Hawkes P.W., Spence J.C.H. (Eds.) Science of Microscopy. V.1 and 2

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Chapter 9 Photoemission Electron Microscopy (PEEM) 665

transmission. A multipole stigmator–deﬂ ector is located close to the

back focal plane of the objective lens to compensate astigmatism and

alignment errors. The intermediate lens and projective lens transfer

and magnify the image onto a microchannel plate or a phosphor

screen. This phosphor transforms the electron into a visible image,

which is captured by a CCD camera.

3.1 The Objective Lens

The objective lens is the most important component of the electron

optical system. Triode and tetrode cathode lenses have been used.

Chmelik et al. compared several types of cathode lenses (Chmelik et

al., 1989). Rempfer et al. analyzed different objective lens types based

on extensive electron optical bench measurements (Rempfer et al., 1999;

Rempfer and Grifﬁ th, 1989).

The lens evaluation occurs in four steps: (1) calculation of the ﬁ eld

distribution of the lens, (2) calculation of the electron trajectories and

the electron optical properties, (3) evaluation of the interaction of the

lens with other parts of the electron optics, and (4) tolerance analysis

in regard to mechanical errors. Often an analytical treatment is not

possible and numerical methods must be used, which beneﬁ t from the

rapid progress in computer technology. A charged particle optics

(CPO) program was developed by Read for solving charged particle

optics problems using the surface charge method (Harting and Read,

1976).* Munro has developed a series of electron optics programs using

the ﬁ nite element method (Hawkes, 1973; Orloff, 1997). Perhaps the

most popular program used in electron optics is SIMION 3D (http://

www.simion.com) which was developed by Dahl (Dahl, 2000). SIMION

is an electrostatic and magnetic ﬁ eld modeling program that solves the

ﬁ eld using a ﬁ nite element method and traces the motion of electrons

using a fourth-order Range-Kutta integrator. Figure 9–7 shows the

equipotential contours of a tetrode objective lens, calculated using

SIMION. The acceleration ﬁ eld is approximately uniform in the vicin-

ity of the optical axis except close to the anode aperture. The aperture

acts as a thin diverging lens (Rempfer and Grifﬁ th, 1989; Davisson and

Calbrick, 1932).

The objective lens can be characterized by the trajectory of the prin-

cipal rays, the ﬁ eld ray and the axial ray (Figure 9–8). The calculation

treats the electrodes as a single lens because the ﬁ elds overlap, but it is

instructive to describe the objective lens optics as being composed of

(1) the accelerating ﬁ eld, (2) a weak diverging lens, and (3) a unipoten-

tial lens. The acceleration ﬁ eld forms a virtual image at unit magniﬁ ca-

tion at twice the sample distance l. The weak diverging lens created by

the aperture of the ﬁ rst electrode forms another virtual image at dis-

tance 3/l behind the sample. It is demagniﬁ ed by M = 2/3. Finally, the

unipotential lens forms a real image with a typical magniﬁ cation of

M = 10–40 (Anders et al., 1999; Schneider and Schöhense, 2002; Watts

et al., 1997).

* Student versions of the program are available at no coot from http://www.

electronoptics.com/.

666 J. Feng and A. Scholl

3.2 The Intermediate and Projective Lens

The intermediate and projective lenses magnify the image of the objec-

tive lens without distortion to reach a ﬁ nal magniﬁ cation typically

between 30 and 4000. The angle and energy-dependent aberrations of

the intermediate and projective lenses are negligible because the energy

and angular spread of the electron beam are small here. However, the

distortion aberrations need to be minimal because the object size after

magniﬁ cation by the objective lens is signiﬁ cant, fractions of a milli-

meter or more. Therefore, the intermediate and projective lenses are

designed to work at minimum focal length. As the excitation of the

electrostatic lens is increased, its focal length decreases and the object

side focal point enters the lens ﬁ eld. The distortion coefﬁ cient can reach

zero at minimum focal length. This type of lens has been examined in

detail by Rempfer et al. (1991) who studied rotationally symmetric,

three electrode, unipotential lenses (Figure 9–9). The term unipotential

means that the image and object side electrodes are on the same poten-

tial. Important lens parameters are the thickness of the center electrode

t, the interelectrode spacing S, the diameter of the center electrode D,

the diameter of the end electrode aperture A, and the radius of the

center electrode R. Starting with parameters chosen from diagrams

Figure 9–7. Equipotential contours of the electric ﬁ eld of an electrostatic

PEEM objective lens at 20 kV sample voltage.

