg the specimen, the contrast due to mlcrofields at the specimen surface is reduced

3 3 AberraUons of the objecuve lens

3 4 The combined aberrattons o f the accelerating field and the objectwe lens The combined aberration coefficients for the accelerating field and the objective lens are 4 Cs = Csa-[- ( 5 )

The objectwe lens used In the low-energy imaging modes discussed here has a focal point well outside its boundary m order to accommodate the acceleratmg gap and to focus on the virtual specimen at a distance of 4l on the other side of the anode As a consequence the spherical and chromatic aberration coefficients C~l and Col of the lens are much larger than for a TEM objecttve lens (For example, an electrostatic objectwe lens which we use m PEM has C~l = 19 cm and Ccl = 1 9 c m ) Furthermore, the object for the objective lens has a lateral magnification of 2 and the aberration coefficients must be referred to magnlficatton 1 to be combined with the aberratton coefficients of the accelerating field As a result C~l is multlphed by (5) 3 4, and Ccl by (5) 3 2 The spherical and chromatic aberration coeffl-

Csl,

(13)

2

Cc : Cca + (5) Cc,

(14)

For PEM and L E E M Csa ts given by eq (6) or (6'), and Cc~ by eq (8) or (8') For MEM C~. = Cc~ = - 2 l Numerical values for the aberration coefficients are given in table 1 for selected values of the electron optical parameters

4. Intensity distribution approach LEEM, and MEM

in PEM,

The intensity distribution approach to resolution calculations is described in ref [24], and elsewhere in this issue [25], for TEM and other systems in which the object is imaged without an intervening accelerating stage We have applied

Table 1 Aberration coefflcmnts for PEM, LEEM, and MEM Mode PEM/LEEM

MEM

Vc (V) 0 25 05 100 00

Csa = Cca

C s = Csa + (-~)4Csl

C e = Cca + (~)2Ccl

(cm)

(era)

(era)

104 73 5 164 - 06

200 169 5 1124 95 4

108 77 7 206 36

The aberration coefflcmnts C~a and Cca for the accelerating field, and C, and C c for the field combined with an electrostatic objectwe lens, are given for an accelerating voltage Va of 30 kV, a specimen-to-anode distance l = 3 mm, and selected values of the emission (or reflection) energy e Ve The aberration coefflcmnts for the objectwe lens are Csl = 19 cm and Col = 1 9 cm

42

G F Rempfer, 0 H Gnfftth /Emtsston mtcroscopy and related techmques

this approach also to systems which have an accelerating stage, in particular PEM [19] Here we summarize this approach and extend ~t to include L E E M and MEM As discussed In section 2, the object for the objective lens in the low-energy electron microscoples, PEM, LEEM, and MEM, is not the specimen itself, but a virtual image of the specimen formed In the accelerating process In section 3 the aberrations of the accelerating field, the diverging effect of the anode aperture lens, and the aberrations of the objective lens, referred to virtual specimen space at magnification 1, are combined m the overall aberration coefficients Cs and C c given by eqs (13) and (14) The optical system can then be considered equivalent to an aberration-free virtual specimen at magnification 1 being imaged by a lens with the combined aberration coefficients C, and C c of the accelerating field and objective lens The intensity distribution approach is applied to this equivalent system The virtual object for the equivalent system is chosen to be a small disc of radius a centered on the axis m the virtual specimen plane, at a &stance of 2/ from the anode This object corresponds to an actual specimen disc of radius a at a distance l from the anode It is assumed that the current density leaving the actual specimen is uniform and the energy range has a parabolic &strlbut~on The angular distribution is assumed to have a cosine form This choice Is appropriate for non-crystalhne specimens such as cell surfaces in biology and amorphous surfaces in materials scmnce However, the highly ordered plane surfaces of crystalline specimens studied by L E E M / L E E D techniques in surface physics reflect electrons in narrow diffracted beams rather than in a continuous range of angles In either case, the angular distribution of the low energy electrons leaving the s p e o m e n is compressed by the accelerating field into the much narrower angular distribution of the rays associated with the wrtual specimen The large angular aperture of the rays leaving the specimen, along with the subsequent compression by the accelerating field, constitutes the main difference between the analyses w~th and without the acceleration stage The

procedure used in ref [19] for calculating the intensity distribution in the image is outhned here Each set of rays leaving the object dlsc m a given direction (defined by the polar angle a.t and the azimuthal angle ~b) forms its own image of the object In the presence of spherical aberration and defocus the image discs are in general not centered on the axis The intensity in the image plane depends on how these individual images are distributed as a function of a . For a given energy of electrons leaving the specimen, the displacement G of an image disc from the axis, referred to virtual specimen space at magnification 1, is r . = - ( A z + c , aZ)a~,

