Cs-corrector.

Felix de Haas

Content

• Non corrector systems

–Lens aberrations and how to minimize?

• Corrector systems

–How is it done?

Lens aberrations

• Spherical aberration

• Astigmatism

• Coma

• Chromatic

• The aberrations of

electron optical lenses

defined a barrier that

has limited the

performance of the

electron microscopes

for a long period

Quality of electron lenses !

• the focal length is given by:

7

beam

K : constant

U : accelerating voltage

N : windings

I : lens current

2 ) ( I N

U K f

=

Lenses

Lenses

• Gaussian Law

8

F ’

f ’ f

s s ’

F

1 1 1 1

f f s s = = +

' '

Spherical aberration

What is Astigmatism

Astigmatism

Astigmatism

Astigmatism

Rotation center

Rotation center

Rotation center

Rotation center

Rotation center

Rotation center

Coma free

Coma free

Coma free

Coma free

Coma free

Coma free

Coma free

Coma free

Coma free

Coma free; Illumination passes through a symmetric path through the lens

Lenses: Chromatic Aberration

• Blurring due to energy spread in electron beam and lens current fluctuations

• Specimen thickness (mean-free-path)

31

a

Plane of least

confusion

P

D +

D =

I

I

E

E C c c

2 a d

What did we learn?

• How to optimize the microscope for;

– Astigmatism

» Stigmator for objective lens

– Rotation center

» Direct alignments

– Coma

» Direct alignments

» or script to tilt beam and observe FFT

» AutoCTF

– Chromatic aberration

» do not use areas where ice is too thick

• What and how?

Cs Corrector

BIMR Workshop

2007

History of the Transmission Electron Microscope

1931, Knoll and Ruska

Electrostatic lenses,

1933 magnetic lenses

Limited by lens defects: aberrations d=0.66Cs

1/4l3/4 resolution =1-2 Å

From H. Rose

37

Correcting Cs

• The idea of correcting for Cs: • To create a field which has an opposite character; i.e. the strength (or refraction) of this field

should decrease with increasing distance to the optical axis – which means negative Cs.

• Why not a concave electron lens? There are no concave electron lenses.

Scherzer theorem: Cs and Cc are always positive for:

• round lenses

• no charge on the axis

• systems where the field do not vary with the time

Therefore: Cs can only be corrected if the rotational symmetry is given up!

•Hexa- or dodeca-pole lenses

•Cs c

orre

cto

r

•Spherical Aberration (Cs) Correction

39

Hexapole Probe CS Corrector

Obj

Obj

Obj

Obj

Cs-corrector

40

A CS corrector corrects for coherent aberrations, in particular CS. Incoherent

aberrations (vibrations, instabilities) or Cc are not improved by a CS

corrector!

• BUT, once CS is corrected, other aberrations become important/dominant.

• Therefore, a corrector not only corrects for CS but for a whole series of

coherent aberrations; like astigmatism and coma and aberrations produced

by the corrector itself (S3, D4).

What’s not the purpose of a corrector:

• A CS corrector does not compensate for a misaligned column!

• Having a corrector, the optical axis of the microscope is given – the column

has to be aligned such that it meets this axis, NOT vice versa!

The Purpose of a CS Corrector

Contrast transfer (Correction)

9/8/2015 41

• Example

Optical axis

u f

Sample

𝐼(𝑥, 𝑦)

Lens

Ab.func. 𝜒 Back focal plane

ℱ 𝐼 𝑒−𝑖𝜒

Camera

ℱ−1 ℱ 𝐼 sin( 𝜒𝑒𝑣𝑒𝑛 𝑒−𝑖𝜒𝑜𝑑𝑑]

𝜒𝑒𝑣𝑒𝑛 𝑞 =2𝜋

𝜆(12𝜆

2𝑞2Δ𝑓+14𝐶𝑠𝑞

4)

𝜒𝑜𝑑𝑑 𝑞 = 0

Show result (e.g. as Thon

ring positions overlay)

Input image

Power spectrum Fitted Thon rings defocus + astigmatism value

•Hexa- or dodeca-pole lenses

•Cs c

orre

cto

r

•Spherical Aberration (Cs) Correction

•Defocus C1

•Twofold astigmatism A1

•Axial coma B2

•Threefold astigmatism A2

•Spherical aberration C3

•Star aberration S3

•Fourfold astigmatism A3

•……

44

The phase plate image

17mrad = 1.3 Å

Confidence limit

of detail

represented with

smaller than 45°

phase change.

10.5 mrad = 2.1 Å

6.7 mrad = 3.4 Å

1.3Å

A2: Threefold

astigmatism

Next parameter

suggested to

focus on with

optimization.

Calculated from determined parameters.

46

• Sall: 3.216nm Sused: 3.216nm (1.669%)

• C1: -5.261nm (95%: 4.31nm)

• A1: 5.626nm / -147.5deg (95%: 4.78nm)

• A2: 109.4nm / +162.7deg (95%: 49.4nm)

• B2: 58.97nm / +117.3deg (95%: 41.8nm)

• C3: 805.3nm (95%: 2.63um)

• A3: 2.746um / +13.7deg (95%: 508nm)

• S3: 1.578um / +151.7deg (95%: 231nm)

• A4: 48.34um / +123.9deg (95%: 13.1um)

• D4: 30.57um / +51.5deg (95%: 10.3um)

• B4: 60.34um / -156.9deg (95%: 17.8um)

• C5: 11.06mm (95%: 1.98mm)

• A5: 1.918mm / +86.4deg (95%: 377um)

What else do you get?

Fast tableau (15 mrad)

Standard (18-20 mrad)

Enhanced (30-35 mrad)

Adjustable

Factory adjustable or fixed

Error bar

Azimuth angle

Measured value

•2 nm

•Cs = 0.0015 mm •Cs = 1.2 mm

•Cs Corrector at work

•Atomically sharp gold specimen edge

•ON •OFF

2 nm2 nm 2 nm2 nm

Image : B.Freitag

Cs-corrected HR-TEM Non Cs corrected HR-TEM

Clear interpretation of atomic structure (Every atomic distance is transferred with the same contrast,[ see CTF])

HR-TEM on Nb7W10O 47.5 TITAN image Cs-corrector vs. non-Cs corrected TEM @ 300kV