Chapter 16 - High-resolution Tem.pdf

High Resolution Transmission Electron Microscopy P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne Switzerland Pierre.Stadelmann@epfl.ch 30 janvier 2009 P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

What is HRTEM ?

1

TEM imaging technique at atomic resolution.

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

What is HRTEM ?

1 2

TEM imaging technique at atomic resolution. Atomic columns.

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

What is HRTEM ?

1 2 3

TEM imaging technique at atomic resolution. Atomic columns. Very thin samples.

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

What is HRTEM ?

1 2 3 4

TEM imaging technique at atomic resolution. Atomic columns. Very thin samples. High symmetry orientation.

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Example 1 : ZrO2

Questions : where are the atoms ? Do we see the oxygen atoms ? P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Example 2 : Au particle

Question : do the atoms appear as white or dark spots ? P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Nano crystals

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Indium oxide

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

PbTiO3 : CM300 Tm = 205 °C

4 nm

Tm = 300 °C

4 nm

Tm = 400 °C

4 nm

Tm = 452 °C

4 nm

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Ray optics : principal rays BFP

FFP o

i Fi

z

O

Fo

f

f

In the back focal plane of the lens T & D beams converge to points (secondary spherical sources). Why is the objective lens so most important lens of the TEM ? P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Perfect optical system PE

PS
PS


PE

ye

O

-ye

ys

z

-ys

I

PE entrance and PS exit pupils. O (object) and I (image) are conjugate points (what does it mean ?). P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Spherical aberration PE

PS P M

Ao



Ai 




ri A'i

Wavefront is deformed −→ inclined incidents rays are not focused at same point. Coefficient of spherical aberration Cs is close to the focal length f of the objective lens. P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Important properties of optical system S

The optical system has two important properties : 1

Linearity.

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Important properties of optical system S

The optical system has two important properties : 1 2

Linearity. Space invariance.

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Important properties of optical system S

The optical system has two important properties : 1 2 3

Linearity. Space invariance. 1 + 2 −→ optical system characterized by a transfer function T˜ (k ).

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Important properties of optical system S

The optical system has two important properties : 1 2 3

4

Linearity. Space invariance. 1 + 2 −→ optical system characterized by a transfer function T˜ (k ). What does it mean ?

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Linearity S {a1 Ψ1o (x ) + a2 Ψ2o (x )} = a1S {Ψ1o (x )} + a2S {Ψ2o (x )} S {a1 Ψ1o (x ) + a2 Ψ2o (x )} = a1 Ψ1i (x ) + a2 Ψ2i (x ) Allows to : decompose the object wavefunction into points sources. consider only the transfer by the optical system of a point source (impulse). Ψo (x ) =



∞

−∞

Ψo (u )δ(x −u )du

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Impulse response Image wavefunction Ψi (x ) is :  ∞  Ψi (x ) = S Ψo (u )δ(x −u )du −∞

By linearity property (Ψo (x ) → coefficients real or complex) : Ψi (x ) =



∞

−∞

Ψo (u )S {δ(x −u )}du

Optical system impulse response t (x ;u ) : t (x ;u ) = S {δ(x −u )} By linearity one has to consider only how the optical system transfer a spherical wave (point source). P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Space invariance : point spread function t (x ;u ) = t (x −u ) Transfer is a convolution integral of Ψo (u ) and the point spread function t (x ) : Ψi (x ) =



∞

−∞

Ψo (u )t (x −u )du = Ψo (x ) ⊗ t (x )

In Fourier space (or reciprocal space) :  (k )  o (k )T  i (k ) = Ψ Ψ 

T˜ (k ) = e

−2π ı

2 Cs λ3 k 4 Δf λ k − 2 4



Electrical systems have the property of time invariance. P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Abbe image formation : transfer function Plan focal image

Diffusion & Interference

S1 u

Objet

z -u S-1

f ~


Remember

π 2

~

(u) T(u)


(x)

phase shift of the diffracted beams !