(15)

where Az is the defocus distance, posltlve for overfocus (away from the lens) and negative for underfocus (toward the lens), and a~ is the angle of the accelerated ray C s is the overall spherical aberration coefficient referred to virtual specimen space at magnification 1 given by eq (13) For values of the angle ae at the specimen such that cos a t is not very different from 1, C~ is approximately constant, and eq (15) can be treated as a cubic equation in a a For negative values of Az the displacement r. exhibits a retrograde behavior, at first increasing with a a in the incident azimuth and then, after reaching a maximum, going to zero and increasing in the opposite azimuth The maximum displacement P in the retrograde &rectlon is the radius of the caustic envelope due to spherical aberration From the derivative of eq (15), the maximum occurs for an angle c~ given by Az = - 3 C ~ a j,

(16)

and the maximum displacement is

,~ = 2c, a•

(17)

The relation between the caustic • and the defocus &stance A z is AZ = -3(CsP2/4)

l/'~

(18)

G F Rempfer, 0 H Grtffith /Emtsston mtcroscopy and related techmques

The displacement r goes through zero at a a = V/3-~a, and through - P at 2~ a In applying the mtenstty distribution approach to PEM, LEEM, and MEM, the angle a a of the tangent to the parabohc trajectory after acceleration corresponds to a of eq (10) in ref [25], Ve and V~ are the beam voltages before and after acceleration, and a e is the angle of the ray leavmg the spectmen, J0 ts replaced by the emtsston or reflection current denstty Je of lOW energy electrons from the specimen, and Cs IS the combined spherical aberration coefficient of the objectwe lens and accelerating field, referred to virtual specimen space at magmfication 1 The intensJty at a point P in the tmage plane, at a distance rp from the axis, depends on the range of the angles a a and ~b for which the pomt ts tllummated by the tmage disc, and on the angular distribution function As described in refs [19,24,25], increments of sohd angle, corresponding to small increments of aa, are multtphed by the angular distribution functJon and summed to obtain the relatwe Intensity j(rp) at the point A series of intensity distributions for PEM or LEEM, plotted as a functton of rp, 1s shown m fig 5 for selected depths in the ~mage These distributions were calculated for a single energy and a cosine angular distrIbutton They are referred to virtual specimen space The object rao dius is 5 A, the emission or reflection energy is 0 25 eV, the accelerating gap Js 3 ram, and the accelerating voltage ~s 30 kV The spherical aberration coefficient of the objectwe lens referred to 3 4 virtual specimen space is (~) Cs1 = 96 cm, and the spherical aberration coefficient of the accelerating field Cs~, calculated from the approximate formula eq (6'), is 104 cm, resulting in a combined aberration coefficient of 200 cm ThJs value is close to the figure based on the exact expression for Cs~, eq (6) The exact expression was used in calculating the intensity d~stribut~on curves The plane which has the h~ghest intensity on the axis (HI plane) has a defocus for which the caustic radius P is equal to the object radms a For the given numertcal parameters the defocus distance of the HI plane is - 1 503 /xm Eq (18) based on the approximate formula eq (6') for Cs~ gwes the defocus d~stance with P = a as Aznt =

43

A z = - B 67 ~m

MARGINAL PLANE FOR o ( a = 2 m r

cf) AZ=-65~'---

LEAST~" C O N F U S I O N

PLANE OF

[e)

Az=-5

0 /am

(cl]

A z=-3

5

(c)

/N z : - I

5 /am

HIGH INTENSITY PLANE

(b) A z = - O 5 ,~m PARAXIAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PLANE {et)

Az=+0

5

Fig 5 Intensity distributions at various depths in the image of a small object In P E M or L E E M The intensity distributions are for a single energy and are referred to virtual specimen space at magmflcat~on 1 The object is a disc of radius 5 A centered on the axis The emission or reflection energy is 0 25 eV and the accelerating voltage is 30 kV The spherical aberration coefficient is approximately 200 cm Th~s series of intensity distribution curves shows clearly that the intensity in the image is not uniform and that it varies with the defocus of the image plane The intensity on the axis is highest m a plane (the HI plane) where the radius of the caustic envelope due to spherical aberration is equal to the object radius, m this case at a distance of 1 5 /zm toward the lens from the paraxlal plane The resolution in the HI plane is better than in the plane of least confusion The optimum image plane is the HI plane for the smallest object detail which provides the required current m the image Adapted from ref [19]

-3(Csa2/4) 1/3= - 1 500 /zm, which Js essentially the same as the exact value In the HI plane the caustic envelope (which is thlckened by the diameter of the object) coalesces with the axial caustic to form an especially high intensity peak (HI peak) The HI peak ts nearly triangular m profile, as shown in fig 5c For larger underfocus dtstances, figs 5d through f, the causttc region enlarges, the maximum due to the causttc enve-