Chapter 9 Photoemission Electron Microscopy (PEEM) 667

Figure 9–8. Principal electron rays in an objective lens: dashed line, axial ray;

solid line, ﬁ eld ray; F, back focal plane; I, image plane.

Figure 9–9. Electrode

geometry of a symmet-

ric unipotential lens.

668 J. Feng and A. Scholl

published by Rempfer (1985), the lens properties can be optimized for

a particular application using numerical methods.

3.3 Transmission and Spatial Resolution of PEEM

The microscope transmission and spatial resolution determine the

quality of a PEEM. Bauer (1991) has used these two quantities to deﬁ ne

a quality factor. The spatial resolution determines how well spatial

details are preserved, and the transmission determines the noise level

in an image and the acquisition speed (Jacobsen et al., 1992; Schneider,

1998). The sample is an integral part of PEEM optics, and the optimum

resolution can be achieved only on ideally ﬂ at surfaces. The effects of

roughness and sample tilt have been previously discussed (Marcus,

2001; Nepijko et al., 2000).

3.3.1 Transmission

The microscope transmission is deﬁ ned as the fraction of emitted elec-

trons that reaches the detector. To reduce aberrations most microscopes

contain several apertures of different size that can be moved into the

electron beam. Thus transmission can be traded for resolution. The

aperture conﬁ nes the electron beam to the paraxial region and can also

limit the energy acceptance of the microscope. Using E/(E + w

)

for

the electron energy distribution for a work function w

and using

Lambert’s law for the electron angular distribution, the transmission

of PEEM can be evaluated in a simple closed form

=−

[( )( )]af U/W/

(1)

Here, a is the aperture radius, U is the sample potential, and f

* is the

image side focal length of the objective lens. A high sample potential

and a large aperture improve transmission, whereas a long focal length

and large work function reduce transmission (Feng et al., 2002).

Figure 9–10 shows SIMION simulations of the angle and energy

distribution of the electrons transmitted through apertures of different

size. Every point represents one electron. Both the energy spread and

the transmission decrease with decreasing aperture size. Note that

high-energy, paraxial electrons still pass through the aperture. The

resolution is dominated by electron diffraction when the aperture size

is lowered beyond a certain minimum value. Therefore, apertures with

a diameter larger than 10 µm are usually used.

3.3.2 Spatial Resolution

The spatial resolution of microscopes is characterized by a spread func-

tion. The spread function describes the microscope image of a chro-

matic point source and a diverging electron beam. Diffraction, spherical,

and chromatic aberrations are regarded as independent contributions

and are added quadratically. The total resolution is given by

r rrr

tot s

=++

(2)

where the third-order spherical aberration is deﬁ ned as

= C

(3)

Chapter 9 Photoemission Electron Microscopy (PEEM) 669

Figure 9–10. SIMION simulations of the angle and energy distribution of the

emitted electrons that are accepted by X-PEEM when apertures of different size

are used. (a–d) The angle; (e–h) the energy distribution.

the ﬁ rst-order chromatic aberration is deﬁ ned as

= C

αδE/(E) (4)

and the diffraction disk is deﬁ ned as

= 0.61λ/α (5)

670 J. Feng and A. Scholl

Here, C

and C

are the spherical and chromatic aberration coefﬁ cients,

λ and E are the electron wavelength and the electron energy, δE is the

electron energy spread, and α is the half angle of the pencil beam

passing the back focal plane aperture. Though Eqs. (2)–(5) can be easily

solved, the results are often misleading because the contributions of

aberrations and diffraction are not Gaussian and therefore the equations

give only a rough estimate of the microscope resolution.

An alternate way of characterizing the spatial resolution is by deter-

mining the spatial frequency transfer of the microscope. The intensity

distribution of an image produced by a point object under incoherent

illumination is called the impulse response function h(x, y). The object

intensity distribution f(x, y) and the image intensity distribution g(x, y)

are related by the convolution equation

gxy f x y hx xyydxdy() (,)( , ), =−−

∫∫

11 1 1 11

(6)

After a Fourier transformation one obtains

G( f

, f

) = F( f

, f

) ⋅ H( f

, f

) (7)

G, F, and H are Fourier transforms of g(x, y), f(x, y), and h(x, y). The

function H( fx, fy) is the complex optical transfer function (OTF), with

real and imaginary parts

OTF( f ) = |H( fx, fy)|e

iφ( f )

(8)

The real part |H( fx, fy)| is called the modulation transfer function

or MTF, and the function φ( f ) is called the phase transfer function.