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Understanding HRTEM image formation Very simple object model : weak phase object approximation (WPOA). Crystal potential V (r ) (constant over dz small enough), wavevector is (e E + V (r )) :  k=

2 m e (E + V (r )) h2

Imagine a plane wave arriving on a crystal. How does the wavefront deform ?

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Refraction index n Refraction index of object n −→ ratio object wavevector and wacuum wavevector (|V (r )| << E :  km E + V (r ) V (r ) n= = ≈ 1+ E 2E kv

=⇒ phase change dϕ as a function of object thickness dz : χ dϕ = (km − kν ) · dr = (n − 1)|kν |dz = V (r )dz 2E with χ = |kv |. Question : value of n (compare to glass) ? P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Phase object transmittance For object of thickness Δz, phase shift Δϕ written as : 

χ z +Δz V (x ; z )dz Δϕ = 2E z χ Vp (x ; z )Δz = 2E Transmittance function of phase object over Δz is : Ψo (x ) = e 2πıΔϕ χ = e 2πı 2E Vp (x ;z )Δz = e ıσVp (x ;z )Δz with σ =

πχ E

=

π λE .

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

WPOA : weak phase object

Ψo (x ) = e ı σVp (x ;z ) ≈ 1 + ı σVp (x ; z )Δz In the back focal plane of the objective lens (Fourier transform) : p (k; z )Δz  o (k ) = δ(k ) + ı σ V Ψ

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

 i (k ) image wavefunction (back focal plane) Ψ Abbe image formation model :  (k )  o (k )T  i (k ) = Ψ Ψ  o (k )e −2πıχ(k ) = Ψ where χ(h ) is (defocus Δf and spherical aberration Cs ) : Cs λ3 (k · k )2 Δf λ(k · k )  − χ (k ) = 4 2

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Ψi (x )Ψi∗ (x ) image intensity (image plane) In objective lens back focal plane :

       Ψi (k ) = δ(k ) + ı σVp (k; z )Δz ][cos 2πχ(k ) − ı sin 2πχ(k ) Choosing sin 2πχ(k ) = −1 et cos 2πχ(k ) = 0 for diffracted  i (k ) becomes : beams k , Ψ p (k; z )Δz  i (k ) = δ(k ) − σV Ψ Image intensity (Ψi (x )Ψi∗ (x )) given by : I (x ) = (1 − σVp (x ; z ))(1 + σVp (x ; z )) ≈ 1 − 2σVp (x ; z )Δz Dark dots at the position of the atomic columns ! P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Remarks π 2

1

Transfer function compensates the diffraction.

2

Direct interpretation of HRTEM micrographs possible.

3

Spots are darker when projected potential is important (heavy atoms).

4

Spots are darker when specimen thickness increases.

5

phase shift due to

WPOA approximation only valid for very thin crystals (Au ≤ 1 unit cell !).

sin 2πχ(k ) = −1 is selected by changing the specimen defocus Δz . P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

WPOA : contrast inverted micrographs Choosing sin 2π χ(h ) = 0 et cos 2π χ(h ) = −1, (Ψi (x )Ψi∗ (x )) is since : p (h; z )Δz ]  i (h ) = [δ(h ) + ı σ V Ψ I (x ) = (1 − ı σ Vp (x ; z )Δz )(1 + ı σVp (x ; z )Δz ) = 1 + σ2 Vp2 (x ; z )Δz 2 White dots at the position of the atomic columns ! Contrast no more proportional to projected potential. Very small Δz change necessary to get contrast inverted micrographs.

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Model : Ti2Nb10O29

P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Image simulation : Ti2Nb10O29

Questions : where are the atoms ? Do we see the oxygen atoms ? P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy

Aberrations corrected TEM

Transfer function limited by gun and objective lens current instabilities (and incident beam convergence). P. Stadelmann CIME-EPFL Bˆat. MX-C, Station 12 CH-1015 Lausanne High Switzerland Resolution [email protected] Transmission Electron Microscopy