44

G F Rempfer, 0 H Gnfflth / Emission microscopy and related techmques

lope separates from the central maximum created by the axml caustic, and the height of the peak decreases Fig 5f shows the intensity dlstrlbutlon in the plane of least confusion for a~ = 2 mrad Although all of the rays pass through the circle of least confusion, and the intensity outside is zero, the intensity distribution inside the circle is not optimum for resolution The central maximum is not as high as for the HI peak, and the area over which there is considerable intensity IS larger than for the HI distribution For smaller amounts of underfocus than for the HI plane, the intensity also falls off On the other hand the shape of the distribution at a defocus of - 0 5 /xm, curve (b) represents the object better than does curve (c) Curve (a) is the intensity distribution in a plane on the other side of the paraxlal plane with an overfocus of + 0 5 /xm This distribution shows a loss of both intensity and resolution

4 1 Intensity and current m the hzgh-mtenslty tinage peak The HI peak has a radius at half-maximum of about 1 1 times the object radius a, and a radius at the base of 2a The point on the axis in the HI plane is illuminated over the range of angles a~ = 0 to 2k a The corresponding angle ae at the specimen is given by sin cte = 2&a VafgJ//~ For a cosine angular distribution at the specimen the relative intensity j(0) is equal to sln2ae = (Va/Ve)(2t~a) 2 F r o m eq (17), with P = a, (2a.) 2 = (4a/Cs) 2/3, and J(0)H , = (4a/Cs)z/3Va/Ve

(19)

The area of the image peak out to the radius % = 2a is illuminated for angles up to about 2 2 a a The fraction l of the emitted current which is received in the image peak is roughly (2 26a)2Va/V~ On substituting again for 26 a we get t = (1 1)2(4a/C,)2/3Va/Ve

(20)

Since the total current emitted or reflected from

the object 1s 7ra2Je, the current in the image peak for a single-energy beam 1s

I = 3 8a8/3(4/C,)2/3( Vd/Ve)J e

(21)

The current depends on the 8 / 3 power of the object radius and on the inverse 2 / 3 power of C, In eqs (19) to (21) the factor Va//Ve reflects the fact that the fraction of the emitted beam which ~s included in the angular aperture after acceleration is larger for smaller emission energies For known values of Je, C~, and Va/Ve, eq (21) for a single-energy beam can be used to calculate the object radius needed for a required current m the image peak In a beam with a range of energies, chromatm aberration causes the HI planes for the different energies to occur at different distances from the lens The optimum image plane in this case can be approximated by the HI plane for the average energy However the image peak is not as high as it would be if the HI distributions were all superimposed in one plane Fig 6 shows the intensity distributions for several energies around an average energy of 0 25 eV for an obJect of radius a = 5 A The distributions were calculated in the HI plane for the average energy In this plane the single-energy curves, and their sum, have radii at half-maximum which are not very different from the object radius Summed intensity distributions for several object sizes and for beams having different energy &stributlons are presented elsewhere [19] The heights are closely proportional to a, as given by

j(O) = Cja

(22)

where the coefficient Cj depends on the energy distribution of the emitted or reflected electrons For V e = 0 - 0 5 0 V, C j = 6 8 × 1 0 - 3 / , ~ = 6 8 × 105/cm, and for Ve = 0 - 1 0 V, C ~ = 3 3 × 1 0 - 3 / A = 3 3 × 105/cm The intensity peaks are higher for Pe = 0 25 V than for V_e= 0 5 V, even though C s and C c are lower for Ve = 0 5 V than for V~ = 0 25 V (see table 1) The main reason for this difference is that, as mentioned earlier, the percentage of electrons within a given angular aperture after acceleration is larger for the smaller emission energy, or to put it another way

G F Rempfer, 0 H Grtfftth / Emission microscopy and related techniques i

ge~, the range of validity of these equations is given by

(aa)ma x =

i

5 /~m

Z~z=-I Va : 0 0

45

25

V

i0

2 2(a/2Cs) '/3 < ~/~/V~

(24) o

For example, for a = 2 0 A and C , = 2 0 0 cm, 2 2& = 175 mrad, whereas for Ve = 0 25 V and Va = 3 0 kV, (aa)m~ × = 2 9 mrad, so eq (24) is satisfied In this example 2 2c~d corresponds to sm a e = 0 6, so 36% of the current from the oobject ends up in the image peak For a = 5 0 A and C S = 100 cm, eq (24) is barely satisfied, with all of the current included in the image peak (For large values of a e the expression for C, which includes the effect of a e in C~d, eq (6), gwes a more accurate value for the current m the image peak )

t Q-

-x~ 11v" 0 05

..~l

0 -20

-10

a v;

I

I

0

10

20

%(in A) Fig 6 The intensity distributions in the image of a small object, for several values of the em~ssmn or reflectmn energy, e Ve The object radius a is 5 ,~ The average energy of the electrons leaving the specimen is 0 25 eV and the accelerating voltage is 30 kV The intensity dlsmbutlons were calculated in the high intensity plane for the average energy and are referred to virtual specimen space at magnification 1 The radu at half maximum of all of the Image peaks are approximately equal to the object radius a Because of chromatic aberration the high intensity planes for energies above or below the average energy do not fall in this image plane, so the sum of the intensity distributions is not as high as it would be without chromatic aberration Adapted from ref [19]

the brightness of the object (current per unit area per unit solid angle per volt) for the same current density is larger for Ve = 0 25 V than for Ve = 0 5 V The current in the image peak is given by