Figure 9–11. Measured (black dots) and calculated (solid line) modulation

transfer function (MTF) of PEEM-2 at the Advanced Light Source.

Chapter 9 Photoemission Electron Microscopy (PEEM) 671

Experimentally, the MTF is measured as the contrast with which spatial

frequencies of an object are transferred to an image. Higher frequencies

correspond to smaller object features. Figure 9–11 shows the measured

and calculated MTF of PEEM-2 at the Advanced Light Source using an

extraction ﬁ eld of 20 kV and a 12-µm aperture (Doran et al., in prepara-

tion). It starts with unity modulation transfer at low frequencies and

gradually drops at higher spatial frequencies. A transfer of 1 indicates

that the spatial detail of an object at that frequency is perfectly trans-

ferred, while zero transfer indicates a complete loss of information.

Values between 0 and 1 indicate partial transfer of spatial frequencies

and a reduced contrast. The Rayleigh criterion for spatial resolution is

equivalent to a 9% contrast transfer, which is reached for a frequency

corresponding to a feature size of 35 nm. Often a higher contrast is

required in experiments. At about 80 nm PEEM-2 reaches 40% contrast

transfer. The agreement between experiment and simulation over a

wide spatial frequency range conﬁ rms the accuracy of the simulation.

4 Aberration-Corrected PEEM Microscopes

The improvements in resolution of light and electron microscopes over

the past two centuries are compared in Figure 9–12 (Rose, 1994). At the

beginning of the twentieth century the light microscope had reached

its best resolution after the invention of aberration correction by Abbe,

using a combination of a concave and a convex lens, called an achromat

lens. The electron microscope soon surpassed the resolution of optical

microscopes because of the much smaller wavelength of kilovolt elec-

Figure 9–12. History of microscopy techniques and evolution of spatial reso-

lution. Shown are signiﬁ cant steps in the development of optical microscopy,

electron microscopy, and X-PEEM. (From Rose, 1994).

672 J. Feng and A. Scholl

trons, effectively removing the effect of diffraction, if electrons of suf-

ﬁ ciently high energy are used. Today, the electron microscope has not

reached the diffraction limit and lens aberrations still dominate. Among

these aberrations, astigmatism, coma, and ﬁ eld distortions can either

be easily corrected or they are small close to the optical axis of the

microscope*. Early in the development of electron optics theory, Scher-

zer showed that simple electron lenses always suffer from two other

important aberrations: chromatic and spherical aberrations (Scherzer,

1936). The electron-optical analogue to a light-optical achromat does

not exist. Different approaches have been tried to remove these aber-

rations by relinquishing one of the conditions of Scherzer’s theorem,

which applies only to simple lenses (Hawkes et al., 1996). Multipole

aberration correctors, which use a nonrotationally symmetric ﬁ eld,

have been successfully used in a low-voltage scanning electron micro-

scope (SEM) (Zach and Haider, 1998), in transmission electron micro-

scopes (TEMs) (Haider et al., 1995, 1998), and in scanning transmission

electron microscopes (STEMs) (Dellby et al., 2001; Batson et al., 2002).

X-PEEM has much larger aberration coefﬁ cients than SEM or TEM,

because of the large energy width of the secondary electron distribu-

tion, the low initial electron energy, and the large ﬁ eld of view. There-

fore, the aberrations of X-PEEM are corrected using an electron mirror,

which has approximately opposite chromatic and spherical aberrations

of the objective lens.

4.1 Aberrations of an Electron Lens

The electron trajectories in an electron lens are affected by spherical

and chromatic aberrations (Figure 9–13). The Gaussian image plan is

deﬁ ned as the intersection of the paraxial ray with the optic axis. The

spherical aberration refers to the variation of the focal distance with

the angle of a ray from the paraxial ray. An electron passing the lens

far from its center experiences a higher ﬁ eld and is focused more

strongly. The focus lies before the image plane. The chromatic aberra-

tion refers to the variation of the focal distance with the electron energy.

Slower electrons experience the ﬁ eld for a longer time and are focused

more strongly. The focus lies before (slow electrons) or behind (fast

electrons) the image plane.