I = C, a3Je,

(23)

where C, = 6 C j If C~ IS known for the range of energies in the beam, and the emission current density is known, eq (23) can be used to calculate the object radius needed to provide the required current in the image In the calculations for eqs (19)-(23) it was assumed that the range of o/a was large enough to include the value 2 2C~a, the effectwe angular aperture contributing appreciable current to the HI image peak Since ~ = ( a / 2 C s ) 1/3, and

4 2 The geometrical resolution hmtt The usual criterion for resolution of two image details requires a drop of 25% in intensity between the peaks In the case of a single-energy HI peak or an intensity distribution for a range of energies in the average HI plane, this criterion is satisfied by a center-to-center distance referred to object space of 2 4a For a dark background, the geometrical resolution limit is the center-tocenter distance between the smallest objects which satisfy the current requirement in the image, rg =

2 4amm

(25)

If the background is not dark, the smallness of the object may be limited by either the contrast requirement or the current requirement The image contrast depends on the ratio of the current density of the object to the current density of the background It also depends on the angular aperture Whereas the angle illuminating the image peak ranges from 0 to ~ 2 2 t ~ , if there is no aperture stop the background is illuminated by the entire range of angles in the beam In order to avoid unnecessarily degrading the contrast in the Image, when there is appreciable background intensity it is important to limit the angular aperture to approximately what is used in the image peak, or to as small a value as compatible with a

G F Rempfer, 0 H Gnffith /Emtsston mtcroscopy and related techmques

46

tolerable diffraction hmlt The resolution in the presence of background intensity is not as good as for a dark background Several examples are worked out in ref [19] In the present paper we consider only the dark background case

4 3 The dlffractton hmttatton The diffraction hmitation is r d = 0 61)t/sin c~,

(26)

where A is the electron wavelength, and a is the angular aperture The wavelength is given by the de Broghe formula A = h/mt~, where h is Planck's constant In terms of the electron's beam voltage V = ½mvZ/e the wavelength (in ~ngstroms) is gwen by A = lX/q~-/V For an accelerating voltage V~ very much greater than the emission voltage Ve, the wavelength is Aa = 1 5 0 ~ , and the ray angle after acceleration is given by sin a a ~ee/Va sin o~e ~ a a The diffraction limitation can then be written r d = 0 61Aa/a

(26')

a

For the numerical parameters V~ = 30 kV, Ve 025 V, and sin a e = 1, we find /~a = 0071 A, a a = 2 9 mrad, and r d = 1 5 /k In terms of the quantities at the specimen before acceleration, r d = 0 61Ae/Sln

= 0 61~/sin

a e = 0

61h/(mv e

a¢

sin

ae)

(26")

The second form of the expression for r d in eq (26") calls attention to the fact that the diffraction limitation depends on the transverse momentum mv e sin ae Thus, even if the axial component of velocity goes to zero, as it does in MEM, this does not mean that the diffraction error goes to infinity In PEM, Ve is the emission voltage 1 2 (TroVe~e) of the photoelectrons, and in LEEM, Ve is the landing and reflection voltage supplied by the electron gun In MEM, V~ is also supplied by the electron gun, but since z e is zero on reflection, ole is always 90 °, and V~ = a2V. If the accelerating voltage and the angular aperture for the imaging beam are the same in L E E M and

MEM as m PEM, the diffraction hmltatlon will be the same in each mode

4 4 The workmg resolutton The working resolution combines the geometrical resolution hmlt and the diffraction hmlt In the intensity distribution approach, the geometrical resolution limit is determined by the aberration coefficients of the accelerating field and objective lens, and the current needed to record the image m a realistic time The diffraction hmltation can be taken into account approximately by increasing the effective size of the object, as in a recent paper by us [19], or by combining the geometrical and diffraction errors quadratically, as in

6 = ~/~-g2+ r0Z,

(27)

where r d IS defined by eq (26') or (26"), and rg is given by eq (25) The results of these methods for including the diffraction effect do not differ very much Here we will use eq (27), which 1s the simplest method This expression has the same form as the equation used to combine the spherical and diffraction errors in the circle of least confusion method However, there is an important difference In the circle of least confusion approach the two errors are interdependent since both terms depend on the angular aperture a The optimum angular aperture is obtained by minimizing the resultant resolution In the intensity-distribution approach the geometrical error has already been optimized to provide the best resolution for the required image current An aperture stop does not affect the high-intensity Image peak for angles larger than those which actually contribute to the peak However, an aperture stop is needed to preserve contrast in the presence of background illumination as mentloned in section 4 2

4 5 Numertcal examples We illustrate the procedure for arriving at the working resolution with some numerical examples The following electron optical parameters

47

G F Rempfer, 0 H Gnffith / Ermsston mtcroscopy and related techmques

are used throughout the examples Va = 30 kV, 1 = 3 mm, also Csi -- 19 cm and Ccl= 1 9 cm for an electrostatic objective lens A current of 10 electrons per second in the image peak is assumed (this would give a sufficiently non-noisy image in 30 s) The electrons leaving the specimen have a cosine angular distribution The angular aperture of the accelerated beam is a a = 2 9 mrad, which allows a transverse energy of 0 25 eV and a diffraction limit of 15 Other things being equal, the resolution In L E E M is the same as in PEM for the same current density and angular distribution The current density which can be produced at the specimen by the electron gun in L E E M and MEM IS much higher than the current density which at present can be achieved in PEM We calculate the resolution In PEM and LEEM, and m MEM, with the presently achievable current density in PEM, about 5 x 10-5 A / c m 2, and also with ten times this current density, 5 × 10 -4 A / c m 2, realistic for L E E M and MEM, and hopefully in the future for PEM with improved illuminating sources and specimen emissivities Two different Initial energy ranges are considered for PEM and LEEM In U V PEM the photon energies are close to threshold, and the emission energies are of the order of 0 5 eV or less In the first example we calculate the resolution in PEM and L E E M for the ideahzed case of a single emission or reflection energy of 0 25 eV For this energy C s in PEM and L E E M is 200 cm (see table 1) First, eq (21) for a single-energy beam is used to calculate atom, the radius of the smallest object whose image receives the reqmred current of 10 elect r o n s / s = 1 6 × 10-18 A The geometrical workmg resolution rg is then obtained from eq (25), and the diffraction effect is included according to eq (27) to give the resultant resolution 8 The numerical values of these quantities for PEM and L E E M with Je = 5 × 10 -5 A / e m 2 are o

am,n = 1 7 5 A ,

o

rg=42A,

rg--18.~,

o

atom = 14 5 A,

o

o

rg = 35 A,

8 = 38 A,

(28c)

and with Je = 5 X 10 -4 A / c m 2, are am,n=612.~,

rg=147.~,

~=21A

(28d)

While the resolution figures for MEM appear to be slightly better than for the other modes, these figures do not include the effect touched on earher, that whereas paraxlal electrons just reach the specimen surface, non-paraxaal electrons are turned back without reaching the surface, and in effect image potential surfaces which do not have as much contrast and sharp detail as the specimen surface itself The reformation which the non-paraxlal electrons carry therefore dilutes the information carried by the paraxml electrons A specimen bias such that the electrons are split between M E M and L E E M modes may be optimal for resolution. We next calculate the resolution in PEM and L E E M for a range of emission energies from 0 to 0 5 V We assume that the Illuminating electron beam in L E E M and M E M has that range of energies With the specimen biased so that paraxial electrons at the low end of the distribution can reach the specimen, L E E M electrons have a range of energies form 0 to 0 5 eV However, there are some off-axis electrons In the lower part of the energy distribution which are M E M electrons In this example we will ignore any difference the MEM electrons might make Since there is a range of energies we use eq (23), where C~ = 6C~, and Cj = 6 8 × 105/cm for V~ = 0 25 V For PEM and LEEM, with Je = 5 × 10 -5 A / c m e, we get

o

8=45A,

(28a)

and with J. = 5 × 10 - 4 A / c m 2 are am~.=74.~,

Imaging in the M E M mode takes place with the specimen biased so that paraxlal electrons just reach the specimen The marginal electrons, however, retain transverse energies of e V e , and hence are turned back before reaching the specimen With C s = 95 4 cm for the M E M mode, and Ve = 0 25 V, the values obtained from eq (21), with Je = 5 × 10 -5 A / c m z, are

8=23.~

o

am,n = 2 0 A ,

o

rg=48A,

o

8=50A,

(28e)

a n d w i t h Je = 5 × 10 - 4 A / c m 2, w e get

(28b)

am~n=93.~,

r~=223~,

8=27.~

(28f)

G F Rempfer, 0 H Gnfftth / Ermsston microscopy and related techmques

48

For the MEM mode the chromatic aberration coefficient is low enough that the single-energy expression, eq (21), can be used, giving us the same values as in eqs (28c) and (28d) However, with the specimen biased so that only paraxial electrons with the highest energies can reach it, the range of distances from the specimen at which the remaining electrons are turned back is about twice as great as in the single-energy case, and the contrast and resolution are further diluted Again, for MEM, and for very low energy LEEM, some mixture of modes may be optimal As a third example we calculate the resolution in PEM and L E E M for V~ = 10 V, which is about the middle of the L E E M range of reflection voltages, and Ace = 1 V This energy range is narrow enough that eq (21) can again be used, this time with C s = 112 cm and Va/Ve = 3 X 103 For PEM and LEEM, with Je = 5 × 10 5 A / c m 2, o

o

am,n = 61 A,

rg = 146 A,

o

6 = 147 A,

(28g)

and with Je = 5 × 10 - 4 A / c m 2 am,n=26,~

,

rg=62,~,

8=64A

(28h)