Figure 9–14 shows the resolution of a typical X-PEEM with a 2-mm

acceleration gap and 20-kV acceleration ﬁ eld for a 4-eV work function,

broken down according to the source of the aberration. Dominant are

the aberration caused by the accelerating ﬁ eld and the chromatic aber-

ration of the lens. Although the resolution can be improved by using

a very small aperture, the usefulness of this approach is limited because

of the severe reduction in microscope transmission and the onset of

diffraction at very small aperture sizes. An ultimate resolution of close

to 20 nm has been achieved in several instruments (Anders et al., 1999;

De Stassio et al., 1999; Bauer, 1997; Schönhense and Spiecker, 2002).

* A simple electrostatic lens is rotational symmetric, uses static ﬁ elds, is space

charge free, and the electronic lenses does not reverse its direction.

Chapter 9 Photoemission Electron Microscopy (PEEM) 673

Figure 9–13. Illustration of spherical (top) and chromatic (bottom) aberrations

of an electron lens.

Figure 9–14. Contributions of various aberrations to the total resolution limit

on uncorrected PEEM. Sample voltage 20 kV, gap = 2 mm, work function w

4 eV. tot, total aberration; acc, aberration of accelerating ﬁ eld; chr, chromatic

aberration of objective lens; sph, spherical aberration of objective lens; diff,

contribution of diffraction.

674 J. Feng and A. Scholl

Aberration correction promises to signiﬁ cantly improve the ultimate

resolution beyond this limit and also to increase the transmission of

X-PEEM at moderate resolution by a factor of 100 (Bauer, 2001).

4.2 An Aberration-Corrected Microscope Using an Electron Mirror

The idea of using an electron mirror to correct the chromatic and spheri-

cal aberration of a round lens dates back more than half a century

(Zworykin and Kosma, 1945; Ramberg, 1949). Extensive studies of elec-

tron mirrors have been performed by Kel’man (1973, 1974), Dodin (1981),

Rempfer (1990, 1992, 1997), Shao (1990a, 1990b), and Rose (1995, 1997),

and co-workers. An electron mirror uses a reﬂ ecting electrode with a

sufﬁ ciently high negative potential to reverse the propagation direction

of the electron beam. In 1990, Shao and Wu proposed using an electron

mirror with more than two electrodes to be able to freely adjust the

aberration coefﬁ cients.

Through numerical analysis of a four-

electrode electric mirror, they showed that spherical and chromatic

aberrations can be adjusted electrically without changing the image

distance of the mirror. Later, a more sophisticated theoretical model

using a time-dependent perturbation method was developed by Rose

and Preikszas (1995) to fully understand the mirror system. Integral

expressions for the aberration coefﬁ cients of an electron mirror were

derived and higher order aberrations were studied. Currently, two

aberration-corrected PEEM microscopes using electron mirrors are

being built. The ﬁ rst is called SMART (SpectroMicroscope for All

Relevant Techniques) in Germany and has been designed as an ultra-

high-resolution spectromicroscope for BESSY II, Germany (Fink et al.,

2003). The second is called PEEM-3 at the Advanced Light Source, USA,

and has been designed for the study of magnetic materials and poly-

mers. Both microscopes rely on the correction of chromatic and spheri-

cal aberrations using a hyperbolic electron mirror, which was pioneered

by Rempfer and co-workers (1997). Here, we will focus on the electron

optical system of the aberration-corrected PEEM-3 experiment, although

many aspects of the two aberration-corrected microscopes are similar

and are based on the Rose design.

Figure 9–15 shows a schematic overview of the electron optics of

PEEM-3. The electrons travel through the microscope along the dashed

lines. The objective lens, together with the ﬁ eld lens, forms a telescopic

round lens system. This allows the mirror to run in the so-called sym-

metric mode in which ﬁ rst-order chromatic distortion and third-order

coma vanish and curvature of ﬁ eld effect is reduced (Rose and

Preikszas, 1992). A set of two electric dodecapoles steers the beam into

the magnetic separator and corrects astigmatism. For a single deﬂ ection

of 90°, the beam separator images its entrance plane 1 : 1 onto its exit

plane, without introducing aberrations of second order or dispersion of

the ﬁ rst or second degree. The electron mirror images the mirror-side

exit plane of the beam separator back onto itself. To cancel coma gener-

ated by the mirror, the magniﬁ cation of the mirror is chosen to be −1

(Wan et al., 2004). The separator then transfers the aberration-corrected

image to the projector optics, which magnify the image. A pair of elec-

tromagnetic dodecapoles in the mirror arm steers the beam into the

mirror and back into the separator. A CCD camera behind the mirror is