The foregoing numerical results for the working resolution are summarized in table 2 We note again that the larger current density Je = 5 × 10 . 4 A / c m 2 is not yet realistic in PEM However, in L E E M and MEM the values of Je can be higher than those used in the examples, with an accompanying improvement in the calculated values of the working resolution In using the same cosine angular distribution for the electrons leaving the specimen in the three imaging modes, we considered the specimen to be amorphous, wlth or without topographical relief However, in surface physics, L E E M has been applied mainly to highly-ordered plane surfaces of crystalline metals and semiconductors [4] With this type of specimen the electrons are reflected in a set of narrow diffracted beams, rather than in a continuous angular distribution In the case of a 10 eV reflected electron, the wavelength at the specimen is Ae = 3 87 .~ For a lattice spacing of, say, 3 .~ only the (0, 0) order occurs and the angular distribution depends on

Table 2 The working resolution in PEM, LEEM, and MEM for a reqmred current in the image, calculated for several values of the emission or reflection current density and energy distribution A~

(v)

(v)

0 25 0 25 0 25 0 25 10 0

00 05 00 05 10

6(A) = 5X10 -5 A/cm 2 38 38 45 50 147

Mode ~ = 5 × 1 0 -4 A/cm z 21 21 23 27 64

MEM MEM PEM/LEEM PEM/LEEM PEM/LEEM

The working resolution 8 is calculated for a current of 10 electrons per second in the image peak The average emission or reflection voltage and the range are denoted by V~, and AV~ In the mirror mode Vc is the transverse reflection voltage for the marginal rays, a2Vd Jc is the current density at the specimen, the higher value 5 × 10 -4 A / c m 2 Js not at present available in PEM

the angular aperture of the illuminating beam Smce the reflected electrons are concentrated in this one beam, the intensity is high On the other hand, if the specimen surface were perfectly unlform (with only the 3 ,~ periodicity), and were imaged with only the narrow (0, 0) diffracted beam, the diffraction error would be large and the image would contain no details However, the specimen is not completely uniform, and the angular aperture outside the 3 ,~ (0, 0) diffracted beam ~s not empty, but is occupied by small-angle diffracted beams from regular structures w~th larger periods, or by beams produced by steps and other features on the specimen surface It is these beams which determine the resolution In principle, for conditions such that the firstorder diffracted beams can be included in the imaging beam, it should be possible for L E E M to do lattice imaging, as in TEM An advantage of this type of imaging is that spherical aberration is essentially avoided A larger reflection voltage, say 20 V, could be used to produce the first-order diffracted beams for a lattice spacing of 3 ,~ However, the angle which the diffracted beam makes with the axas would be quite large, 24 mrad after acceleration through 30 kV and before divergence by the anode aperture lens For larger lattice spacings the lens system would not have to

G F Rempfer, 0 H Grtffith / Emtsston mtcroscopy and related techmques

49

Fig 7 Photoelectron mlcrograph of colloidal sdver demonstrating 70 ,~ resolution To increase specimen brightness the colloidal sdver was briefly exposed to c e s m m vapor In photoelectron imaging the limiting factor is frequently emission current density rather than instrument resolution

accept such large angles, and lower reflection voltages could be used The first-order diffracted beams for a 25 A spacing would just be included m the 2 9 mrad aperture used in the numerical examples The diffraction hmltatlon in this case Is

25 A, The difference between this figure and the 15 .~ figure for the cosine distribution has to do with the fact that the angular aperture is occupied only at the diffraction angles in the first case, but is occupied throughout the 2 9 mrad

50

G F Rempfer, 0 H Grtffith / Emission mtcroscopy and related techrtlques

aperture m the second case A wide spectrum of periodicities can be found in both physical and biological structures An example in surface physics is the (7 × 7) diamond-shaped unit cell on the silicon (111) surface, with an edge length of 26 ,~ Examples from biology include the crystalhne m e m b r a n e proteins bacteriorhodopsln, gap junctions, and cytochrome c oxldase, with periodicities ranging from 60 to 100 .~ for the protein positions in the lattice, down to 1 5 ~. for the indwldual C - C bonds within each protein, and DNA, which has a 3 4 A periodicity created by the bases and a 34 ,~, helical repeat

5. Experimental test of resolution in PEM For the PEM resolution test, colloidal silver was prepared by the method of Carey Lea [26] as described by Frens and Overbeek [27] In this procedure the colloid is formed by reducing silver nitrate with a mixture of sodium citrate and ferrous sulfate The reducing agent was formed by mixing 3 5 ml of a stock aqueous solution of trlsodlum citrate 2 H 2 0 (400 m g / m l ) with 2 5 ml of a freshly prepared stock solution of FeSO 4 7 H 2 0 (300 m g / m l ) The reducing solution (6 ml) was then added with vigorous stirring to 2 5 ml of a silver nitrate solution (100 m g / m l ) forming a brownish-black precipitate, which was isolated by centrlfugatIon at 100000 × g for 1 h After removal of the supernatant, the pellet was redlspersed in water, reprecipItated by addition of an equal volume of 40% trisodlum citrate, and recentrifuged The redIspersal-centrlfugatlon process was carried out a total of three times with the final pellet obtained being dispersed in 1700 ml of distilled water (Redispersal in a smaller volume of water resulted in the coalescence of the colloidal particles into large aggregates) By transmission electron microscopy, the colloidal Sliver particles ranged between 6 and 22 nm m diameter with most of the particles being approximately 8 nm The P E M specimen mount was a 5 m m glass covershp coated first with 5 nm chromium to make it conductive, and then overcoated with a thin layer of spermldine-derwatized dextran pre-

pared by the procedure described previously for dlamlnohexane-derlvatlzed dextran [28] The glass disc was floated on the top of a 10 izl drop of the colloidal silver suspension for 1 h, followed by extensive washing with distilled water Excess water was drawn off and the disc was allowed to air dry The specimen was placed in the work chamber of the P E M and dried under vacuum It was then exposed briefly to cesium metal vapor, as described elsewhere [29] The cesium greatly enhances the photoemlsslon, probably by formation of a A g - O - C s photocathode [30] Finally, the specimen was transferred directly into the PEM without exposing It to air The PEM is an oil-free ultrahigh vacuum instrument described in detail In refs [31,32] A pair of photoelectron mlcrographs is shown in fig 7 The bright objects are the colloidal silver particles, or aggregates of silver particles The images were recorded with about 1 rain exposure The mlcrographs demonstrate an Instrument resolution of about 7 0 / k

6. Discussion Calculating the resolution of PEM, L E E M and M E M is somewhat more involved than for a T E M The accelerating field has both spherical and chromatic aberrations The spherical and chromatic aberration coefficients are large because some of the electron optics takes place when the electrons have very low energies and the spread in energy associated with the axial direction is of the same order as the average energy The spherical and chromatic aberrations of the objective lens are larger than in T E M because the focal distance must be long enough to focus on the virtual object on the other side of the accelerating gap The aberrations of the anode aperture lens are negligible However, including the effects of the anode aperture are important because it acts as a weakly diverging lens and forms a virtual image at magnification 2 / 3 In order to combine the aberrations of the objective lens with those of the accelerating field at magmficatlon 1, the spherical and chromatic aberrations of the lens consequently must be multiplied

G F Rempfer, 0 H Grtffith /Emtsston mtcroscopy and related techntques 3 2 (7) , respectively Especially in PEM, where the emission current densities are low, the effect of object size cannot be ignored For these reasons It is important to consider all of the following factors in calculating the resolution the spherical and chromatic aberrations of the accelerating field, the spherical and chromatic aberrations of the objective lens, the effect of the anode aperture, the intensity distribution in the image, the location of the optimum image plane, specimen brightness and the size of the object detail needed to obtain the required current in the image, and, finally, the diffraction limit The geometric approach was selected for this calculation of resolution of PEM, LEEM, and MEM because it reveals characteristics of the intensity distribution which could go unnoticed in a wave optics approach The early treatments by Henneberg, Recknagel, Langmmr, and Artslmovich established the importance of the aberrations inherent in a uniform accelerating field [7-11] In these treatments the resolution hmlt imposed by the accelerating field was found to be proportional to the energy of the emitted electrons and inversely proportional to the electric field at the surface of the specimen Taking into account the spherical aberration of the accelerating field and allowing all values of the emission angle a e (1 e no aperture), the equation for the circle of least confusion for a monoenergetic beam of electrons having an initial kinetic energy e = e V e is

by (3)4 and

r~c = 0 6( V e / E ) = 0 6 l V J V a,

(29)

where E = V a / l IS the electric field strength at the surface of the specimen and Va is the accelerating voltage Eq (29) is often called the Recknagel formula For an accelerating gap l = 3 mm, Ve = 0 25 V, and V~ = 30 kV, this equation gives a value of rlc = 150 A Wu [12] appears to have been the first to publish an intensity distribution treatment of the accelerating field This work, based on geometrical optics, takes into account the spherical and chromatic aberrations of the accelerating field From a consideration of the Intensity distribution as a function of depth in the image, he locates

51

the optimum poSltlOn of the image plane for both a point source and for objects of finite (non-zero) size Wu arrives at an estimate of the geometrical resolution which is a factor of ten smaller than given by eq (29) We became aware of this paper only after completion of the present calculations Where there is overlap we are in agreement, but our calculations carry the intensity distribution approach further We include the aberrations of the objective lens and the diffraction effect We also derive analytical relationships for the height of the high-intensity image peak and for the current included in the image peak This enables us to define a working resolution which reqmres a practical current in the image Storbeck carried out a full wave-optical calculation of resolution in PEM which included the object size, chromatic aberration, and a contrast criterion, but not a current requirement [13,14] Our results are consistent with Storbeck's, if account is taken of the differences in electron optical parameters in the two treatments Storbeck's treatment did not take into account the aberrations of the objective lens and did not reveal the interesting effects resulting from the combination of spherical aberration, defocus, and object size Engel calculated the resolution of PEM taking into account the energy distribution of the emitted electrons His model began with a point source and made use of the fact that the aberrations of the accelerating field do not depend on the transverse velocity but only on the axial component of the emission velocity [15,16] For his choice of parameters, Engel calculated a theoretical resolution of 120 ,~ He achieved this resolution in an instrument of his design [15,16] At about the same time, Uchlkawa and Maruse calculated the chromatic aberration of the accelerating field and objective lens, and combined them linearly [17] They studied the chromatic difference of magnification as well as the chromatic difference of the image position Recently, Llebl [18] published a dlscusslon of the aberrations of the uniform acceleration field and the effect of the aperture stop from the point of view of energy filtering Bauer performed the first resolution calculations specifically for L E E M and compared it to emission electron microscopy (e g PEM) [20]

52

G F Rempfer, 0 H Grtfftth /Emtsston mtcroscopy and related techmques

Bauer's calculation is for a point source It is based on the geometrical aberrations of the accelerating field combined with the diffraction hm~tatlon He did not include the aberrations of the objective lens Bauer assumed different conditions for emission microscopy than for LEEM, which led to a poorer estimation for resolution in emission microscopy compared to L E E M It can certainly be argued that the practical resolution of L E E M may be better than in emission microscopy because the current density available at the specimen is larger (one can always turn up the beam current in LEEM, whereas the available hght sources for PEM usually do not prowde as high a current density, unless a synchrotron or laser xs available) However, for the s a m e conditions, the resolution in emission microscopy and in L E E M should be the same since the aberrations for emission microscopy and L E E M are the same (see table 1) One difference in the two treatments is that Bauer shows the resolution improving as the reflection voltage Increases for a constant accelerating field The reason for this is that the spherical and chromatic aberrations for the accelerating field, and hence the overall aberrations, decrease as the reflection voltage increases (see table 1) However, as the reflection voltage increases the fraction of the rays that are included in a given angle after acceleration decreases, and hence the intensity decreases The aberrations of the accelerating field depend on the square root of the reflection voltage, and the intensity depends on the first power Therefore, the net effect of the increased reflection voltage is to decrease the resolution in the intensity distribution approach, where a given current is required In the image Both treatments predict that the resolution increases with increasing field strength, as does the Recknagel formula, eq (29) We are in general agreement with Bauer's main conclusion, namely, that for LEEM, resolutions m the nm range should be practical [20] Another calculation of resolution m L E E M pubhshed recently [21] takes into account the aberrations of the accelerating field and the objective lens and the diffraction effect The resolution hmltatlons are combined by means of an approximate wave optical method A resolution

limit as small as 3 nm is calculated However, a sign error in eq (8) of ref [21] leads to a negative value for the chromatic aberration coefficient of the accelerating field, eq (14) of ref [21], whereas the chromatic aberration coefficient is actually positive (the image moves forward in the direction of the beam as the electron energy is increased) In contrast to what is stated in ref [21], the chromatic aberration coefficient is positive for electron lenses as well The effect of object size is not considered and the effect of the anode aperture is not mentioned explicitly, although it could be included as part of the objective lens The aberration coefficients assumed, C~ = 30 mm and C c = 15 mm, of the objective lens are unrealist~cally small, considering the long object distance (6 mm + the distance from the anode to the optical center of the lens) For example, the magnetic lens designed by Engel for emission microscopy has a spherical aberration coefficient referred to virtual specimen space of 300 mm, ten times as large [16] There have been relatively few calculations of resolution in MEM [5,6,22,23], although all of the above references are relevant since MEM also utilizes the accelerating field combined with an objective lens In the early studies of MEM the ~mage formed was not a focused image, but rather a type of projection image This simple type of mirror microscope works surprisingly well up to a point, but it has basic shortcomings Since the image ~s not focused, the ultimate resolution is hm~ted by the effective s~ze of the projection source, and by diffraction Also, contrast is reduced by the energy spread in the beam, with higher energy electrons impinging on the specimen and causing wash-out, and lower energy electrons not getting close enough to the spec~men to be affected slgmficantly by the mlcroflelds Another drawback is that after entering the mirror the beam is diverging, and rays at different distances off-axis do not come equally close to the specimen What is changing this situation for the better ~s the continuing development of beam separating systems, which makes feasible a focused image of high quahty The same beam-separating system is used for LEEM, MEM and optionally with PEM (PEM does not